{"id":2591,"date":"2020-04-21T12:57:54","date_gmt":"2020-04-21T10:57:54","guid":{"rendered":"https:\/\/artenio.net\/wp\/2020\/04\/21\/full-text-of-mathematics-for-modern-life\/"},"modified":"2020-04-21T12:57:54","modified_gmt":"2020-04-21T10:57:54","slug":"full-text-of-mathematics-for-modern-life","status":"publish","type":"post","link":"https:\/\/artenio.net\/wp\/2020\/04\/21\/full-text-of-mathematics-for-modern-life\/","title":{"rendered":"Full text of “Mathematics for Modern Life”"},"content":{"rendered":"

OBSOLETE STOCK 0jc UBBIS I Digitized by the Internet Archive in 2017 with funding from University of Alberta Libraries https:\/\/archive.org\/details\/mathematicsformoOOmcco MATHEMATICS FOR MODERN LIFE BY JOSEPH P. McCORMACK Chairman of the Department of Mathematics, Theodore Roosevelt High School, New York. Author of “Plane Geometry”, “Solid Geometry”, etc. NEW YORK D. APPLETON-CENTURY COMPANY INCORPORATED Copyright \u2014 1’937, 1942 \u2014 by D. Apple t on – Cen t u ry Company, Inc. All righls reserved. This book, or parts thereof, must not be reproduced in any form without permission of the publisher. 463 This book has been manufactured in accordance with the regulations of the War Production Board UBMRY OF THE UHIVEBSITY #F ALBERTA IN THE UNITED STATES OF AMERICA. PREFACE Years ago it was generally assumed that there was a complete transfer of training from school subjects to the problems of life, and little or no attempt was made in the schools to aid this transfer. Then the pendulum swung to the opposite extreme and a prominent psychologist startled the. educational world by asserting that there was very little transfer of training. Now, however, we have gotten away from both extremes, and it is reasonably well established that the teacher can get as much transfer of training as he tries to get, but that he must teach his subject in such a way that it will transfer. He must con- sciously seek this transfer, and the best way to accomplish this is to make as many contacts with life situations as the nature of the subject-matter will permit. The search for these contacts with life situations should not be left for the teacher who has neither the time nor the oppor- tunity for such investigation. This is clearly the duty of the author. The modern textbook should be so written that (1) it interests the student, (2) it furnishes him with those contacts with life that will enable him to carry over his mathematical training into his later life, and (3) it develops in him those qualities and ideals which will better fit him for the life he is to live. It was with such objectives in mind that this book was written. This book aims to interest the student. Its approach is natural. He is led from the arithmetic he already knows to the techniques of high-school mathematics by simple steps in which the new is constantly tied up with the old. Many topics begin with a review of his arithmetic, and the connection of the new work with the old is constantly emphasized. The sequence of topics is such as to create a need for the new topic before the study of it is begun. .The explanation of the new topic is simple and clear and expressed in words that he understands. The book is written to the pupil. Nothing kills his interest more 111 IV PREFACE quickly than meeting difficulties that he cannot overcome, and nothing arouses his interest more enthusiastically than the knowledge that he is learning how to do something that he wants to do. Any subject is interesting to the student if it shows him how to solve the problems that he wants to solve. In this book he is shown from the very start the purpose of algebraic sym- bolism and its advantage in solving his problems over the clumsier methods he has previously used. He is led through a great wealth of practical problems which he either has already wanted to solve or knows that someone needs to solve as part of his daily life. And all this time the problems are kept within his ability so that he need not meet the discouragement of failure. This book furnishes the student with those contacts with life that will enable him to carry over his mathematical training into his later life. Every topic that he studies is immediately ap- plied to the problems of life. And great stress is laid on those problems that interest him now. For the boy, problems for the Boy Scout, finding heights and distances, baseball, radio, chemistry, aviation, building bridges, and all such problems that boys want to solve. For the girl, problems in the home, in cooking and sewing, in sports, problems for the Girl Scout, and all such problems that girls want to solve. In addition, both are shown that mathematics is the foundation of nearly every sub- ject of study. They apply their new tools to the problems of the professional man, the physician, the chemist, the astronomer, the pharmacist, the physicist, the civil and electric engineer, to the problems of the business man, the banker, the account- ant, the economist, the merchant, the broker, to the problems of the artist, the musician, the painter, to the problems of the farmer, and to a wide variety of the problems of everyday life. Whenever the topic furnishes a shortcut to the processes of arithmetic, the student is immediately given exercises in which this shortcut is applied, and it is not left to him to attempt to use in later life what he has never been taught to use. His mathe- matics will transfer to life situations because he is taught to transfer it to them and is trained in applying it. We all believe that mathematics is the very foundation of our modern civ- ilization. This the book attempts to bring out. If we tell the PREFACE V Student that nearly everything in life is based on mathematics, he may believe us, but he can never realize it as he will if he himself applies it to nearly every subject. This book attempts to develop in the student those qualities and ideals which will better fit him for the life he is to live. In addition to those problems that bring out the advantages of right living as opposed to carelessness, disobedience and crime, such as statistics on accidents, on punishment of crime, on cost and value of education, etc., and to those problems that tend to make the student a better citizen because he knows how to apply the advantages of mathematics to the problems of his life, the objectives that mathematics is best fitted to develop are constantly kept in mind. Neatness and orderly thinking are demanded, for example, by the arrangement of the analysis of a problem; accuracy and confidence in his conclusions and results are developed by an insistence on checking, training in analytic thinking by presenting problems about which he will want to think even outside of his school requirements and whose solution will mean something to him; training to work up to the level of his ability by problems graded on three levels so that the teacher can select those best fitted to the individual needs of the pupil or the class, habits of reflective thinking by thought problems and by the difficult exercises at the end of each chapter, and so on. The usual breach between cultural and vocational mathe- matics is avoided in this text by the expedient of using the problems of practical mathematics as the material for the processes of cultural mathematics. There is no good reason why the useless should be more cultural than the useful. Although this book is not a commercial arithmetic, it applies the prin- ciples of mathematics to so many of the problems of business life that the pupil has a better business foundation than he is likely to obtain from a purely business arithmetic. It is not a vocational text, but it uses such a variety of problems from the various vocations that it gives the student the training he needs as a foundation for any vocational pursuit. And at the same time it presents him with that mathematics most needed as a foundation if he continues his study of the subject as preparation for college. The best preparation for college is VI PREFACE preparation for life. In fact, it goes further in this direction than is usual as, for example, in the treatment and use of logarithms which are customarily relegated to a second course in algebra. Some of the features of this book are: 1. The development approach to new topics. This resembles the Investigation Problem of the author\u2019s geometries. Before the student is shown how to solve a new problem, he is asked a series of questions that should enable him to work out his own solution through his own activity. 2. The systematic grouping of related topics, as for example, the formula, graph, table, variation and dependence, or ex- ponents, logarithms and slide rule. 3. Methods of thinking emphasized. In the demonstrative geometry, the object is training in fundamental methods of thinking rather than arriving at a particular theorem. 4. Emphasis on those basic concepts, the graph, the formula, the language of mathematics, the equation, the construction and solution of problems, relationship and dependence, ratio and proportion, generalizing and the general solution, and their application to the real problems of life. 5. Exercises graded on three levels of ability, and even graded within each level so that the teacher can easily select what is best fitted for his class. 6. Explanations that the pupil can understand. The book is written to the pupil and for the pupil, and the wording is kept down to his level. 7. Culture through the useful. Each mathematical principle is applied to the problems of modern life. Business problems and other problems that someone in his daily life needs to solve make mathematics real to the pupil and arouse his interest. 8. Geometry introduced as material for algebraic problems. The principal concepts of geometry, congruence, parallelism, similarity, the right triangle, areas, indirect measurement and numerical trigonometry, are all made familiar to the student through use. 9. The arrangement of the text is such that the material most likely to function in the life of the student is placed in the earlier chapters, with the more technical material postponed to later PREFACE vii chapters for those classes that have time for a more extended course. 10. A suitable treatment of symmetry. 11. A new method of introducing logarithms that brings them down to the level of the immature student, and that makes him enjoy using them. 12. The slide rule as an application of logarithms. 13. New-type tests and reviews with each chapter. These tests are graded from the extremely simple exercise at the be- ginning to the rather difficult final exercise. 14. This text covers all topics required by the syllabi in ele- mentary algebra of the different states, and it meets completely the recommendations of the Joint Committee of the Mathemat- ical Association of America and the National Council of Teachers of Mathematics.^ The author wishes to acknowledge his indebtedness to Emory F. White, Mary Doran, and J. Hood Branson for valuable sug- gestions, and to Mrs. Edna S. Whare, of Mont Pleasant High School, Schenectady, N. Y., and Paul C. Roundy, of Western Reserve Academy, Hudson, Ohio, for reading the manuscript. J. P. McC. ^ The Place of Mathematics in Secondary Education, Report of the Joint Committee of the Mathematical Association of America and the National Coimcil of Teachers of Mathematics, Fifteenth Yearbook of the National Council of Teachers of Mathematics (New York, Bureau of Publications, Teachers College, Columbia University, 1940). .’S ‘ F u’ : \u2018 ‘.jS’ \u25a0 \u25a0\u25a0: h-V \u25a0:\/ -.\u2019 -i :\u25a0^ \u2022 ho’ >\u25a0 ‘ :’k j I ,4\/’;: :.i’i\u2018L” ”li “” *’\u2022 ‘jl rj., , – ^ – … ^ \u2022’ \u2018 – \u2018 t ?t t u pjC-^kU’.fi’r. i4>:^i4^\u2018- -iT – ‘ ‘.^tWI l^f ‘ fiV ” ^i^^u >M \/ Vi\u2018.\u201c”.\u2018A\u00ab\u2019Vi \u2019 ~ ‘\u2019A -k -” ^ ‘kt,4pin t^nij ; ,1\/ ;,i;:,,-;i ,, -vj:- ^ \u2019Vt ‘* ^ \u25a0’\u25a0’\u25a0\u25a0 \u25a0 \u2018:nu\/ \u25a0,.,’ -rj^i ;;*,.^ 1,1 \u2022 – \u2022\u25a0 ‘ ^ . V ‘ ‘\u2022 . \u25a0\u2022 ‘ \u2019 .v. ‘ Chapter 1 STATISTICAL GRAPHS Why we study graphs. If someone talked to you for a long time describing the appearance of a man, you would not have a good idea of how he looked until you glanced at his photograph. To a lesser degree the same thing is true of statistics. Most people, in reading a scientific article or a daily newspaper, either skip the columns of figures altogether or at most get only a hazy idea of their relationships. But when these figures are pictured for them, they quickly notice many facts that they could have found in the figures only after considerable study. A picture showing these relationships of statistics is called a graph. Graphs are used for a great variety of purposes and are becoming more and more important every year. The better class of daily newspapers carry graphs of business conditions. Magazines and books of reference contain them; nurses use them to record a patient\u2019s condition; and every large business today has a statistical department in which a large force of clerks makes graphs to show how business is changing and how the corporation\u2019s money is being spent. THE BAR GRAPH Perhaps the simplest kind of graph is the bar graph such as is shown here. This graph appeared in a daily news- paper. Can you understand it? It shows the population of the United States for each census since 1860. The figures on the left tell the number of millions of people. For 1 2 STATISTICAL GRAPHS example, 60 means 60,000,000. The figures at the bottom tell the year in which the census was taken. 1. In 1890, the population was about 63,000,000. How does the graph show this? Millions 130 2. What was the population in (a) 1880? (b) 1930? (c) 1910? 3. At what census did the population pass 100,000,000? 4. Has the population in- creased every decade? How does the graph tell you? 5. How many times as large was the population in 1930 as in 1860? 6. When was the population about half as large as it was in 1930? 7. How much did the popu- lation increase from 1920 to 1930? 8. Was the population in- creasing faster or slower in 1930 than in 1870? What can we learn about the bar graph? 1. Are the bars all the same width? 2. Are the spaces between bars the same width? 3. Are all bars the same length? On what does the length of the bars depend? 4. Notice the numbers at the left. What number is at the bottom? Do you think that this is important? 5. Does the same difference in space always represent the same difference in millions on this graph? 6. If the largest population had been 240 million instead of 120 million, how many million should each interval repre- sent if we wanted to obtain a graph of about the same size as this one? Population of the United States. THE BAR GRAPH 3 7. Do the numbers at the bottom increase as we go toward the left or toward the right? 8. Do the numbers at the left increase as we follow them toward the top or toward the bottom? How to make a bar graph. 1. Draw a vertical line near the left side and a horizontal line near the bottom of the sheet. These are your axes. 2. Decide on a unit of such a size that all of your numbers will fit on the graph, and that the graph will be about as large as the space will allow. 3. Make all bars the same width and the spaces between them the same width. 4. Write the numbers at the left, beginning with 0 at the bottom and increasing as you go toward the top. Numbers differing by the sanie amoimt should always be the same number of spaces apart. 5. The numbers or names along the bottom should be equally spaced, beginning at the left. The bar graph may be drawn with the bars running hori- zontally instead of vertically. In this case the time might be written along the vertical axis. This is the only kind of graph in which the time is so placed. This graph shows the sales of a business house for a year. 123456789 1011121314 Hundreds of Dollars 1. In what month were the sales the greatest? The smallest? 2. What was the approximate amount of sales in April? In August? 3. In what month were the sales nearest $1000? $800? 4. By what amount did the sales in May exceed those in April? 5. In what month were the sales about twice those in March? 4 STATISTICAL GRAPHS 6. Were the sales greater in summer or in winter? 7. In which month were the sales the greater {a) January or November? {b) April or September? (c) May or August? 8. Was there as great an increase in sales from May to June as there was from April to May? The double bar. Sometimes we wish to compare two kinds of data by putting two different graphs together. In this case the two sets of bars are shaded differently. This graph compares the exports and imports of the United States for the years 1910 and 1930 for cer- tain products. Exercises Make a bar graph illustrating the data in each of these exercises. 1. The casualties in the World War in millions were: Russia, 9; France, 6; British Empire, 3; Germany, 7; Italy, 2; Austria-Hungary, 7; and Turkey, 1. 2. The highest peak on each continent in thousands of feet is: North America, 20; South America, 23; Europe, 18; Asia, 29; Africa, 20; Australia, 7; and Antarc- tica, 15, 3 . The population (1930) of New York was 6,900,000; of Chicago, 3,400,000; of Philadelphia, 1,900,000; of Detroit, 1,600,000; and of Los Angeles, 1,200,000. (Make your scale read in hundred thousands, and then omit 5 zeros from each of the populations.) 4 . The longest suspension bridges are : Golden Gate, 4200 ft. ; George Washington, 3500 ft.; San Francisco, 2300 ft.; Ambassa- dor (Detroit), 1800 ft.; Delaware River (Philadelphia), 1700 ft.; Bear Mountain (Hudson), 1600 ft.; Williamsburgh (N.Y.), 1600 ft.; and Brooklyn, 1600 ft. THE BAR GRAPH 5 6. The weight in pounds needed to crush 1 cu. in. of these stones is: Sebastopol limestone, 1100; Vermont marble, 13,000; Caen limestone, 3600; granite, 16,000; and Berea sandstone, 9000. 6. The number of children killed and injured in New York in one year was: crossing streets not at crossings, 2373; crossing Buruju uf Ca^uaUj and iiurau Undcnirtiers, A DANGEROUS SPORT Coasting on city streets costs the lives of thousands of children every year. Play in the parks and play safe. against traffic lights, 1003; stealing rides, skating, riding on bicycles, etc., 1129; playing games in street or running into street, 2913; stepping from behind parked cars, 460; collision of vehicles, 1362; all other causes, 1783. In what way was the largest number of children injured or killed? The second largest number? What was the total number of children killed in New York that year? How can boys and girls help to avoid such a loss of life? 7 . The largest navies, estimated in the number of tons, are: U.S., 1,000,000; Great Britain, 1,200,000; Japan, 800,000; France, 600,000; and Italy, 400,000. 8. The principal crops of the United States in millions of 6 STATISTICAL GRAPHS acres are: wheat, 47; corn, 102; oats, 36; barley, 10; hay, 66; and cotton, 30. 9. The census shows that 8% of families consist of only 1 person, 23% of 2 persons, 21% of 3 persons, 17% of 4 persons, 12% of 5 persons, 7% of 6 persons, and 12% of more than 6 persons. 10. The lengths of the longest rivers are: Mississippi, 4200 mi. ; Amazon, 3900 mi.; Amur, 2900 mi.; Lena, 2800 mi.; Nile, 4000 mi.; Congo, 2900 mi.; Yangtze, 3100 mi.; and Ob, 3200 mi. 11. The cost of educating each pupil per year in certain cities is: Boston, $130; Chicago, $95; Cleveland, $97; Minneapolis, $89; New Orleans, $53; New York, $150; Salt Lake City, $70; San Francisco, $129; Washington, $118; and Yonkers, $158. If you go to school about 160 days a year, how much money is being spent per day on your education? Are you doing your part so that you get that much value from it? 12. Make a bar graph for the number of crimes committed in New York City, and shade the part of the bar that represents the number of arrests. Kind of Crime Cases Arrests Murder 1396 711 Robbery 1251 921 Burglary 2980 1506 Assault 2457 2142 Other felonies. . 2681 2323 Misdemeanors . 2702 2595 About what fraction of those committing robbery is cap- tured? Assault? Misdemeanors? Do you think that crime is a safe way to get rich? 13. Make a bar graph for these: A large store offered these shoes at a sale. Sh 4 5 5h 6 6h 7 7h 8 9 A 2 7 21 48 50 58 50 51 48 26 9 B 11 33 41 60 81 94 88 81 54 46 34 10 C 2 11 22 30 36 39 34 35 18 10 1 THE BAR GRAPH 7 14. Here are the world\u2019s largest cities. Make a bar to repre- sent the metropolitan area, and shade the part that represents the city proper. City Proper Metropolitan Area New York London 6.900.000 4.400.000 2,000,000 3.400.000 3.900.000 2.900.000 2.600.000 1,900,000 10,900,000 8,200,000 5.300.000 4.400.000 4.300.000 3.800.000 3.700.000 2.900.000 Tokyo Chicago Berlin Paris Moscow Philadelphia . . . 16. Make a double bar graph to compare the death-rates of the principal countries for 1920 and 1930. 1920 1930 United States . . 13.0 11.3’ England 12.4 11.4 France 17.2 15.7 Geraiany 15.1 11.1 Italy 18.8 13.8 Japan 25.4 18.2 Spain 23.8 17.4 Which countries have low death-rates? The pictogram. Shortly after the World War this pic- ture graph was shown by The Literary Digest. It represents the sizes of the armies of the principal nations of the world as they were at that time. Such a graph is called a picto- gram. Its advantage is that it appeals to people who would not look at a bar graph. The size of the army is represented only by the height of the figure. For that reason it is very deceptive, for we ordinarily think of a person as three-dimensional, and a figure twice as high as another gives the impression of being 8 times as large. You know a six-foot man is far stronger 8 STATISTICAL GRAPHS than two three-foot boys. The big fellow representing China appears strong enough to sweep away all of the little fellows. In reality, however, he represents a force only about equal to that of the British Empire and Russia com- bined. This type of graph is used extensively in advertising for one or both of two reasons: (1) because it has an appeal for the mass of. the people, and (2) because the unscrupu- lous advertiser can give the impression of a greater superi- ority over his competitors than he actually possesses. A better type of pictogram is that shown here; all the figures are the same size, but the comparison is shown by CHILD CARE Eoch child represents 1000 children the number of figures. Here two figures appear twice as large as one, and consequently the graph gives a more accurate picture of the facts. How many children does this graph indicate for 1929? For 1935? How much greater is the number in 1935 than that in 1929? THE BROKEN-LINE GRAPH 9 THE BROKEN-LINE GRAPH If, instead of drawing in the bars in a bar graph, you had simply marked a point where the top of each bar would reach and had then joined these points in order with straight lines, you would have had a broken-line graph. Let us make a broken-line graph from these statistics. The Excelsior Motor Car Company sold 30,000 cars in Jan., 28,000 in Feb., 34,000 in Mar., 52,000 in Apr., 56.000 in May, 50,000 in Jun., 42,000 in JuL, and 40.000 in Aug. First draw a horizontal line and a vertical line meeting at the lower left-hand corner of the paper. These lines are called the horizontal and vertical axes, and the point where they meet is the origin. Next we must choose suitable scales. t 50 5 . N \/ \/ 7 a 1. As there are 16 squares along the horizontal axis and we have 8 months to place, how many squares apart can we place them? 2. We have 15 squares along the vertical axis and the largest number is 56,000, so we should let each square represent what number? We can then omit the 3 O\u2019s by writing at the left \u201cthousands of cars\u201d so that 22 will mean 22,000. 3. On what vertical line shall we put the point that represents the January sales? 4. If each square Represents 5000, how many squares up must we count to represent 30,000? 5. Where shall we put a point to represent 28,000 in Feb- ruary? 6. How far up shall we put the point to represent the 42,000 in July? Will this be a whole number of squares? 10 STATISTICAL GRAPHS STOCK-MARKET GRAPHS In the stock market, because of the fact that prices go up and down very rapidly, often many times a day or week, a type of graph is used that shows the range through which the stock varied for that day or week. The lower end of the bar indicates the lowest price of the stock and the upper end the highest price. Sometimes a cross-bar is added to show the final price of the day. 1. When were the stocks the highest? 2. Can you tell from this graph when the depression began? 3. Which were higher, railroad or industrial stocks? 4. What was the lowest price the railroad stocks reached? When? 5. What was the price of industrials on June 1, 1935? 6. How much did industrial stocks drop from the highest point in 1929 to the lowest point in 1932? 7. What were the highest and lowest prices of industrial stocks in 1929? THE BROKEN-LINE GRAPH 11 Exercises Make a broken-line graph to illustrate the data in each of these exercises. 1. Charles received these test marks in algebra: 75, 60, 80, 70, 75, 80, 45, 70, and 90. 2. Dorothy saved her money in the school bank. At the end of each month her balance was: Oct., $2.10; Nov., $3.80; Dec., $.90; Jan., $1.70; Feb., $2.80; Mar., $2.20; Apr., $1.50; and May, $2.60. 3. In the U.S., the number of people killed by automobiles was: 1923, 14,000; 1925, 18,000; 1927, 21,000; 1929, 27,000; 1931, 30,000; 1933, 29,000; and 1935, 34,000. 4 . In a certain year the number of inches of rainfall in Mem- phis was: Jan., 4.8; Feb., 4.4; Mar., 5.3; Apr., 4.8; May, 4.2; Jun., 3.6; Jul., 3.0; Aug., 3.4; Sept., 2.8; Oct., 2.7; Nov., 4.2; and Dec., 4.5. 6. If all the money in the U.S. had been divided equally in the years given, each person would have had: 1914, $1900; 1916, $2500; 1918, $3800; 1920, $4600; 1922, $2900; 1924, $3000; 1926, $3100; 1928, $3000; 1930, $2600; 1932, $1900; and 1934, $1400. 6. Mr. King\u2019s electric-light bill shows that he used the follow- ing number of kilowatt hours of electricity: Jan., 26; Feb., 24; Mar., 20; Apr., 16; May, 14; Jun., 11; Jul., 12; Aug., 15; Sept., 19; Oct., 20; Nov., 23; and Dec., 27. 7 . According to the New York Police Department the number of juvenile delinquents (children committing crimes) arrested was: 1930, 7114; 1931, 6332; 1932, 6264; 1933, 6269; and 1934, 4849. 8. Make two broken-line graphs on the same axes to compare the average monthly temperatures at Chicago and Atlanta for a certain year. Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sepi. Oct. Nov. Dec. Chicago. . 24 26 35 47 58 67 72 72 65 54 40 29 Atlanta. . 44 45 52 61 70 76 78 77 72 63 52 45 In what month was the temperature the highest? The lowest? In what month was the difference in temperatures at Atlanta 12 STATISTICAL GRAPHS and Chicago the greatest? The least? Is the difference the greatest in the summer or in the winter? \/%M A Courtesy Taylor Instrument Co- THE RECORDING THERMOMETER This graph shows the temperatures for a whole week. The graph paper turned slowly by clockwork and a pen moved up and down on it according as the temperature changed. 9. The exports and imports of the United States in millions of dollars were: Year 1860 1870 1880 1890 1900 1910 1920 1930 Exports. . . 400 530 835 858 1395 1745 8228 3843 Imports. . . 362 462 668 789 850 1557 5279 3061 10. One week the attendance in an algebra class was: Mon. 28, Tues. 32, Wed. 29, Thurs. 30, and Fri. 27. Historical Note. The term statistics comes from state. It was so called because early statistics dealt with state affairs and were collected by the government. SMOOTH-CURVE GRAPHS 13 SMOOTH-CURVE GRAPHS This graph is called a smooth-curve graph. It records the temperatures for a certain day. Compare this graph with the broken-line graph on page 9. Why do you suppose we use a smooth curve here and a broken line there? Re- cording thermometers make graphs like this. When you look at an ordinary ther- mometer, you can tell what the temperature is, but you cannot tell what it was at 6:30 this morning. The ad- vantage of the graph is that you can tell what the tem- perature was at any time in the past. 90 |70 |60 50 3 6 9 AM. 12 3 6 9 12 P.M. 1. Was there a time between January and February when the 12 number of motor cars sold was half way between 30,000 and 28,000, or does the number jump abruptly from one to the other? Was there a time between 6 A.M. and 9 a.m. when the temperature was half way between 50\u00b0 and 58\u00b0, or did it jump suddenly? 2. Do you think the smooth curve more appropriate when the quantity changes suddenly or when it changes gradually? 3. What was the temperature at 6 a.m.? At 10 a.m.? At 2 P.M.? 4. When was the temperature highest? When lowest? 5. About what time was it 70\u00b0? 6. At what time was the temperature rising most rapidly? Falling most rapidly? During what interval was it most nearly stationary? 7. Judging by this graph alone, would you predict that the following day would be warmer or colder? Why? 8. How many degrees did the temperature rise between 9 a.m. and 12 noon? Between noon and 2 p.m.? 9. Why do you suppose a part of the graph below 50\u00b0 is made to appear torn out? 14 STATISTICAL GRAPHS This graph shows the weights of boys and girls from 4 to 21 years of age. Notice how, instead of showing the whole scale from 0 to 30, it appears as if a part had been torn out. This is a better form than simply beginning at 30. 1. How much heavier is a boy than a girl at the age of 6? At 16? 2. Are boys heavier than girls at all ages? 3. Which is gaining weight more rapidly at 12? At 14? At 16? At 20? 4. Between what ages is the girl the heavier? 5. What is the average weight of a 13-yr.-old girl? 6. If a normal boy weighs 60 lbs., about how old is he? 7. About how much difference is there between the weights of boys and girls from 6 yrs. to 10 yrs.? 8. A girl of 11 weighs 70 lbs. Is she heavier or lighter than the normal girl of her age? 9. Is a girl growing faster from 9 to 11 or from 19 to 21? 10. When does a boy grow most rapidly? 11. At what age is a normal girl twice as heavy as she was at the age of 7 yrs.? SMOOTH-CURVE GRAPHS 15 Exercises lUtlstmte these figures with a smooth-curve graph. Always lay off time on the horizontal axis. 1 . The temperature was 50\u00b0 at 6 A.M., 47\u00b0 at 8 a.m., 53\u00b0 at 10 A.M , 64\u00b0 at noon, 70\u00b0 at 2 p.m., 68\u00b0 at 4 p.m., 62\u00b0 at 6 p.m., and 54\u00b0 at 8 P.M. 2. The number of persons per hundred thousand who died of tuberculosis was: 1912, 125; 1915, 123; 1918, 107; 1921, 86; 1924, 78; 1927, 71; 1930, 63; and 1933, 53. 3. Helen had a fever. The graph made by the nurse showed: Fri. A.M., 102\u00b0; P.M., 103.5\u00b0; Sat. a.m., 101.5\u00b0; p.m., 103\u00b0; Sun. A.M., 99\u00b0; P.M., 100.5\u00b0; Mon. a.m., 98.5\u00b0; and p.m., 98.5\u00b0. 4 . The population of Hartford was: 1870, 37,000; 1880, 42,000; 1890, 53,000; 1900, 79,000; 1910, 99,000; 1920, 138,000; and 1930, 164,000. 6. The population of the United States in millions was: 1790, 4; 1800, 5; 1810, 7; 1820, 10; 1830, 13; 1840, 17; 1850, 23; 1860, 31; 1870, 39; 1880, 50; 1890, 63; 1900, 76; 1910, 92; 1920, 106; and 1930, 123. From your graph estimate the population in 1925. Make. two graphs on the same axes for each of the following exercises. 6. These populations of Springfield and Seattle are in thou- sands. 1870 1880 1890 1900 1910 1920 1930 Springfield 23 33 44 62 89 130 150 Seattle 1 4 43 81 237 315 366 7. Compare the interest at 6% on $100 at simple interest and at compound interest. Years 0 5 10 15 20 25 Simple interest . . . 0 30.00 60.00 90.00 120.00 150.00 Compound interest 0 33.82 79.08 139.65 220.71 329.18 1 16 STATISTICAL GRAPHS {a) Find the simple interest from your graph for 7 yrs. For 11 yrs. By extending the graph, for 28 yrs. {b) Find the compound interest for 3 yrs. For 8 yrs. For 18 yrs. (c) Find how much the compound interest exceeds the simple interest for 12 yrs. For 18 yrs. {d) At about what time has the compound interest be- come double the simple interest? 8 . Here are the heights in inches of boys and girls. Age 6 7 8 9 10 11 12 13 14 15 16 Boys 44 46 48 50 52 55 57 60 62 65 67 Girls 43 45 48 50 53 56 58 60 61 62 64 {a) Compare your graph with that for weights shown on page 14. Are girls taller than boys during the same years in which they are heavier than boys? {b) At what age are girls growing faster than boys? (c) Between what ages are girls taller than boys? {d) When do boys begin to grow faster than girls do? Depressions In 1929 millions of poor people lost their savings by- speculating in stocks just before the depression began. They did not expect a depression; yet if they had studied a graph like this, they would have known that boom times and depressions follow each other with a fair degree of regularity. One can never tell just when a depression will begin, but he can make a reasonable guess as to about when it is due. This graph shows all depressions and prosperous times since the early days of the United States. 1. How does the length of depression compare with that of prosperity? From 1790 to 1820 about what part of the time did we have depression? 2. How many years did the depression of 1873 last? The depression of 1840? STATISTICS 17 3. Was the good time preceding the depression of 1907 longer or shorter than the depression? 4. Which depression was the worst? Next to the worst? 5. Was the country prosperous or not in 1925? 1894? 1874? 1905? 6. What condition from 1914 to 1919 would have led you to expect a depression about 1920? If you had this graph and your present knowledge in 1929, would you have expected a severe depression? Why? 7. Do depressions occur with exact regularity? Every few years? 8. Do you think there will ever be another depression? Give a reason for your answer. STATISTICS Have you ever noticed a very good bargain in hats, or collars, or shoes in a store window, only to learn on entering that your size was not to be obtained? The merchant had to sell these odd sizes cheap and lose money on them be- cause he did not know when he bought them that few people wore those sizes and that there was so little demand 18 STATISTICAL GRAPHS for them. If he had known something about statistics., he might have avoided that loss. It is a peculiar fact that most things in nature, such as heights or weights of people of a certain age, sizes of collars. Courtesy of International Business Machines Carp. A STATISTICAL DEPARTMENT USING ELECTRICAL ACCOUNTING MACHINES Every big corporation has a large statistical department that collects and organizes information about business. It prepares tables and graphs for executives to study. hats, or shoes, marks in algebra or English, diameters of trees, the time it takes runners to go 100 yds., the number of quarts of milk cows will give, and many other things, follow a very definite law. If you know this law, you can predict in advance how many of a group will be of a cer- tain size or speed or get a passing mark. The study of these facts and the collection and grouping of them are called statistics. How to interpret statistics. Problem: The same test was given to two classes in algebra. In Class A, 1 pupil got 100; 2, 90; 8, 80; 12, 70; 6, 60; 4, 50; and STATISTICS 19 2, 40. In Class B, 1 got 90; 5, 80; 10, 70; 11, 60; and 3, 50. Which class did the better work? Since it is difficult to tell from looking at the numbers, we might find their average. The average of a set of num- bers is found by dividing their sum by the number of numbers. Let us arrange the data in a column. Class A Class B No. of Pupils Mark Total of Marks 1 100 100 2 90 180 8 80 640 12 70 840 6 60 360 4 50 200 2 40 80 35 2400 Average 2400 35 68 – 1 – No. of Pupils Mark Total of Marks 1 90 90 5 80 400 10 70 700 11 60 660 3 50 150 30 2000 . 2000 Average = = 67- So on the basis of average, it appears that Class A is the better. But an easier, and sometimes better, method is to find the median score. The median number is the middle number when the numbers are arranged in order of size. If there is an even number of numbers, the median is half way between the two middle numbers. One advantage that the median has over the average is that it is not so much affected by a number of exceptional size. In Class A there are 35 pupils, so if we divide by 2 get- ting 17i, we see that there are 17 pupils on each side of the middle. Consequently the 18th is the middle one. Begin- ning at the top, 1 and 2 are 3, and 8 are 11, and we need 7 more to make 18. This will not take all of the next 12, so the 18th pupil is among the 12. The median mark for 20 STATISTICAL GRAPHS Class A then is 70,* the mark that each of these 12 pupils got. In Class B there are 30 pupils, so dividing by 2, we find that there are 15 on each side of the middle. The median then is between the 15th and the 16th. This would place the median pupil in the group of 10, and the median mark is again 70. So on the basis of the median, the classes are equal in ability. There is another method of comparing classes, by means of the mode, which is easier to find than the median. The mode is the number that occurs the greatest num- ber of times. It is the most stylish or most fashionable number. In Class A the mark 70 occurred 12 times. Since no other mark occurred so many times, 70 is the mode. In Class B, 60, that occurred 11 times, is the mode. On the basis of the mode, therefore. Class A is the better class. Exercises In these exercises find the average, the median, and the mode. 1. In his 5 major subjects a student received the marks 70, 70, 75, 90, and 95. 2. The girls of Excelsior Camp hiked on successive days 8 mi., 6 mi., 5 mi., 3 mi., 6 mi., and 4 mi. First rearrange the data in order of size. 3. On a test 3 pupils got 60%; 11, 70%; 8, 80%; 6, 90%; and 2 , 100 %. 4. A class was asked to contribute to charity; 15 pupils gave 10^ each; 8, 15^ each; 3, 25^ each; and the teacher gave $3. Do you think the average or the mode better represents the facts? 5. In the final examination 5 pupils received 100%; 20, 90%; 27, 80%; 22, 70%; 20, 60%; 8, 50%; and 3, 40%. * For simplicity we are assuming that these 12 marks are all 70\u2019s and not distributed through the interval 70 to 80. THE FREQUENCY POLYGON 21 6. Charles sells newspapers. On Mon, he sold 46; on Tues., 40; on Wed., 53; on Thurs., 51; on Fri., 40; and on Sat., 64. 7. In weekly tests one term Gladys got marks of 8, 10, 7, 10, 9, 4, 8, 9, 8, 5, 9, 7, 6, 8, 6, 3, 10, and 8. 8. On a single day a telephone exchange handled 80 calls at 5^ each, 20 calls at 10^ each, 6 calls at 20jz^ each, and 1 call at $2.40. 9. In a class test 2 pupils got A, 8 got B, 13 got C, 7 got D, and 2 got F. Find the median and mode. 10. In settling how much damages a defendant should pay, a jury decided to have each member write on paper the amount he thought satisfactory. Here are the results: $75, $200, $200, $2400, $200, $50, $250, $300, $200, $250, $200, and $275. Which would represent better the opinion of the jury, the average or the median? The average or the mode? THE FREQUENCY POLYGON OR HISTOGRAM If we arrange the individuals of a group in order of bar graph showing the weight, height, etc. and make a number of them in each di- vision, the figure is called a frequency polygon, or histo- gram. In making this poly- gon the bars are placed ver- tically and are put together without leaving spaces be- tween them. The number of collars of the different sizes sold one year by a merchant is shown in this frequency polygon. 100 CO 70 w m y\/^ 4 \u25a0\/\/\/\/, ‘7777 * m ii m. 77^ m 1 1 1 1 m m i m 1 1 1 i 1 1 ni m 14 14^ 15 m 16 161- 17 m 18 Sizes of Collars 1. What size of collar had the greatest sale? The second largest sale? The third largest sale? 2. If you were buying collars for a store, would you buy more of the middle sizes or of the very large or very small sizes? 3. Which can you determine more easily from a frequency polygon, the average, the median, or the mode? 22 STATISTICAL GRAPHS 4. If a merchant bought about the same number of collars of each size, of what sizes would you expect to find the largest numbers left on hand? 5. Do many men wear collars of size 13|^? Of size 18? Of size 15? 6. How does the number of men wearing size 14^ compare with the number wearing size 16? If, instead of drawing the bar graph as in the figure above, we had plotted the points and had drawn a smooth- curve graph through them, we would have the figure shown at the right. This graph is called a normal fre- quency curve. If the number of objects is very large, the curve will have the same height on both sides of the median line. Is the highest point of the curve near the middle or near an end? Can you draw a line along which the figure might be folded so that the part on one side would almost fit that on the other side? Exercises Make a frequency polygon for each of these exercises: 1. In the U.S. army intelligence is rated A, B, C+, C, etc. The per cent of white men in each group is: A, 4%; B, 8%; C+, 15%; C, 25%; C-, 24%; D, 17%; and D-, 7%. 2. The intelligence quotients for seniors in a high school were: IQ 76 81 86 91 96 101 106 111 116 121 126 No. of pupils . . 23 36 77 138 192 211 179 131 82 41 22 3. In a test in handwriting given in the eighth grade, 3% of the pupils got 30; 10%, 40; 16%, 50; 22%, 60; 23%, 70; 23%, 80; and 3%, 90. THE FREQUENCY POLYGON 23 4. In a certain city the number of high-school seniors, dis- tributed by age, was: Age 15 16 17 18 19 20 21 22 No. of pupils 80 1420 3120 4100 1950 980 120 80 5. The heights of freshmen in a certain college were: Height in inches. . . . 60-61 62-63 64-65 66-67 68-69 70-71 72-73 74-75 No. of men . . 18 70 245 590 890 640 230 68 6. A large department store furnished these figures. The number of shirts of a particular design sold one spring, grouped by sleeve length on the left and neck-band size on top, was: Neck-Band Sizes jutn^in m 14 14 15 15 16 16 17 17\\ 18 32 1 13 26 23 20 12 0 0 0 0 33 6 16 83 98 67 32 6 5 0 0 34 3 12 30 54 61 26 15 4 5 3 35 0 4 20 32 26 26 12 8 4 3 36 0 1 2 5 8 4 5 2 0 1 Totals 10 46 161 212 182 100 38 19 9 7 Notice the central tendency, that is, that the numbers grow larger as you approach the middle, for all horizontal rows of figures and also for the vertical columns. (a) Make a frequency polygon using the neck-band size on the horizontal axis and the total number sold on the vertical axis. (b) In the same way make frequency polygons for the number sold at each of the different sleeve lengths. (c) Make a frequency polygon using the sleeve length on the horizontal axis and the number of IS^-size shirts sold on the vertical axis. Thought Questions 1. Why is it important that the scale on the bar graph should begin at 0, whereas on the smooth-curve graph it may begin at 24 STATISTICAL GRAPHS any convenient number? What effect would beginning at some other number have on the length of the bars? On the compara- tive length of the bars? On the shape of the smooth-curve graph? 2. Does a set of numbers always have a median? A mode? Which of the three, average, median, or mode, do you think the most satisfactory for comparing your final marks with those of other pupils? Why? 3. Which do you think would give the more regular normal frequency curve, and why; (a) Your final marks in all of the subjects you take this term? (b) The final marks in algebra of all the pupils in a large city school? Review Exereises Decide what type of graph is most appropriate for each of these exercises and draw the graph: 1. The number of million telephones in the United States was: 1902, 2; 1907, 6; 1912, 9; 1917, 12; 1922, 14; 1927, 19; 1932, 17. 2 . The amount of food, that cost $1 in 1926, cost as follows: 1913 1916 1919 1922 1925 1928 1931 1934 $.64 $.76 $1.29 $.88 $1.00 $1.01 $.75 $.60 3. The population of the earth in millions by continents is: Asia, 950; Europe, 550; America, 230; and Africa, 150. 4 . The number of millions of acres used for crops was: wheat, 48; corn, 102; oats, 36; barley, 10; hay, 66. 5 . One year the exports of certain countries in hundred millions of dollars were: United States, 49; France, 21; Great Britain, 40; and Germany, 24. 6. The assets and liabilities of a corporation were: 1910 1915 1920 1925 1930 1935 Assets Liabilities. . 85.000 62.000 115,000 80,000 102,000 78,000 140,000 90,000 165.000 103.000 110,000 85,000 When was the firm most prosperous? When least prosperous? REVIEW EXERCISES 25 7 . One day the temperature readings in a certain town were: Hour 8 9 10 11 12 1 2 3 4 5 6 Temperature .. . 10\u00b0 12\u00b0 16\u00b0 23\u00b0 32\u00b0 40\u00b0 43\u00b0 43\u00b0 41\u00b0 34\u00b0 29\u00b0 (\u00ab) From your graph determine the temperature at 11.30 A.M.; at 4.30 p.m. (b) Between what hours was the temperature rising most rapidly? Falling most rapidly? 8. Here is a table of the weekly wages of boys who leave school at 14 and boys who leave school at 18. Age Wages of Boy Who Left School at 14 Wages of Boy Who Left School at 18 14 4 16 5 18 7 10 20 9 15 22 11 20 24 12 24 26 13 30 How does the pay of the boy who left school at 14 com- pare with that of the boy who remained in school until 18 when the two boys are 24? 30? Do you think that education pays? 9. The number of people per square mile in the United States was: 1870 1880 1890 1900 1910 1920 1930 13 17 21 26 31 35 41 10. Lucy had a fever. The nurse recorded these temperatures. (The normal temperature is 98.6\u00b0.) Tues. A.M., 100\u00b0; p.m., 102\u00b0; Wed. A.M., 100.5\u00b0; p.m., 103.5\u00b0; Thurs. A.M., 102\u00b0; p.m., 104\u00b0; Fri. A.M., 100.5\u00b0; p.m., 102.5\u00b0; Sat. A.M., 99\u00b0; p.m., 99.5\u00b0. 11. Arrange these classes in order of rank, make a frequency 26 STATISTICAL GRAPHS polygon of each, and find its average, median, and mode. Then from your results determine which class is better. Class 1 Name Mark Name Mark JBA. . ….75 ARN. . . ..80 CHB.. . …65 RNP. . . ..60 RMB. ….80 DMR.. . .100 ELC. . ….70 KHR. ….85 JND.. ….85 RPR.. . . . .85 RDD. 95 AKS.. , ,…80 WHF. . . . .75 RWS. . …70 LBH.. ….60 CLS… ….50 RFH.. . …90 JSS…, ….80 JWL. . ….80 HCT. . . . .85 JJM.. . . . .55 JSW. . . …80 WSM. . . . .95 MRW. . . .85 FWM. ….80 KSY. . . . .75 Class 2 Name Mark Name Mark FLB. . . …70 JBP. . ….60 HHC. . . . .80 FJP. . ….90 AME. . . . .95 JHR.. ….50 JOF. . ….70 FSR. . ….75 FDG. , , . . .85 WJS.. . …80 WBG. . . . .60 HNS.. ….70 MJH. ….75 RAS.. ….65 RWJ., , . . .65 WBT. ….80 CBL.. . . . .90 cwv. ….95 FSL. . ….80 KJW. ….55 SHM. . . ..85 HPW. . . . .60 LSM. . . . .60 RRW.. ,…80 RMN., . . . .70 12. A grocer\u2019s record of sales of peaches for a certain year showed that at $1.25 a basket he sold 10 baskets; at $1.00 a basket he sold 30 baskets; at $.75, 70 baskets; at $.60, 85 bas- kets; and at $.50, 90 baskets. Draw the graph, and from it estimate the probable demand when the price is $1.10 a basket. 13. The temperatures recorded for a certain day were: Hour 8 9 10 11 12 1 2 3 4 5 6 Temperature.. . 50 52 56 63 72 80 83 83 81 74 69 14. This table shows the earnings and spendings of a boy from Apr. to Oct. Apr. May Jun. Jul. Aug. Sept. Oct. Earnings. . . $2.00 $3.20 $4.80 $10.40 $12.00 $4.40 $5.20 Spendings. . 3.20 3.60 2.40 2.00 1.60 8.40 4.00 (a) Read the points where the graphs cross and tell what they mean. (b) Did the boy spend his vacation at work or at play? How is this shown by the graph? TESTS 27 Matching Test Write the numbers from 1 to Q in a column. After each number, write the letter before the word that corresponds to the sentence after the number. 1. The name of the number obtained by- dividing the total by the number of items. 2. The kind of graph to use when the data change gradually from one value to the next. 3. The name of the number that occurs the largest number of times. 4. The kind of graph to use when we are comparing the sizes of different things at the same time. 5. The name of the number that is equally distant from both ends when the data are arranged in order of size. 6. The kind of graph to draw when the data change abruptly. Interpreting Graphs Test 1. This frequency polygon shows the marks the pupils of a class received in a test in algebra. Copy the sentences, inserting the correct word in the blank spaces. % \u00a7 {a) The number of pupils ($ 6 getting 80% was …. {b) The mode was … . ^ s (c) Seven pupils received a i mark of . . . . {d) If 60% was the passing mark, the number of pupils who failed was … . 2. This graph (page 28) shows a girl\u2019s final marks in English, algebra, French, and biology. {a) She received the lowest mark in . . . . (&) She received the highest mark in . . . . (c) Her mark was about 90% in . . . . {d) Her mark in algebra was … . 55? 55= 55 555 55: 555 F 55 5?? % 55 55 55 55 55 55 55 55 5. 55 55 5^ 5^ 55 55 55 55 100 90 80 70 60 50 40 Mark’ Received A. Median. B. Mode. C. Average. D. Smooth-curve. E. Bar graph. F. Broken- line graph. 28 STATISTICAL GRAPHS 1 i 1 1 1 i i fc. Cq Ex. 2 3. This graph shows the temperatures at Denver for a single day, as shown by a recording thermometer. (\u00ab) At 6 A.M. the temperature was about … . (b) The temperature rose above 32\u00b0 at about . . . o\u2019clock. (c) The temperature was rising most rapidly about … . (d) Was this graph recorded in January or July? … (e) From the information given by this graph alone, would you predict that the following day would be warmer or colder? . . . (\/) The temperature was highest at about . . . o\u2019clock. (g) The temperature at noon was about … . (h) The temperature was falling most rapidly about … . 4. This graph shows the assets and liabilities of a business firm for a period of 5 years. (a) The firm was most prosperous the . . . year. (b) The assets at the end of the fourth year were … . (c) If capital equals assets minus liabilities, at the end of the first year the capital was … . (d) During the fifth year the capital reduced to . . . . 3 S ~ ^ J4ssetsj 1 LiabiUties 1 \/ r N 7 s r \/ \/ \/’ Years Chapter 2 THE FORMULA W E all like short methods of expressing ourselves. We cut down \u201ccannot\u201d to \u201ccan\u2019t,\u201d \u201cautomobile\u201d to \u201cauto.\u201d We use abbreviations, as for example, S.D. for South Dakota, C.O.D. for cash on delivery, U.S.A. for United States of America, T.R.H.S. for Theodore Roose- velt High School, and so on. And we study shorthand so as to be able to write more words per minute. Now in algebra we shall learn many short cuts for processes that were long and clumsy in arithmetic, and the most im- portant of these short cuts is the formula. We shall now discover what a formula is. Investigating the formula. How would you find the area of a rectangle if its length were 12 and its width were 10 ? What procedure did you follow in getting your answer? But this rule is long. Could you abbreviate it by writing only the initial letters of the important words, omitting the other words altogether, and using such signs as =, X, etc.? If you have succeeded in this, you have a formula, for: A formula is a shorthand way of writing a rule. How to make a formula. Rule. The area of a rectangle equals the product of its length and width. Using only the important words, we have: Area = length X width 29 30 THE FORMULA Now we can shorten it still more by writing only the ini- tial letters of these words: A = Ix w And if we agree that, whenever two letters are written together with no sign between them, we are to under- stand that the product of the quanti- ties is meant, we can still further shorten our rule by omitting the times sign. Our rule now becomes A = lw This we call a formula. In this formula, I, w, and A all represent numbers. If the rectangle is 8 in. long and 6 in. wide, I represents the number 8, w represents 6, and A represents 48, for: A = Iw = 8X6 = 48 l Rectangle Exercises 1. Using the formula, A = Iw, find the value of A if: {a) I is 7 and w (c) I is 21 and m; is 3 (&) I is 14 and m; is 3 (d) \/ is 7 and is 6 2. How does the value of A change when: (\u00ab) I is doubled but w remains the same? (b) I is trebled but tv remains the same? (c) \/ remains the same but w is doubled? 3 . Using the same formula, find A when: (a) I = 384 and w = 226 (b) I = 17i and w = Hi (c) I = .0013 and w = .0007 4 . Can I stand for any number you wish it to? Can wl Can A? Can all three of these letters at the same time stand for any three numbers you can select for them? Make formulas for these rules: 6. The diameter of a circle is twice the radius. THE SHORTHAND OF ALGEBRA 31 6. The distance a car travels is the product of the rate and the time. 7. The number of inches (i) is 12 times the number of feet (\/). 8. The cost of a piece of meat is found by multiplying the price per pound by the number of pounds. 9. The number of rods (r) in a certain distance is 320 times the number of miles (m) in it. 10. The wages a man receives (w) equal the product of the price per hour (p) and the number of hours (h). 11. The volume of a box is the product of its length, width, and height. 12. The cost of sending a package by air mail is 5 times the number of ounces (z). 13. The number of seats in the room (s) is the product of the number of rows (r) and the number of seats in a row (n). 14. The cost (c) of 3^ stamps is 3 times the number of stamps bought (s). Thought Question If y stands for a number of years and m for the number of months in the same period of time, which is correct : (a) y = 12 m? (b) m = 12 y? Practice in the shorthand of algebra. Illustration. The width of each board is b. What is the width W of three such boards? We can add b b b, or we can multiply 6 by 3. In either case the answer is IT = 3 1. If the thickness of a book is b, what is the thickness T of 5 such books standing together on a shelf? 32 THE FORMULA 2. Each edge of a square is s. What is the distance around the square? S S 3. A field has 6 equal sides each of length \/. Find the length \u00a9f the fence around it. 4. A stairway has 15 steps each of height h. What is the total height (H) of the stairway? Express each of these in shorter form: 5. p = e-{-e–e + e S. A = ab ab + ab 6. m = 5^ + 3\/?+2\/? 9. ^=5s + 3s + 4s 7. M = 3w–w–4tv–iv 10. V = 3 Bh 5 Bh Bh The perimeter of a figure is the distance around it, or the sum of its sides. 11. {a) How many sides has a square? If 1 side is s, what is the sum? Make a formula for the perimeter of a square whose side is s. ib) Find p when s = 5; when s = 10. (c) When s becomes twice as large as before, does p become twice as large as it was? {d) Find the value of p when s = 5 \u00ab. 12. (\u00ab) Make a formula for the perimeter of the rectangle whose length is I and whose width is w. How many sides of length I has it? How many of width tr? s (b) Find p when \/ = 14 and w = 10. When \/ = 28 and w = 10. Rectangle Ex. 12 s s A s Square Ex. 11 THE FORMULA IN GEOMETRY 33 (c) When \/ becomes twice as large, does p become twice as large? {d) Find the value of p when I = Sx and w = 3x. 13. (a) Make a formula for the perimeter of an equilateral triangle whose sides are each s. (b) Find p when s = 4; when s = 12. (c) By what is p multiplied when s is multiplied by 3? 14. {a) Make a formula for the perimeter of an isosceles triangle whose base is b and whose other sides are each s. ip) Find the value of p when & = 10 and s = 18. 15. {a) Make a formula for the perimeter of a triangle whose sides are a, b, and c. (b) Find p when \u00ab = 10, 6 = 8, and c = 9. (c) Find p when a = 20, b = 16, and c = 18. THE FORMULA IN GEOMETRY The parallelogram. GBCD is a parallelogram. The sides GD and BC run in the same direction, and the sides GB and DC run in the same direction. The parallelogram has both pairs of sides parallel. ^ We can change it into a rec- tangle by cutting off triangle DCF and replacing it by triangle GBE. 1. What is the area of the rectangle EBCF’i 2. Does triangle GBE have the same area as triangle DCF? 34 THE FORMULA 3. Does parallelogram GBCD have the same area as rectangle EBCF? 4. What then is the area of GBCD, using b for base and h for height? The area of a parallelogram equals the product of its height and its base. A = hh Exercises Find the value of A if: 1 . & = 8 and ^ = 5 4 . & = 12 and h = A 2. & = 8 and ^ = 10 5. & = 24 and h = A 3. & = 8 and h = 6. & = 6 and ^ = 8 7. What change takes place in A when h is doubled? Trebled? 8. What change takes place in A when b is doubled? Halved? 9. What is the effect on H if & is halved and h is doubled? The triangle. 1. What is the area of the parallelogram whose height is h and whose base is bl ‘ 2. How does the area of the triangle GBC compare with that of the parallelogram? 3. Make a formula for the area of the triangle, using A for area, h for height, and b for base. The area of a triangle equals half the product of its height and base. Exercises Using this formula, find the value of A when: 1 . = 10 and b = A 5. h = 5 and b 2. h = 10 and & = 8 G. h = 20 and b 3. h = 10 and & = 12 7. h = 20 and b A. h = 10 and b = 2 S. h = 5 and b = A = A = 8 = 8 THE FORMULA IN THE HOME 35 9 . How is the value of A changed if h remains the same but: (a) b is doubled? (b) b is trebled? (c) b is halved? 10 . How is the value of A changed if: (a) b remains the same but h is doubled? (b) both h and b are doubled? (c) b is doubled but h is halved? The’ Formula in the Home 1. How many square yards of linoleum are needed for the floor of a kitchen 12 ft. long and 9 ft. wide? 2. Find the cost of paving a sidewalk 30 ft. long and 6 ft. wide at $.20 a sq. ft. 3 . How many gallons of paint will be needed for the 4 walls of a building 40 ft. long, 25 ft. wide, and 20 ft. high, if 1 gal. of paint will cover 400 sq. ft.? 4 . A house whose width is 30 ft. has 2 gabled ends, and the ridge is 20 ft. above the plate at the eaves. How many sq. ft. of boards are needed to cover both gables? 6. Mr. Brown buys a field 64 rds. by 50 rds. at $25 an acre. If an acre equals 160 sq. rds., what price should he pay? Practice with Formulas Using initial letters, make formulas for these rules: 1 . {a) The volume F of a rectangular box equals the prod- uct of its length, width, and height. (b) Using your formula, find the volume of a box whose length is 12, width 8, and height 5. (c) Find V when 1 = 7, w = 5, and h = 3. 2. (a) The distance a car goes equals the speed multiplied by the time. (b) Using your formula, find the distance when the speed is 28 mi. an hr. and the time is 4 hrs. (c) Find d when s = 15 and t = 12. 36 THE FORMULA Photograph from Philip D. Gendreau, N.Y. THE PYRAMIDS OF EGYPT More than 3,000 years ago, the Egyptians used mathematics in designing and constructing these great monuments. f PRACTICE WITH FORMULAS 37 3. (a) The volume of a pyramid equals i the product of its height and base. (b) Find V when b = 42 sq. in. and A = 8 in. (c) Find V when b = 16 and h = 10. 4. (a) When the base of a pyramid is a rectangle, we can find the volume by taking i the product of the length, v\/idth, and height. (b) Using your formula, find the volume of a pyramid if its height is 300 ft. and the length and width of its base are 200 ft. and 170 ft. (c) Find V when h = 22, I = 18, and w = 10 . 5. (a) The interest on a sum of money is found by multiplying the principal by the rate by the time. (b) Using your formula, find the interest on $300 at 4% for 6 yrs. (c) Find i when p = 250, r = .06, and t = 4. Another short cut in writing formulas. The exponent. The square is a rectangle. Consequently its area is found by multiplying its length by its width. A = ss Instead of writing ss, we write (read s-square). Our formula then becomes A = s^. Similarly sss = (read 5-cube) and ssss = (read s-fourth). Quantities multiplied together to form a product are called factors of that prod- uct. A product of equal factors is called a power of one of the factors. A small number written to the right and above a quan- tity to tell how many times it is taken as a factor is called an exponent. Be careful that you do not make the mistake of thinking that the exponent 2 in means that the value of s is to be A s Square 38 THE FORMULA multiplied by 2. It means that the value of s is to be multi- plied by itself. For example: If s = 3, then s2 = sXs = 3×3 = 9 If s = 5, then s2 = sXs = 5X5 = 25 Can you discover a reason for calling the exponent 2 “\u2018square\u201d and the exponent 3 \u201ccube\u201d? Exercises Write each of these expressions in the shortest form: 1. XXX 4. xxyyy 7. 3 X 3 X 3 10. 2 aa 2. yyyyy 5. aaaaabb 8. 2 X 2 X 2 X 2 11. 3 xxyyy 3. AAA A 6. xxxyzz 9. 4x4x4x4x4 12. 4 xxyzzz If s = 4, find the value of: 13. 14. 16. 16. 3 The exponent belongs only to the letter after which it is written, xy^ means xyyy, not xxxyyy. 5 means 5 A A, not 5X5 AA. When ^ = 2, 5 = 5 x 2 X 2 = 20. When r = 5, find the value of: 17. 5 r 18. 2 19. 4 20. 21. 2 Using the formula, A = s^, find the value of A when: 22. s = 6 23. s = 3 24. s = 1.2 25. s = 2|- 26. s = 12 27. How many times as great does A become when s is dou- bled? Halved? Formulas for Familiar Forms 1. You learned that the volume of a rectangular box was V = huh Now a cube is a rectangular box having all of its edges equal. (a) Letting e represent the edge of a cube, write a formula for the volume of the cube. FORMULAS FOR FAMILIAR FORMS 39 (b) Using your formula, find the volume of a cube whose edge is 6 in. (c) Find the value of V when e = 4; when e = S. (d) By what number is the volume of a cube multiplied when its edge is doubled? 2. How many faces has a cube? Are they all equal? Are they all squares? What is the area of a square whose edge is e? Now write a formula for the total surface of a cube. (a) Find the surface of a cube whose edge is 10 in. (b) Using the formula S = 6 find S when e is 4, 8, 12, 2 (c) By what number is S multiplied when e is doubled? Trebled? Halved? (d) When e increases, would you say that S increase? at the same rate, faster, or slower than e? 3. The circumference of a circle is the length of the circle or the distance around the circle. The circumference of a circle is 2 TT times the radius, where tt is Sy or more accurately 3.1416. (a) Express this rule as a formula. (b) Using TT = 3 y, find c when r = 7; r = 14; r = 21. (c) By what number is the value of c multiplied when we double r? When we treble r? 4. The area of a circle equals x times the square of its radius. (a) Express this area as a formula. (b) Using your formula, find the area of a circle whose radius is 5 in. (x is always approximately ^ or 3.1416.) (c) Find A when r = 4. 5. The common vegetable or fruit can is a cylinder. Its top and bottom are circles. What is the area of its base if the radius 40 THE FORMULA is r? The volume of a cylindrical can is found by multiplying tt by the square of the radius by the height. {a) Write a formula for the volume. {b) Using your formula, find the volume of a can of fruit if the radius is 2 in. and the height is 3 in. (c) Using TT = 3.14, find the volume of these cylinders: r = 2, h = 2 r = 2,h = A r =4, h = 2 r =4, h = 4 (d) Which has the greater effect on the volume, doubling the radius or doubling the height? (e) By what number is the volume multiplied when the radius and the height are both doubled? (\/) If you were buying a can of tuna fish, would you select one whose height was twice as great or one whose radius was twice as great, other conditions being equal? 6. The volume of a cone is found by multiplying -jx times the square of the radius times the height. (a) Write this rule as a formula. (b) Find the volume of a cone if r = 7 and A = 10; if r = 14 and \/? = 10; if r = 7 and h = 20. (c) If you could have your choice in selecting an ice cream cone, would you take one whose height was twice as great or one whose radius was twice as great? Why? Sphere Ex. 7 Cone Ex. 6 7 . The volume of a sphere is f x times the cube of its radius. (a) Express this rule as a formula. (b) Find the volume of a sphere whose radius is 7. (c) Using X = 3.14, find the volume of a sphere if r = 10. (d) Does the volume increase when the radius increases? (e) By what number is V multiplied when r is doubled? SQUARE AND CUBIC MEASURE 41 Square and cubic measure. Do you 3 ft. know the number of square inches in a square foot, or square yards in a square rod? Probably you have forgotten, but it does not matter. You do not need to remember. The formula 5 = will tell you. (5 is the number of square units isq.yd. units of length.) Illustration. How many square feet are there in a square yard? Since there are 3 ft. in a yd., \/ = 3. Then 8 = 1\u201c^ becomes s \u2014 = 9. There are 9 sq. ft. in 1 sq. yd. In the same way by using c = you can make your own cubic meas- ure table. and I the number of 1 cu. yd. Exercises Using the table at the left as a guide, complete the other tables on a separate sheet of paper. Linear Measure 1. 12 in. = 1 ft. 3 ft. = 1 yd. yds. = 1 rd. Square Measure … sq. in. = 1 sq. ft. … sq. ft. = 1 sq. yd. … sq. yds. = 1 sq. rd. Cubic Measure . cu. in. = 1 cu. ft. . cu. ft. = 1 cu. yd. cu. yds. = 1 cu. rd. 2. 10 millimeters (mm.) = 1 centimeter … sq. mm. = 1 sq. cm. 10 centimeters (cm.) = 1 decimeter … sq. cm. = 1 sq.dm. 10 decimeters (dm.) = 1 meter (M.) … sq. dm. = lsq. M. . . . cu. mm. = 1 cu. cm. . . . cu. cm. = 1 cu. dm. . . . cu. dm. = 1 cu. M. 42 THE FORMULA How to express other relations in formulas. We have learned that multiplication in algebra is generally expressed by putting two letters together with no sign between them. We may also use the or- dinary times sign of arithmetic (X), or we may write a dot be- tween the quantities half way up the height of the letter. Often the quantities to be mul- tiplied are put in pa- rentheses ( ) with no sign between the paren- theses. Two numbers cannot be put together JOSEPH LOUIS LAGRANGE to indicate multiplica- (1736-1813) . V, f French mathematician who took a prominent becaUSe they form part in establishing the metric system of a neW number that is weights and measures. He also wrote on the solution of equations, but his most famous nOt the prodUCt work was his Analytic Mechanics. ^ times sign, the dot, or the parenthesis must be used. 2-3-4= 24 2X3X4 = 24 (2)(3)(4) = 24 To express addition or subtraction, we always write the plus (+) or minus (-) sign. 2x-f3y ba \u2014 b To express division we can use the ordinary sign (-f) as in arithmetic, but more often we write the number we are dividing by (the divisor) under the number we are divid- ing it into (the dividend) with a line between them just as in fractions. _ 6 \u201c 3 18 6 = 3 THE PARENTHESIS 43 Puzzle. Here is a simple example in arithmetic. Try it. 2+2×4-6^3×2 + 5×24-2-1 What answer do you get? The correct answer is 13. Can you discover how to get it? It is evident that you can get a large number of different answers depending on what you do first. So if we are going to agree on an answer, we must first agree on the order in which we shall perform the operations. Mathematicians have agreed on the following order: Order of operations. 1. First perform all multiplica- tions. 2. Next perform all divisions from left to right. 3. Finally perform all additions and subtractions in order from left to right. Now try the puzzle again, and see if. you can get the right answer. Need for the parenthesis. If we do not want the multi- plication done first, how could we write the example? Suppose, for instance, we wished to add 2 and 5 and multi- ply the result by 3. We could not write 2 + 5 X 3 for this would mean that only the 5 is to be multiplied by the 3 and then the 2 added. We write it this way: (2 + 5) 3 or better 3(2 + 5) which equals 3 X 7 = 21 A parenthesis ( ) is used to show that the operations in- side it are to be performed first, or that the quantities in it are to be treated as a whole. Exercises Find the value of the following: 1 . 7 X 3 + 4 2 . 8 – 2 X 3 3 . 3 + 4 4 – 2 4 . 5 X 6 ^ 3 6. 8 4 – 2 \u2022 2 6. 4 + 5X3-1 7 . 6×2-44-2 8. 3 + 2 X (3 + 4) 9. (2 + 5)(3 – 1) 10 . 18 4 – 3 4 – 3 + 4 44 THE FORMULA 11. 2^-2×4-4-^2.2+2×4-^2-2 12. 8 – (2 + 3) + 3(5 – 1) – 6 3 + 1 ,, 5(9+3) ,, 6(2+8) 3(14 + 7) 4(15 + 10) 13. 2 5 7 10 17. In Exercises 15 and 16, would you get the same answer if you divided 7 into 7 and 10 into 10 before adding? Try it. Parentheses in Formnias The trapezoid is a four-sided figure, two of whose sides are parallel, that is, run in the same direction, as ED and BC. It can be cut into two triangles h EBC and EDC. Do you think that these triangles have equal altitudes (heights)? What is the area of triangle EBC if h is the altitude and b is the base? Of CDE if h is the altitude and a the base? The area of a trapezoid equals half its altitude multi- plied by the sum of the two parallel sides. Formula: A = h{a + h) 1 . Find + if: {a) h = 10, a = 7, b = 11 (c) h = 9, a = 12, b = 14 (b) h = S, a =4hb =6^ {d) h = 7, a = 5, b = S (e) If k, a, and b are all doubled, by what is A multiplied? 2. Here is the cylinder again. What is the Shape of the top and bottom? If the radius is r, what is the area of the bottom? Of the top? Of both? E a D h PARENTHESES IN FORMULAS 45 If this cylinder were made of paper and we cut it straight down from top to bottom and spread it flat, it would form a rectangle. What line of the cylinder would become the altitude of this rectangle? What line would become its base? Can you write a formula for finding c if r is known? The formula for the surface of a cylinder can be put in the form: S = 2 7rr(r + h) Find the surface of a cylinder if r = 10 and h = A. 3. The formula for the surface of a cone (page 40) is: S == 7rr(r + s) Find the surface of a cone if: (a) r = 7 and s = 9 (c) r = 7 and s = 18 ip) r = 14 and s = 9 ( 0 11 1 Honor Work 21. K = + AQ e from e = 0 to g = 5 22. R = .001 + 273)2 for ^ = 0 and t = 27 23. PV = 840 fromP = ItoP = 8 24. L = 100(1 + .03 \/ + .0001 P) from \/ = 0 to \/ = 40 by lO\u2019s THE FORMULA GRAPH John Hunter got a position at the Post Office just before Christmas. He wanted to be able to tell customers quickly the cost of sending parcels. He found that for the third zone the rate was for the first pound and 2^ for each ad- ditional pound. John wrote the formula c = 9 + 2{n \u2014 V) and then he changed it to c = 2 w + 7. Can you see how he got this result? Using this formula, he next made the table: n 1 2 3 4 5 6 7 8 9 10 c 9 11 13 15 17 19 21 23 25 27 THE FORMULA GRAPH 57 Finally he decided to make a large graph so he could read results directly from it. So he laid off the values of n at equal intervals on the horizontal axis and selected a scale for c for the vertical axis. Then he marked points. For w = 1, c = 9, he put a point directly above the 1 on the n axis and at a height of 9. Above 2 he put a point at a height of 11 and so on. Finally drawing a line through all his points, he obtained this graph. Engineer\u2019s Square-Root Graph Engineers use graphs for many purposes. Here is one from which they can find squares and square roots of numbers. Let y = Fromthis we can make the table: X 1 2 3 4 5 6 7 8 9 10 1 4 9 16 25 36 49 64 81 100 Laying off x on the horizontal axis and y on the vertical axis we get this graph. 58 THE FORMULA r prom the graph estimate the squares of 3.5, of 6.3, of 4.8. The square roots of 30, of 54, of 20, of 43. In making graphs, we generally follow these rules: When the formula is solved for one letter, that is, when one letter is alone on the left side of the equation, we use that on the vertical axis. When one of the quantities is time, we measure that off on the horizontal axis. Mr. Wright found that the test he gave his class was too hard so he decided to mark a pupil who got 50% up to 65%, but to leave 0 and 100 unchanged. So he drew this graph, making the horizontal axis the mark the pupil received and the vertical axis the mark the pupil deserved. What mark should he give a pupil who made 80% ? 40%? 65%? 50%? 90%? 55%? Class Exercises Draw a graph for each of these exercises. 1. i = At 4:. c = Afr 7. c = 2 w + 6 2. A = IQh 5. c = 2^ i 8. c = 20 m + 10 3. V =32 1 6. \/?= 2 ^ 2 – Find Vj_ when p^ = 15, p^ = 80, and – 120 . 19. m = Y~Er^ ‘ 1 . p — dp ^ 13.60, d = 20%, and P = 15% 20. T = irr^ + Tfh. Find h when T ~ 471, x = 3.14, and – 10. 21. H = (Wi + W2)it2 – t i)r – (x +:v + =30, w = 20, L and 2 = 21 .55 u, = 50 . 87, = 62, r = 11, X = 22, j = 34, 22. ^ j(F -b) == k. Find k when P = 80, F = 10. a = .01, and b = .05. Review of Formula Graphs 1. {a) Draw the graph of F = Ah from ^ = 0 to \/? = 6. (&) By extending your graph, find the value of F when h = 7. (c) When h increases, does F increase or decrease? (d) When h becomes 3 times as large, does F become ex actly 3 times as large as before? 2. (a) Draw the graph of A = 200 + 12 t from ? = 0 to t = 5. (b) By extending your graph find A when t = 6. (c) From your graph find t when A = 242. {d) When t increases, does A increase or decrease? 62 THE FORMULA (e) When the value of t is doubled, is the value of A dou- bled? 12 3. (\u00ab) Draw the graph of U = \u2014 from P = 1 to P = 12. {b) From your graph find the value of V when P = 9. (c) From your graph find P when F = 5. {d) When P increases, does V increase or decrease? (e) When P is doubled, what happens to the value of V? (\/) Can you find the value of V when P = 0? 4. (a) Draw the graph of C = f (P \u2014 32) from P = 32 to P = 95. (b) From your graph find C when P = 68. (c) When P increases, does C increase or decrease? (d) Is C doubled when P is doubled? 6. (a) Draw the graph of A = from s = 0 to s = 10. (&) From the graph find A when s = 6.5. Find s when A = 20. (c) When s increases, does A increase or decrease? (d) When s becomes 3 times as large, by what number is A multiplied? (e) Which is increasing the faster, s or ^4? Test in Evaluating Formulas Formula Given Values Answer 1. – a + b c a = 12, b ^ 8, c = 9 p = . . . 2. A = bh b = U,h = 10 4 =.. . . 3. .4 = khib + t) h = 15, b = 13, \/ = 7 4 = . . . 4. 4 = P + Prt P = 500, r = = .04, \/ = 6 4 = . . . 5. P = mv^ m = 14, V = 9 E = . . . 6. r = 7rr2 -f- Trrh TT = ^,r = l,h = 12 T = . , . 7. 5 = n \u2014 k n = 26, k = 00 t\u2014 * 4^ 5 = . . \u2022 p – k 8. V = vm-^ V = 50, m-^ = 12, = 17 F – %. h = s \u2014 ^ ^ j. ^ 5, s = 88 h = . . . 7r(r + 3) 10 . w = 800 kn^ k = 6.6, n = : 3, r = 6 w = . . . r2 +40^2 TESTS 63 Test in Making Formulas from Rules Make formulas using initial letters of the important words: 1. The perimeter of a regular hexagon is 6 times an edge. 2. The amount of money at interest equals the sum of the principal and interest. 3. The sum of a certain group of numbers is i the number of them multiplied by the sum of the first and last numbers. 4. When 2 boys balance on a seesaw, the weight of 1 equals the product of the weight of the other and his distance from the fulcrum divided by the distance of the first from the fulcrum. 5. A married man\u2019s income tax is found by taking 1% of the remainder after deducting $2500 and $400 times the number of his children from his income. Matching Test in Recognizing Formulas Here are important formulas that you should recognize. Write the numbers 1 to 14 in a column and after each number, write the letter of the formula that corresponds to the state- ment after the number. The formula for finding the: 1. Area of a rectangle b. V = Iwh 2. Perimeter of a triangle 3. Volume of a box 4. Area of a circle 5. Amount at simple interest 6. Volume of a cube 7. Circumference of a circle 8. Percentage 9. Area of a triangle 10. Perimeter of a rectangle 11. Area of a trapezoid 12. Weight on a seesaw 13. Distance a stone will fall 14. Capital of a corporation i. A = 7rr2 j. V = e^ k. A ^ ^hb l. p = br m. c = 2 irr e. c? = i gt^ f. A = Iw n. A = ^h{h^ + hf) c. A = P + Prt d. C = A – L g. p = a + b + c h. p =21 + 2 w Chapter 3 GEOMETRIC MEASUREMENT M r. Deems, a surveyor, says that he ran a line at an angle of 87\u00b0 with the road. What did he mean? If you wish to understand the language of surveyors, or if you are interested in astronomy, architecture, or any kind of engineering, you need some knowledge of angles and lines. You know what a straight line is, but do you know that a mathematician thinks of that line as having no end but yj ^ going on and on? The part from A to R he calls a line segment. Do you know that a line has no width or thickness, but only length? It is like the upper edge of our line AB. We read a line segment by naming the two capital letters at its ends, as AB, or by reading one small letter placed somewhere near its middle, as line segment a. When the small letter is used, it generally stands for the number of units in the length of the segment. For example, a could stand for 7 in. or 16 ft. When no confusion can result, we generally call either a line or a line segment simply a line. Exercises 1. Draw a line on paper. Is it a line or a line segment? Ex- plain. 2. Is it a true line segment? Has it width? Thickness? 3. Which do you think better represents a line segment, a pencil, a piece of wire, or a piece of spider web? Why? 4. If the end of a line segment is a point, how large is a point? 6. Make a point on your paper, and draw a straight line 64 THE STRAIGHT LINE 65 through it. Can you draw other straight lines through that same point? 6. Make two points on your paper, and draw a straight line Photograph from Philip D. Gendreau, N.Y THE ENGINEER AT WORK Drafting plans for large buildings requires a thorough knowledge of mathematics. through both of them. Can you draw other straight lines that pass through both points? 7. How many of its points must you know to be able to tell just where a straight line goes? We say that two points determine a straight line because only one such line can pass through both of them, and we can tell which dine is meant when two of its points are named. 8 . Point out the segment AB, BC, DE. a t> ^ n 9. What segment equals AB + BCl ^ CD + DEI AB + BC + CD? 6(> GEOMETRIC MEASUREMENT 10. What segment equals AC \u2014 A B’i AE \u2014 AB \u2014 DE? AD – CD? How to lay off a line segment. To lay off a length on BC equal to segment a, open your compasses until the K B ‘ C points will rest on the ends of a. Then putting a point of the compasses on B, make a mark across BC. When we copy a figure accurately, using only compasses and ruler, we say that we construct it. If we also use other drawing instruments, of which you will learn later, we say that we draw the figure. Exercises 1. Draw a segment, and mark it a. Now construct a segment 3 times as long as a. 2. Draw two segments k and n. Construct a segment equal to their sum. 3. Draw three segments r, s, and Construct r + s + \/. 4. Draw two segments a and h. Construct 2\u00ab + &, 2\u00ab + 3&, 2 \u00ab – &, 2 a + 2 3 a – 2 6. The angle. An angle is formed by two straight lines that meet at a point. The lines are the sides of the angle, and the point is its vertex. We generally read an angle by a capital letter near its vertex, as angle A (written ZA). However when two or more angles have the same vertex, we cannot tell by this method which of them is meant. We then read three letters, one at the vertex and one on each side, the letter at the vertex being read between the other two, as Z ABC THE ANGLE 67 or Z.CBA. The letters are read in the order you would come to them if you drew the angle with one continuous stroke of your pencil. We can also read an angle by a small letter inside the angle. Z ABC can be read Ax. We may think of A ABC as having been formed by turning a line around b* point 5 as a pivot from the initial posi- tion BC to the final position BA. The size of the angle depends on the amount of this turning and not on the length of the sides. Illustration. Lay your book closed on the desk. Now open the front cover without moving the book. Notice that the cover turns around the back edge as the book opens. The size of the angle that the lower edge of the cover makes with the lower edge of the first page depends on the amount the cover has turned. Courtesy of Missouri State Highway Dept. MODERN CROSSROADS BECOME GEOMETRICAL Nowadays superhighways are made more safe for drivers by elimination of intersections; In the picture above 45\u00b0 angles are used to solve the prob- lem. In others complete circles are utilized. 68 GEOMETRIC MEASUREMENT Exercises 1. Draw lines AB and CD crossing at point E. Read the angles formed by these lines. 2. There are six angles in this figure. Can you name all of them? 3. Which of the angles ABC and DEF is the larger? If you lengthen the lines BA and BC, does A ABC become larger? 4. Which is larger, ZAFB or A AFC? 5. If ACFD is added to A AFC, what angle is the result? 6. If AAFB is subtracted from AAFD, what angle is the result? 7. Is there any angle in the figure larger than AAFD? 8. If ADFE is added to AAFD, what angle is the result? 9. Draw a straight line AB. Now draw another straight line CD meeting A 5 at Z) so that 2 equal angles are formed at D. What is the name of this kind of angle? CLASSIFICATION OF ANGLES When the line AB has rotated around B until it takes the position BF in a straight line with CB, the angle formed is 2i straight angle (st. A). ACBFis a straight angle. When line AB had turned half that amount to the position BD, the angle it formed with CB was a right angle. A right angle (rt. Z ) is half a straight angle. ACBD is a right angle. CLASSIFICATION OF ANGLES 69 Acute Two lines that form a right angle are perpendicular to each other. DB is perpendicular to FC. An angle smaller than a right angle is an acute angle, as ZA. An angle larger than a right angle but smaller than a straight angle is an obtuse angle; as AB. The degree. The ordinary unit for measuring angles is 60 ‘ = 1\u00b0 the degree (\u00b0) which is of a right angle. 90 \u00b0 = 1 rt. z The straight angle then equals 180\u00b0, and a 180 \u00b0 = 1 St. z complete revolution about a point equals 360\u00b0. A degree may be divided into tenths, hundredths, etc., or into 60 equal parts called minutes (‘). How to measure an angle. A protractor is an instru- ment for measuring the number of degrees in an angle. It parts called degrees. To find the number of degrees in an angle, we place the protractor so that its middle point is on the vertex A of the angle and so that the straight edge of it fits along one side AC. Then the point where the other side AB crosses the scale tells us the number of degrees in the angle. In the figure the angle is about 33\u00b0. Can you see how we found that answer? The principal instruments used in engineering and navigation, such as the transit and sextant, are funda- mentally protractors. 70 GEOMETRIC MEASUREMENT Exercises with the Protractor 1. Draw an acute angle BAC. Now using a protractor as shown in the figure, measure the number of degrees in your angle. 2. Draw an angle as near a right angle as you are able without using the protractor or compasses. Now measure it with a protractor, and note how many degrees it differs from a true right angle. 3 . In the same way draw and afterwards measure angles of 45\u00b0, 135\u00b0, 30\u00b0, 60\u00b0, 120\u00b0, 22^\u00b0, 671\u00b0, 20\u00b0, and 40\u00b0. In each case compare your estimate with the measured result. 4 . Draw an angle whose sides are each 2 in. long, and meas- ure it. Now lengthen the sides, making one of them 3 in. and the other 4 in. long. Measure the angle again. Has its size changed? 5. Draw two angles as nearly equal as you can estimate. Measure them, and determine the number of degrees by which they differ. 6. Draw an angle ABC. Now draw a line BD that you think will divide \/.ABC into two equal parts. Measure the two parts, and determine their difference. 7 . Draw an angle of 45\u00b0, and extend one of its sides through the vertex. Measure both angles. Find their sum. 8. Make the same drawing and measurements, starting with angles of 135\u00b0, 30\u00b0, 60\u00b0, 120\u00b0, 150\u00b0, 221\u00b0, and 40\u00b0. Could you have computed the number of degrees in each of the angles formed by extending the line, without measuring? How? 9. Draw two lines that cross each other. Measure two op- posite angles. How do they compare in size? How to copy an angle. To make an angle at D equal to AB: 1. Place the point of your compasses on B, and with an^i convenient opening make an arc cutting AB at F and BC at G. CLASSIFICATION OF ANGLES 7i 2. Keeping the same opening of the compasses, put the point on D, and draw an arc cutting DE at H. 3. With your compasses measure GF. Then put the point on H, and make an arc cutting the other arc at 1. 4. Using your ruler, draw DL AD will equal AB. Exercises with Compasses and Ruler Remember we say we construct a figure when we draw it accurately using compasses and straight-edge ruler only. 1. With your protractor draw an angle of 70\u00b0. Now construct an angle equal to it. Check with your protractor. 2. Draw the following angles, and construct an angle equal to each: {a) 15\u00b0, (&) 110\u00b0, (c) 90\u00b0, {d) 45\u00b0. 3. Draw an angle of 35\u00b0. Now construct an angle twice as large. 4. Draw any two angles, and then construct an angle equal to their sum. 5. Draw an angle of 20\u00b0. Construct an angle {a) 3 times as large, (&) 4 times as large, (c) 6 times as large. 6. Draw three unequal angles, and construct ah angle equal to their sum. Check with your protractor. Vertical angles. When two straight lines cross, the opposite angles are called vertical angles, as Ax and Supplementary angles are two angles whose sum is a straight angle, as Z;r and Ay, or as Av and Aw. Each angle is the supplement of the other. 72 GEOMETRIC MEASUREMENT Class Exercises 1. If the line AB turns around point B in the direction op- posite to that in which the hands of a clock move until it lies in a straight line with BC, name in order the four kinds of angles it forms with BC. 2. What kind of angle equals the sum of a right angle and an acute angle? What kind of angle equals their difference? ^ 3. If an acute angle is subtracted from a right angle, what kind of angle is the result? B C 4. If an acute angle is subtracted from a straight angle, what kind of angle is the result? 6. Compare an acute angle with its supplement. 6. If an acute angle grows larger, how does its supplement change? How large must the angle become to equal its supple- ment? To exceed its supplement? 7. What kind of angle is f of a straight angle? f of a straight angle? 8. Is ^ an obtuse angle necessarily an acute angle? Is twice an acute angle necessarily an obtuse angle? 9. If 3 times an acute angle is an obtuse angle, what number of degrees must the acute angle exceed? 10. In this figure there are 6 pairs of supplementary angles. Can you name all of them? There are also 2 pairs of ver- tical angles. Name them. 11. What is the angle formed by the hands of a clock at 3 p.m.? At 6 p.m.? At 4 p.m.? At 2 p.m.? At 5 P.M.? 12. Through how many degrees does the minute hand of a clock turn in 1 min. of time? 13. How many degrees are there in the angle formed by the hands of a clock at 42:30 p.m.? At 2:30 p.m.? 14. Through how many degrees does the minute hand pass in 20 min.? In 13 min.? 15. If the earth turns completely around in 24 hrs., through how many degrees does it turn in 1 hr.? In 2 hrs.? In 15 min.? How long does it take the earth to turn through an angle of 60\u00b0? Of 1\u00b0? CLASSIFICATION OF ANGLES 73 Optional Exercises 16. Find the supplement of 50\u00b0. Of 67\u00b018′. 17. Find the angle that is f of its supplement. 18. If a part of a circle (an arc) has the same number of degrees as the angle at the center that cuts it off, how many degrees are there in a whole circle? 19. How many degrees are there in a half circle? In a quarter circle? 20. Using the formula c = 2 wr, find the circumference of a circle whose radius is 7, 10, 1000, z. 21. What part of a circle is cut off by an angle at the center equal to: 90\u00b0? 60\u00b0? 45\u00b0? 20\u00b0? 57\u00b0? n\u00b0? 22. If the radius of the circle is 7, what is the length of an arc of 180\u00b0? 90\u00b0? 60\u00b0? w\u00b0? Honor Work 23. Using the formula A = find the area of a circle whose radius is 7. 24. What part of the circle is shaded if Z 0 s: 90\u00b0? 60\u00b0? 100\u00b0? w\u00b0? 25. If r = 7, find the area of the shaded part (sector) when Z 0 = 90\u00b0, when Z 0 = 60\u00b0. To bisect means to cut in two equal parts or in halves. How to bisect a line segment. To bisect AB: 1. With A as center and with a radius more than half of AB, construct arcs on both sides of AB. 2. With B as center and with the same radius, construct arcs cutting the first arc at C and D. 3. With your ruler draw CD cutting AB at E. 4. E is the middle point of AB. Note. CD is also perpendicular to AB. It is called the perpendicular bisector of AB. GEOMETRIC MEASUREMENT How to bisect an angle. Draw an angle ABC. With B as center, draw an arc cutting AB at D and BC at E. With D and E as centers and with the same radius, draw two arcs crossing at F. With your ruler draw a straight line from BtoF. bisects \/.ABC. Construction Exercises 1. Draw a vertical line segment, and bisect it. Test with your compasses. 2. Draw a line segment. Then construct a segment times as long. 3. Divide a line segment into 4 equal parts. 4 . By the method learned above, can you divide a line segment into 3 equal parts? 5 equal parts? 8 equal parts? Construct those that are possible. 5 . Bisect an angle of 45\u00b0, 60\u00b0, 120\u00b0. Test each angle either with your compasses or with your protractor. 6. Construct a 90\u00b0 angle by bisecting a straight angle. 7 . Draw an angle and divide it into 4 equal parts. 8. Can you divide an angle into 3 equal parts? 5 equal parts? 8 equal parts? Construct those that are possible. 9 . Construct an angle of 45\u00b0, of 135\u00b0, of 22\u00b030’. 10. Construct an angle li times a given angle. 11 . At a point C on line AB, construct a line perpendicular to AB. Thought Questions 1 . By the methods you have just learned, can you cut a line segment or an angle into n equal parts if n is 2? 3? 4? 5? 6? 8? 9? 16? 20? 24? 32? 64? 2 . If an angle can be cut into n equal parts by these con- structions, can a factor of n be 2? 3? 5? 7? Of what prime factors is n necessarily composed? ft 3. Can you divide an angle into ^ equal parts if w is 1? If n is 3? Can n equal any whole number? SYMMETRY AND BEAUTY 75 Symmetry and Beauty 1 . Make a few heavy lines with ink on a sheet of paper, and fold it quickly before the ink dries. Has it become more beauti- ful? 2. Draw a few lines on paper, and hold a mirror perpendicular to the paper. Does the reflection add to the beauty of the figure? 3. If you can obtain 2 pieces of plane mirror, hold them perpendicular to the paper and perpendicular to each other. Does this still more increase the beauty of your diagram? \/ When a figure can be folded along \/ a line so that the part on one side ^ of the line will exactly fit the part on the other side, we say that it is symmetric with respect to that line. The line is called an axis of symmetry, 4. Look at the picture in the front of this book. Is it beauti- ful? Can you find an axis of symmetry in it? 6 . How many axes of symmetry can you find in: {a) A square? {d) An isosceles triangle? (&) A rectangle? {e) A regular 6-sided figure? (c) An equilateral triangle? (\/) A circle? Square Rectangle Equilateral Triangle Regular Circle Hexagon 76 GEOMETRIC MEASUREMENT 6. Are you symmetrical? Is a ship symmetrical? An auto- mobile? An animal? Where is the axis of symmetry in a front- view photograph of a person? Courtesy American Museum of Natural History. BUTTERFLIES Most animals are symmetrical, but the butterfly is a particularly striking example. 7. The butterfly is a beautiful example of a symmetrical creature. Can you find the axis of symmetry? Photographs by W. A. Bently. SNOWFLAKES Plato said: \u201cGod continually geometrizes. \u201d 8. How many axes of symmetry can you find in these snow- flakes? Are they beautiful? 9. Are the needlework figures on the following page symmet- rical? How many axes of symmetry can you find in them? THE CIRCLE GRAPH 77 SYMMETRY IN NEEDLEWORK DESIGN THE CIRCLE GRAPH The circle graph is used to show how the whole of any- thing is distributed into its parts. It should be used only when all of the data occur at the same time and when the sum of all the data has a definite mean- ing. This graph shows how an average family uses its income. Here the sum of all the parts is the total income. About what part of the income is spent for food? For clothing? For amusements? If the income of this family is $1800, estimate the amount spent for amusement. How much is saved? What is the largest expense? The smallest? How to make a circle graph. Illustration: A college student spends $400 for board, $450 for tuition, $100 for books, $100 for clothing, $60 for travel, and $90 for amusements. 78 GEOMETRIC MEASUREMENT Solution. First find the total amount spent. Then find the part of 360\u00b0 that each number is of. this total. 400 is represented by \u2014 X 360\u00b0 = 120\u00b0. 1200 360\u201d =135″, and SO on. Now lay off at the center of a circle angles of 120\u00b0, 135\u00b0, 30\u00b0, etc. Finally label the parts of the circle, and shade or color them. 400 450 100 100 60 90 1200 120 \u00b0 135 \u00b0 30 \u00b0 30 \u00b0 18 \u00b0 27 \u00b0 360 \u00b0 Notice that the number of degrees N in the arc can be 360\/ found by using the formula N = ^ , where \/ is the value of an item and T is the total of all items. Find N when: {a) I = 240 and T = 480 (c) \/ = 16 and T = 384 {b) \/ = 82 and r = 246 (\/) \/ = 86 and T = 720 Exercises Make a circle graph for each of these: 1. On school days George is in school 6 hrs., sleeps 8 hrs., eats 2 hrs., studies 3 hrs., works 2 hrs., and plays 3 hrs. 2. In an algebra test 7 pupils received a mark of A; 8, a mark of B; 10, a mark of C; 5, a mark of D; and 6 failed. 3. In our high school there are 200 seniors, 250 juniors, 280 sophomores, and 350 freshmen. 4 . A cost accountant found that the expenses in a manufac- turing business, based on each dollar spent, were distributed as follows: wages, 45)zi; cost of materials, 30^; cost of sales, 12)zi; overhead, 10^; and profit, 3^. 5 . In making his income-tax report, Mr. Wells found that he had $2000 salary, $800 interest on investments, and $200 other income. 6. A family budget provides 30% for food, 25% for rent. RECTANGLE DISTRIBUTION GRAPH 79 15% for clothing, 20% for other expenses, and the remainder for savings. 7. In a single year the U.S. spent $960,000,000 for veterans\u2019 relief, $700,000,000 for defense, $400,000,000 to reduce its debt, $600,000,000 interest, $440,000,000 for farm relief, and $1,100,- 000,000 for all other expenses. 8. The land area of the earth is divided as follows: Asia, 17.000. 000 sq. mi.; Africa, 11,000,000 sq. mi.; No. America, 8.000. 000 sq. mi.; So. America, 7,000,000 sq. mi.; Europe, 3,500,000 sq. mi.; and Australia and islands, 3,500,000 sq. mi. 9. The grain crops of the world, in 100,000,000 bu., for a certain year were: corn, 44; wheat, 50; oats, 45; barley, 20; and rye, 21. 10. Here is the way a certain city spent its income one year: education, $78,000; protection, $92,000; interest on bonds, $120,000; other expenses, $70,000. Which is the largest expense? If this expense were not neces- sary, by about what fraction would the taxpayers\u2019 burden be decreased? 11. One year our exports to the Latin American countries amounted to $900,000,000. $180,000,000 of it went to Argen- tina, $100,000,000 to Brazil, $50,000,000 to Chile, $170,000,000 to Cuba, $120,000,000 to Mexico, and the remainder to the other countries. THE RECTANGLE DISTRIBUTION GRAPH Instead of using the circle, we sometimes divide a long rectangle into parts to show how a whole is distributed. This graph shows the distribution of automobile acci- dents. Speeding On wrong side of road Did not have right of way ” Failed to signal Reckless driving Other accidents WTiat do you think is the cause of the largest number of accidents? About what per cent of the accidents does it cause? What is the next most important cause of acci- dents? What per cent of accidents does this cause? 80 GEOMETRIC MEASUREMENT Exercises Make a rectangle distribution graph for these data: 1. The number of million people in the world is: No. America, 150; So. America, 65; Europe, 475; Asia, 1000; Africa, 140; others, 10. 2. The percentage population of the United States according to age is: under 5 yrs., 10%; 5 to 14 yrs., 20%; 15 to 20 yrs., 10%; 21 to 45 yrs., 38%; and over 45 yrs., 22%. 3 . Automobile accidents to pedestrians (people walking) are divided as follows: crossing against signal, 11%; crossing where there is no signal, 15%; crossing not at the corner, 25%; children playing in the street, 18%; coming from behind parked car, 13%; and all other causes, 18%. If accidents happened to about 300,000 pedestrians, about how many children were hit while playing in streets? How many were hit coming from behind parked cars? 4 . Of the world’s oil supply, the United States produces 63%; Europe, 15%; So. America, 12%; Asia, 5%; and the rest of the world, 5%. 5 . In 1930 the number of million people in the United States whose parents were native was 82; whose parents were for- eign, 17.5; one of whose parents was foreign, 8.5; who were themselves born in a foreign country, 14. 6. The cost of building a house was distributed as follows: foundation, 10%; lumber, 25%; carpenter work, 25%; mason and plumbing, 20%; and finishing, 20%. 7 . In a recent year the number of million children in school in the United States was: in public elementary, 21; in public high, 5; in all other schools, 6. Review Exercises 1. Explain how a line segment differs from a line. 2. How many points are needed to determine a straight line? 3. What is the difference between constructing and drawing a figure? 4. How many degrees are there in a right angle? In a straight angle? REVIEW EXERCISES 81 5. What is the name of the kind of angle that has 90\u00b0? 70\u00b0? 130\u00b0? 180\u00b0? 6. Are all acute angles equal? Are all right angles equal? Are all obtuse angles equal? 7. Explain the difference between perpendicular and vertical. 8. Write in short form: a a a a a a ‘ a ‘ a ‘ a \u2022 o, aaaaaaa. 9. Draw a line segment, and construct one twice as long. 10. Draw an angle, and construct one 3 times as large. 11. At 5 o\u2019clock the hands of a watch form an angle of how many degrees? 12. Find the supplement of an angle of 105\u00b0, of 75\u00b0. 13. How can you tell if an object is symmetrical? 14. Is a tree symmetrical? Do you consider a nearly symmet- rical tree more beautiful than one that lacks symmetry? 15. In the formula K = abC, find K when a = 12, b -= 20, and C = .88. 16. In the formula S = c + <3r + find S when c = 5 and r – 3. 17. Make a rectangle graph to show the following: The sales, in the departments of a store were: groceries, $60,000; clothing, $50,000; hardware, $18,000; toys, $32,000; and furniture $40,000. 18. In the formula m = find m when c = $8.80, p ^ 40%, and d = 20%. Honor Work Find the value of ike letter on the left side in these exercises: 19. X – ^ 120 , P 2 = 80, Xi = 9, and \u25a0Cl + Cg , =11 20 . 3^1^ + 2^j^g -j-^2^ 2 ( 2 +\/? 2 ) 21. P = \/) + (\/^ + x)y p ^ 42.5, 10 and ^2 = 15 h = 6.38, X = 5.12, y = 20 22. P = ~ r = 8.76, p = 13.4, = 2.4, i = .05 23. M = ^ p == 3460, \/ = 6.38 X lO\u00ae, e = 2 X 10^ 82 GEOMETRIC MEASUREMENT Computation Test 1. h = d — e <^ = 43 and e = 5 h = 2. A = i 5 = 24 and A = 15 A = 3. F = i -KT^h X = 3i r = 14, A = 12 F = 4 . ^ Fo = 60, = 55, Vo^ =33 = 5. F 18, ^ = 4,and\/ = 2 P = 6. F = Fo Fo=70and^ = 39 F = 7. F = i \/2(&2 + \/2 + 4 M) \/i = 20, & = 7, \/ = 5, M = 13 F = \u00ab\u2022 \u201d = ^ = 8.^ = 10,B = 30, A = 16 w = Matching Test Match the number of the statement with the letter before the word that completes the sentence: 1. The supplement of an angle of 150\u00b0 a. 2. An angle greater than a right angle h. 3. A straight line is determined by . . . c. 4. The point of an angle is called its d. … e. 5. When lines meet at right angles, they \/. are . . . g. 6. A kind of graph used for distributing h. the parts of a whole i. 7. A shorthand way of writing a rule j. 8. Distance around a figure 9. A small number that tells how many times a quantity is taken as a factor 10. An angle less than a right angle Perpendicular Formula Circle 30\u00b0 Vertex Acute angle Exponent 2 points Perimeter Obtuse angle Construction Test 1. Draw an angle of about 30\u00b0, and construct an angle 4 times as large. 2. By starting with a straight angle, construct an angle of 45\u00b0= 3. Construct a line perpendicular to another line at a point A of that line. TESTS 83 4 . Construct an angle li times an angle that you start with. 6. Draw a line segment, and construct a segment 2^ times as long. Test on Circle Graphs 1. This graph shows the distribution of oil among the coun- tries of the world. (\u00ab) The United States produces about … of the oil. (b) The continent that produces the second largest amount of oil is . . . . 2. This graph shows the distribution of a man\u2019s income. His total income from salary, commission, interest, and rent was $6000. (a) His salary was about … . (b) He received about equal amounts from . . . and … . (c) He received about . . . interest. (d) The fraction of his income from rent was about … . 3. Make a circle graph showing these statistics; State taxes are distributed as follows: on property, 22%; on income and inheritance, 11%; on automobiles and gas, 27%; on other licenses, 14%; miscellaneous, 26%. Chapter 4 ALGEBRA AS A LANGUAGE What we can do with letters. To find the perimeter p of the rectangle we use the formula p = I + w I i + w where I means the length and w the width. Now w + w = 2 w ior w ^ alone always means 1 w. In algebra just as in everyday speech, we often ^ omit the number 1. We say \u201cthree books\u201d when^we mean 3 books, but we generally say \u201cthe book\u201d when we mean 1 book. The \u201cone\u201d is omitted. Also \/ + \/ = 2 \/. So our formula becomes p = 2 I + 2 w. We cannot add the I and w until we know what numbers they stand for, so we leave it in the form 2 I 2 w. To multiply w by 3, we write 3 w, for we agreed that when we wrote two quantities together without a sign between them, we meant to multiply them. But when we wish to add 3 to w, we must leave it in the form w 3 until we know the value of w. Quantities that are connected by multiplication only are called factors. 3 and w are factors of 3 w. Each factor of a product is the coefficient of the others, but we more often use the word coefficient to refer to the number only. Therefore, 3 is the coefficient of in 3 w. An expression made up of numbers and letters, con- nected by multiplication and division only, is called a term. Thus, 3 is a term, but ei; + 3 is not a term because the parts are connected by addition. When you come to a plus or minus sign, you have come to the end of a term. There are then two terms mw + 3. When two terms are 84 THE LANGUAGE OF ALGEBRA 85 composed of the same powers of the same letters, we call them like terms. Like terms may have different num- bers as coefficients. For example, 5 ab and 3 ab or a^b^ and 5 a\u2018^\u00a5 are like terms. We have also seen that when we have two or more fac- tors that are equal, we can write the factor once and put a small number to the right and above to tell how many of those factors there are. For example, aa = aaa = G^ aaaa = and so on. A small number written to the right and above a quantity to tell how many times it occurs as a factor is called an exponent. Be careful not to confuse terms with factors, and co- efficients with exponents. If m; = 3, then: Terms: m;-|- 5=3+5=8 Factors :5 m; = 5×3 = 15 m; -f 5 is not the same as 5 m; If ^ = 4, then; Coefficient: 3^=3X4 = 12 Exponent: = 4x4x4 = 64 3 ^ is not the same as Exercises 1. Give the terms in each of these expressions, and also give the factors in each term: (a) 4 a (c) <\/ + 1 (e) 3 ahcde -1+2\/ ip) 6 aH (<?) 2 + 5 (\/) 4 – 2 2. State which numbers are coefficients and which are ex- ponents in Exercise 1. 3. If a = 10 and b = 3, find the value of: (a) 2 b (d) + 71 (g) ab (b) a + 4 (e) A b^ (\/?) 7 a" – 3 a& + 2 (c) b^ (\/) a + 6 – 4 (f) 5 a^ + 2 a& 4 . If a = 1, how much is 3 a? a^? 6. If & = 2, how much is 3 bl b^l 5 6\u00ae? 6\u00ae? 6. If c = 0, how much is 3 c? c + 5? c\u00ae? 86 ALGEBRA AS A LANGUAGE THE LANGUAGE OF ALGEBRA Algebra is a language that can say more in a small space than any other language in the world. Compare it with English. English. To find the amount at simple interest, we multi- ply together the principal, the rate, and the time, and add the result to the principal. Algebra. A = P + Prt Illustration: Write a number that is 5 less than the sum of X and y. Answer. % + y \u2014 5 Exercises 1. What is the sum of a and 3? Of x and y? 2. What is the product of y and 4? Of a and hi 3. Express 7 more than x, a more than 5, c more than d. 4. How many inches are there in 2 ft.? In\/ ft.? In 5 yds.? In y yds.? 5. How many inches are there in 2 ft. and 7 in.? In \/ ft. and 7 in.? In h ft. and c in.? 6. Dorothy had cjzi and spent 15(2^. How many cents has she left? 7. What number is 7 less than nl x less than 12? b less than a? 8. Express in cents: 3 dimes, d dimes, 2 Quarters, q quarters, 2 quarters and 3 dimes, q quarters and d dimes. 9. The difference of two numbers is 5, and the smaller is 8. What is the larger? What did you do to find the larger? 10. The difference of two numbers is d, and the smaller is s. What is the larger? 11. Write in algebra: a increased by b, x diminished by y, 3 times the sum of a and b, twice the product of m and n. 12. John is 3 yrs. older than Robert. Express John\u2019s age if Robert is r yrs. old. Express Robert\u2019s age if John is j yrs. old. 13. How many pounds are there in 48 oz.? In k oz.? 14. How many feet are there in 24 in.? In i in.? 15. What is the quotient of 3 divided by 8? Of n divided by 8? Of a divided by bl THE LANGUAGE OF ALGEBRA 87 16. Find the cost of p pencils at each, of b books at each, of e eggs at 50^ a doz. 17. Gladys saves SOizi a week. How much will she save in 14 weeks? In w weeks? In d days? 18. How far will an automobile travel in 5 hrs, at 35 mi. an hr.? In h hrs. at 35 mi. an hr.? In 5 hrs. at m mi. an hr.? In h hrs. at m mi. an hr.? 19. How many seats are there in a classroom that has 5 rows of 7 seats each? 5 rows of s seats each? r rows of 7 seats each? r rows of 5 seats each? 20. By how much does 17 exceed 12? What did you do to 17 and 12 to obtain the answer? 21. By how much does 17 exceed xl k exceed 12? m exceed w? 22. When you have already expressed two numbers, how do you express the amount by which one exceeds the other? Ex- press the amount by which a exceeds h. 23. What number exceeds 8 by 7? What did you do to 8 and 7 to obtain the answer? 24. What number exceeds 8 by %? a by 7? m by w? 25. When you know one of two numbers and the amount by which the other exceeds it, how do you express the other? Write a number that exceeds h by k. 26. What number is 5 less than 9? Do you subtract 9 from 5 or 5 from 9? 27. What number is X less than 9? 5 less than y? x less than y? When you express \u201cis less than\u201d in algebra, do you write the algebraic symbols in the same order in which the words occur in English? 28. How long will a train take to travel 120 mi. at 30 mi. an hr.? m miles at r mi. an hr.? , Translation Exercises Translate from algebra into English: 1. h = 2 a, i h the number of dolls Helen has and a the number Alice has 2. r = m; -f 5, if r is the number of marbles Roy has and w is the number William has 3. g -b 7 = 3 &, when g is George\u2019s age and b is Ben\u2019s, in years 88 ALGEBRA AS A LANGUAGE V = Iwh, when v is the volume of a box, I its length, w its width, and h its height 5. i = prt, if i is the interest on p dollars at rate r for t yrs. %. C = A – L, when A stands for assets, L for liabilities, and C for capital 1. d = rt, when r is the rate at which a train is traveling, t the time it is traveling, and d the distance it goes 8. \/) = 3 g – 10, when p is the number of cents Paul has and e the number Emily has 9. F = f C + 32, when F is the number of degrees registered on a Fahrenheit thermometer and C the number on a centigrade thermometer 10 . c = 15 + 20 m, when c is the cost of riding m miles in a taxi Translate from English into algebra, using the initial letters of the important words as symbols: 11. The gain (g) is found by subtracting the cost (c) from the selling price (s). 12. The area of a rectangle equals the product of its length and width. 13 . Three times a certain number {n) is 6 more than twice the number. 14 . If 13 is added to 4 times a number, the result is 31 more than the number. 16 . The volume {v) of a pyramid is i the product of its base {h) and height {h). 16 . The cost (c) of sending a package by parcel post for a certain zone is more than the number of pounds (w). 17 . The circumference of a circle equals two tt times the radius. 18 . If a stone is dropped from a height, the distance {d) it will fall is 16 times the square of the number of seconds (0- 19 . The average {a) of three numbers \u2014 x, y, and z \u2014 is found by adding them and then dividing their sum by 3. 20. The length {1) of the belt that will go around 2 equal pulleys is 2 TT times the radius (r) of a pulley added to twice their distance apart {d). THE EQUATION 89 THE EQUATION Away back in the distant past, 1700 years before Christ, an Egyptian priest named Ahmes solved problems by nnn iiiii algebra, but he did not know how to use the beautiful shorthand forms that you are learning now. For example, the equation shown here is simply y + | + ^ + j: = 37 Can you find the denominators 3 and 7 and the num- ber 37? An equation is a statement that two expressions are equal. Formulas, such as A = \/tc, F = Trr% and A = P + Prt, are equations. But there are also other kinds of equa- tions, such asA:-|-3 = 7, 2y \u2014 5 = 4, 3 a: \u2014 2y = 9. Equations are very important. Much of our work in al- gebra will be studying equations and learning how we can use them to help us solve problems. Finding the answer to an equation is called solving the equation, and the an- swer is the root of the equation. You may think of an equation as a question. For ex- ample, a: -|- 5 = 9 may be read, \u201cWhat number added to 5 gives 9?\u201d Evidently the answer is 4, so x = 4. Check the answer by putting 4 in place of a: in the orig- inal equation. 4 + 519 9 = 9 Solve iA: = 3. The question is: One-fourth of what number is 3? Then x must be 4 X 3 or 12. Many easy equations can be solved by arithmetic. We shall, however, learn methods of solving equations that are too difficult to be done this way. 90 ALGEBRA AS A LANGUAGE Class Exercises Solve the j allowing equations by arithmetic, and check your cnswers: 1 . 2 . 3. 4. 6 . 15. 16. 17. 23. 24. 26. X + 4 = 7 6. 7a: = 21 11. 5 a: = 25 a: + 8 = 9 7. a: – 3 = 7 12. a: + 5 = 6 – 1 = 2 8. a: + 2 = 9 13. – = 5 – 3 = 10 9. a: – 7 = 2 2 3 a: = 12 10. a: – 2 = 6 14. CO II Optional Exercises 2 a: + 3 = 11 18. 5 a: – 9 = 6 21. 1 + 2 = 3 3 a: – 4 = 8 19. 7 % + 3 = 17 5 ^ \u2014 5 \u2014 3 2 20. 4y – 1 = 11 22. 6z – 8 = 1 Honor Work 3 X \u2014 5 = X + 6 26. X + a = b 7x – 5 = 2 X + 10 21. cx \u2014 d = 2 3 + X = J + 5 28. ^ = 2 \u00ab 4 Practice in Algebra Shorthand How fast can you write shorthand? Try to write the equa- tion for each of these sentences while it is being dictated by your teacher. 1. What number increased by 11 equals 15? 2. Three times a number diminished by 9 equals 6. 3. Twice a number exceeds 8 by 5. 4. If twice a number is increased by 4, the result is 3 times the number. 5. Twelve exceeds a certain number by 7. 6. If 8 is added to 5 times a number, the result is 13. 7. If 3 times a number is diminished by 11, the remainder iS 10. 8. Five times a number equals 12 diminished by the number. 9. If a certain number is divided by 3, the quotient is 7. THE EQUATION 91 10. If 4 times a number is increased by 5, the result is 7 less than 6 times the number, 11. If I double a number and add 10, the result is 18 more than the number, 12. Three-fourths of a number increased by 4 equals 16. 13. If 7 is added to 3 times a number, the result is 1 more than 5 times the number. 14. Three times a number exceeds 10 as much as the number exceeds 2. 15. If 5 is added to 7 times a number, the result is 13 dimin- ished by the number. 16. A number equals ^ of itself increased by 8. 17. I am thinking of a number. If I double it and subtract 4, I shall have 3 more than the number. 18. .Fifteen exceeds a number as much as the number ex- ceeds 3. 19. Seven times a number lacks 12 of equalling 9 times the number. 20. If 17 is added to 4 times a number, the result is 3 more than 6 times the number. Algebra Shorthand in Geometry 1. The line AB is 10 in. long. After a length a has been cut off 3 times, the part left is 4 in. long, 2. A line AB, 17 in. long, is cut into 2 parts. One part is a and the other part is 3 in. longer than a. u -B a -B A- 3. ZA5C equals 65\u00b0. After 2 equal angles, each x, are cut off, the part left has 25\u00b0. 4. After 2 equal angles, each y, were laid off, it required 40\u00b0 more to make a straight angle. 5. I laid off 5 equal angles, and their sum was a straight angle. 92 ALGEBRA AS A LANGUAGE Review Exercises 1. Translate into algebra: (\u00ab) Twice a number decreased by 13 leaves 7. (b) If a number is added to 3 times itself, the result is 36. (c) Divide a certain number by 3, and you have 10 less than the number. 2. Express in algebra: (a) A number 3 larger than n (b) A number 4 times as large as t (c) Katherine\u2019s age 3 yrs. ago, if k is her age now 3. An angle has 3 x degrees, and its vertical angle has 2 ;c + 50 degrees. Find x, and also the number of degrees in the angle. 4. The change in pressure in a liquid is given by the formula P 2 \u2014 = dgihi \u2014 hi). Find P 2 when Pi = 17, d = .9, g ^ 32, \/?2 = 20, and hi = 4. 6. Draw an angle of about 20\u00b0, and construct another angle 5 times as large. 6. Construct a line perpendicular to another line at a point on the line. 7. Make a formula for: (a) The kinetic energy (k) of a moving body is found by multiplying ^ its weight (w) by the square of its velocity (v). (b) The amount of work (w) done in raising a weight is the product of the force (\/) and the distance (d) through which it moves. 4 ttH 8. Find g in the formula g ^ if \/ = 7 and t = 3. 9. Make a bar graph to illustrate these figures : The specific heat of water is 1; of alcohol, .7; of benzine, .9; of ether, .5; and of sulphuric acid, .3. 10. Make a graph for the formula w = 100 d from <\/ = 0 to is . . . . Construction Test 1. Construct an angle of 45\u00b0. 2. Construct a line 3 times a given line. 3. Cut a line into 4 equal parts. 4. Construct an angle twice a given angle. 94 ALGEBRA AS A LANGUAGE Formula Test Make formulas for these rules: 1. The selling price equals the cost plus the gain. 1. . . 2. The perimeter of a square is 4 times a side. 2. . . 3. In making coffee the number of tablespoonfuls (0 to use is 1 more than the number of people {p). 3. . . 4. To find the profit, subtract the cost and the expenses from the selling price. 4. . . 5. The distance an airplane flies is the product of the rate and the time. 6. . . 6. The average of three numbers, x, y, and z, is their sum divided by 3. 6. . . Test in Solving Equations Find the value of the letter in each equation: 1. X + 2 =b 2. a; + 8 = 33 3. a; – 5 = 9 4. a: – 3 = 16 6. 2 a: = 6 6. 5 a: = 30 9. 2 X + 3 = 11 10. 3 a: + 7 = 22 11. 3 a: – 5 = 7 12. 5 a: – 3 = 12 13. I + 2 = 6 15. 1.4 a; = 14 16. 3.6 a: = 7.2 17. 3 a; = 10 + a: 18. 7 a: = 15 + 2 a: 19. I + ^=10 1-1 = 1 14. I + 3 = 10 20. 2 – I = 5 8.2 = 5 Chapter 5 ALGEBRAIC ADDITION ALGEBRAIC NUMBERS T he thermometer in ordinary use in the United States is the Fahrenheit, on which the freezing point is 32\u00b0 and the boiling point 212\u00b0. But in most of Europe and in scientific work in this country, a different thermometer, the centigrade, is used. This has the freezing point at 0\u00b0 and the boiling point at 100\u00b0. One day while I was travel- ing in France, I noticed that the temperature was 20\u00b0 centigrade. Do you think it was a hot day or a cold day? You can find out by using the formula F = | C + 32 where F is the reading on our Fahrenheit thermometer and C that on the centigrade thermometer. What Fahrenheit reading corresponds to 40\u00b0 centigrade? To 10\u00b0 centigrade? To 80\u00b0 centigrade? And when we wish to find the centigrade reading cor- responding to one on the Fahrenheit, our formula can be changed to the form: C = f (F – 32) Copy the following table on a separate sheet, and complete it, using this formula: F 1 95 \u00b0 86 \u00b0 77 \u00b0 59 \u00b0 41 \u00b0 32 \u00b0 CO 14 \u00b0 c When F = 95, C = |(95 – 32) = f X63 = 35 Now you fill in the others. 95 96 ALGEBRAIC ADDITION When we come to 23\u00b0, we have C = f (23 \u2014 32). What does this mean? Can we take 32 from 23, or is there no centigrade reading corresponding to 23\u00b0 Fahrenheit? Look at the thermometers. When the Fahrenheit reading was 32\u00b0, what was the centigrade? If then the temperature drops below 32\u00b0, where would you expect to find the centigrade reading? On our thermometer, 23\u00b0 F. cor- responds to 5\u00b0 below zero centigrade. Then in some way our formula should express \u201c5 below zero.\u201d We shall now learn how to deal with this new kind of number. A new kind of number. Look at either thermometer. Does the scale end at 0, or are there numbers on the other side of 0? Can you think of other places where the numbers can go beyond 0? Later you will find many uses for a number scale that does not end at 0. In algebra there are numbers on both sides of 0. On one side are the positive numbers. These are the ordinary numbers that you have studied in arithmetic. They may be preceded by a plus sign or they may be written without any sign at all. Numbers on the opposite side of 0 from positive numbers are called negative numbers. They are always preceded by a minus sign. Positive and negative numbers are used to indicate opposites. Such numbers are called algebraic numbers. What is the opposite of: 1. A gain of $20? 2. 40\u00b0 north latitude? 3. A 15\u00b0 rise in temperature? 4. 75\u00b0 west longitude? 5. 3000 ft. above sea level? 6. Going forward 10 yds? 7. Earning $1,75? 8. $800 profit? 9. Adding 7? 10. 422 A.D.? 11. Going 40 mi. east? 12. Gaining 10 lbs. weight? ALGEBRAIC NUMBERS 97 13. In music, what is the opposite of a sharp? What is meant by \u2014 3 sharps? Class Exercises What is meant by: 1. A temperature of \u2014 40\u00b0 ? Of + 17\u00b0? 2. A latitude of \u2014 15\u00b0 if north latitude is positive? 3. The date \u2014 44 if time after Christ is positive? 4. A gain of \u2014 $100? 6. A longitude of \u2014 60\u00b0 if west longitude is positive? 6. A gain in weight of \u2014 2 lbs.? 7. A net change of \u2014 2^ in the price of a stock? 8. A deposit of \u2014 $28 in a savings bank? 9. A gain of \u2014 15 yds. by a football team? 10. A score of \u2014 220 points? Of + 150 points? 11. A man\u2019s property is + $500? Is \u2014 $800? 12. The temperature changed \u2014 12\u00b0? + 10\u00b0? 13. The altitude above sea level of Death Valley, California,, is – 276 ft.? What is the final result of: 14. Winning 200 points in a game and afterwards losing 250 points? 16. Walking 2 mi. north and then 5 mi. south? 16. Earning $2.30 one day and then spending $3.40 the next day? 17. A rise of temperature of 8\u00b0 from ~ 12\u00b0? 18. A ship starting at 25\u00b0 south latitude and traveling 40\u00b0 north? Optional Exercises 19. Will a balloon weighing \u2014 600 lbs. rise if the weight carried is + 400 lbs.? Is + 700 lbs.? 20. A firm has assets of $7000. What is it worth if its liabil- ities are $5000? If $7000? If $10,000? 21. At what rate can Paul row up-stream if he can row 4 mi. an hr. in still water and the current is flowing 3 mi. an hr-? If the current is flowing 6 mi. an hr.? 98 ALGEBRAIC ADDITION 22. Dorothy has 40jzf and owes 65^. What is her balance? 23. Augustus Caesar ruled Rome from the year \u2014 31 until the year + 14. What does this mean? How many years did he rule? 24. At noon the temperature was 24\u00b0. During the afternoon and night it dropped 34\u00b0, but rose 10\u00b0 again the next morning. What was the temperature the next noon? 25. Gladys is given $4, then spends $10, and later earns $9. How much money has she? 26. Here is a section of a stock-exchange report for a certain year as taken from a daily newspaper. The net-change column shows how much higher the stock was at the time reported than at the close of the previous year. Explain each net change, and tell what the price was a year earlier: Last Net Change Allis Chalmers 1901 not + 731 + 351 – 12i American Can American Car and Foundry 98i American Locomotive American Telephone and Telegraph 1081 193 – 31 + 14f 27. A recording thermometer made the following graph on a winter day: (\u00ab) What was the temperature at 3 A.M.? At 6 a.m.? At 12 noon? At 9 P.M.? R.20 10 12 3 6 A.M. 12 3 6 9 12 PM. (b) At what hour was it coldest? Warmest? (c) Tell at what times the temperature was 30\u00b0, 15\u00b0, 10 \u00b0. (d) What was the change in temperature from 3 a.m. to 6 A.M.? Front noon to 3 P.M.? From 6 p.m. to 9 p.m.? From 9 p.m. to midnight? (e) When was the temperature increasing most rapidly? Decreasing most rapidly? 28. When in the stratosphere 14 mi. above the earth, Stevens and Anderson reported the temperatures as \u201c55\u00b0 below zero ALGEBRAIC NUMBERS 99 centigrade outside and \u2014 7\u00b0 centigrade inside.\u201d IfF = fC + 32, what were those temperatures on a Fahrenheit thermometer? 29. Is the temperature increasing or decreasing when it changes from: {a) + 10\u00b0 to + 40\u00b0? {b) + 20\u00b0 to \u2014 30\u00b0? (c) – 18\u00b0 to + 8\u00b0? {d) – 20\u00b0 to – 5\u00b0? 30. Can a man\u2019s wealth increase to 0? Honor Work Deviations from the average. In statistics it is often important to know not only the average of several items, but also how much each item differs from the average. When an item is greater than the average, this deviation is positive; when less than the average, the deviation is negative. In these exercises, find the average, then the deviation of each item from the average, and finally the sum of the deviations: 31. 32. 33. 34. 8 32 18 82 12 43 19 67 9 27 21 91 7 51 23 73 4 36 16 85 11 44 11 96 5 33 18 87 36. For lunch one week Sarah spent 32jzf, 26^, 20^, 30^, and 22^. What was her average, and how much did she deviate from the average on the various days? 36. Mr. Wright\u2019s weekly pay for 2 mo. was $43, $52, $40, $44, $47, $53, $38, and $35. Find his average weekly pay, the devia- tions from the average, and the sum of the deviations. 37. A scientist wished to know a distance very accurately so he measured it 10 times. If his measurements were 42.3 cm., 42.1 cm., 42.4 cm., 42.0 cm., 42.4 cm., 42.2 cm., 41.9 cm., 42.2 cm., 42.3 cm., and 42.2 cm., find the average and the deviations from the average. What is the sum of the devia- tions? 100 ALGEBRAIC ADDITION The graph scale extended. When we studied graphs, we began our scale at 0 and laid off the numbers to the Photograph Try Lewis W. Hine. MACHINIST USING MICROMETER The machinist, as well as the scientist, must measure very accurately, sometimes to the ten-thousandth of an inch. right and upward in order from that point. In dealing with temperatures we used only those that were above 0. But temperatures can be below 0, and it is also convenient to be able to extend our scale to the left of the 0 on the hori- zontal axis. -8 -7 -6 -5 -U -3 -2 -1 0 ~1 i i 1 5 6 7 8 These numbers represent points on a straight line that can run on indefinitely in both directions. If you start at point 2 and add + 3, what point do you reach? Do you go to the right or to the left when you add a positive number? ALGEBRAIC NUMBERS IN SURVEYING 101 If you begin at point + 6 and add \u2122 2, what point do you reach? Do you go to the right or to the left when you add a negative number? Add + 4 to 5. Add \u2014 5 to + 4. Did you end at the same point both times? Exercises By starting at the first number named and going in the proper direction, perform these additions graphically: 1. (+ 2) + (+ 3) 5. 3) + (+ 4) 9. (+ 0) + (- 2) 2. (+ 5) + (+ 1) 6. (- 7) + (+ 3) 10. (0) + 6) 3. (+ 6) + (- 1) 7 . (+ 5) + (-^ 2) 11. (+ 2) + (~ 7) 4. (+ 2) + (~ 5) 8. (- 5) + (- 1) 12. (+ 0) + (- 0) Determine from the graph what you must add to the first number to get the second number: 13. +1, +5 16. \u2122 3, + 4 19. + 5, – 1 14. – 2, + 4 17. – 2, — 6 20. – 6, +3 15. 0, + 6 18. 0,-3 21. – 2, – 2 THE ALGEBRAIC NUMBER IN SURVEYING When a surveyor wishes to make a contour map showing elevations, he first selects a starting point or bench mark, which he marks 0 elevation. Whenever it is convenient, sea level is taken as 0, but any other starting level will do. Then he measures the elevations at equal horizontal dis- tances, say 10 ft., calling the points \u201cStation 1,\u201d \u201cStation 2,\u201d etc. If a point is 5 ft. higher than his starting point, he labels it -f 5; if 3 ft. lower, ~ 3. He records these readings in his note-book as follows: Station Elevation 1 -f- 28 2 + 22 3 + 7 4 – 8 Station Elevation 5 – 10 6 . – 12 7 – 18 8 0 Station Elevation 9 + 6 10 + 7 11 + 15 12 + 16 102 ALGEBRAIC ADDITION Back at the office, this graph is made from his readings. In the graph, 0 is the level of the river. 1. Which bank of the 20 10 0 -10 -20 river is higher? 2. At what station is the deepest channel of the river? 3. Between which sta- tions is the river bed nearly horizontal? Which bank is steeper above water? If the stations are 10 ft. apart, how wide is the river? 4 5 6 7 8 9 10 11 12 4. 5. 6. Would it be safe for children who cannot swim to bathe in this stream? Exercises Make a contour map for these elevations: Station Elevation Station Elevation Station Elevation 1 0 5.. . . . . . – 4 9 . + 29 2 – 5 6 .. . . 0 10 . + 20 3 – 1 7.. . . . . . + 10 11 . + 31 4 + 3 8 …. . . . + 18 12 . + 35 Station Elevation Station Elevation 2 . 1 .+ 100 6 …. . . . + 360 2 .+ 170 7…. . . . + 390 3 . + 230 8 …. . ..+ 90 4 . + 380 9…. . ..- 10 5 . + 490 10. . . . . . . – 100 Historical Note on Negative Numbers: Although the laws of negative numbers were known from early times, these numbers were not really understood until about 1500. Diophantus, about 275 A.D., worked with them, but called the answers absurd. Men like Vieta, Fermat, Harriot, Stifel and Hudde helped clear up the idea, but it was the geometric representation of Descartes that finally gave them a definite meaning. REVIEW OF ARITHMETIC 103 \u00a9 + ^) + Z)(r – s) 48. a~ix^ + Xg) + a^{x^ + x^) + a^ix^ + x^) Note: For the addition of polynomials, see page 359. Algebraic Numbers in Business 1. Principal Hill wished to learn how many books he must order, so he took an inventory. He used plus signs to indicate the number of books a class had more than it needed and minus to tell how many books a class was short. Here is his statement: Class Algebra English Civics Biology [ A + 7 – 28 – 18 + 12 B – 12 + 5 – 15 – 32 C – 10 – 22 – 7 + 9 D + 8 + 14 + 16 + 10 How many of each kind of book must he order? 108 ALGEBRAIC ADDITION 2. Mr. Jones measured a room of his new house to see how many square feet of plaster he would need. He found that all walls were 9|- ft. high. Here is his ^ record of the dimensions in feet. Can you find the gi answer for him without multiplying out each num- 5 x her separately? Subtract 60 sq. ft. for doors and win- 1 X 9| dows. 3 X 9| 3. Mr. Banker had the following accounts outstand- ing. Find the total interest due him. {a) $387 X .05i 246 X .05i 492 X .051 175 X .051 {h) $256 X .04f 762 X .04f 244 X .04f 338 X .04f (c) $581 X .031 432 X .031 174 X .031 413 X .031 4. Chester Green, a broker, found these net changes each day last week in his stocks: Stock Mon. Tue. Wed. Thu. Fri. Sat. Week American Gas. . + 1 + i – 2i \u201e 1 4 \u201d i + f Standard Coal. . – 2 3 4 + u 3 4 + 1 + i United Can …. 3 4 + h + 1 2 + If – 1 Find the net change in each stock for the Oct. 1. . .71 T at $ 6 | week. Oct. 8 . . . 83 X at 64 5. The Black Coal Company bought qS 16 58 Tat 6 ^ this coal during the month of October. Oct 21. ‘ ^93 T at 6 ^ Find the total amount due for the month. Oct. 25. . .70 T at 6 | SIMPLE EQUATIONS Problem : Smith and Brown buy suits at $16. At what price must they sell them to make 20% of the selling price? Let s = the selling price The profit is i of the selling price, so ^ s = the profit SIMPLE EQUATIONS 109 If we take the profit from the selling price, the remainder will be the cost, so s – I s = 16 This is an algebraic equation. If we could learn how to solve such equations, that is, to find the answer to them, we could do many problems that are very difficult or even impossible by arithmetic. Perhaps you can do the problem given here by arithmetic or even get the answer in your head, but there are many problems that you could not do in either of those ways. The algebraic equation is of use in solv- ing such problems. The equation is a bal- ance. Think of the equa- tion as a balance for weighing. Suppose there are s lbs. of sugar in the left pan and 12 lbs. of weights in the right pan, and that they balance. If I wish to keep them in balance and: 1. If I add 2 lbs. of sugar to the left pan, what must I do to the right pan? 2. If I take 5 lbs. of sugar from the left pan, what must I do to the right pan? 3. If I put 3 times as much sugar in the left pan, what must I do to the right pan? 4. If I leave only i as much sugar in the left pan, what must I do to the right pan? 5. If I replace the two 5-lb. weights by a 10-lb. weight, will it remain balanced? How to solve an equation. You can see that the cor- rect answer to the equation s | s = 16 is s = 20, for 20 \u2014 i X 20 is 16. Notice that in the answer, s = 20, s is all alone on the left side of the equals sign and 20 is on 110 ALGEBRAIC ADDITION the right side. To solve an equation then, you must get the letter on the left side and have nothing else there. You must get rid of any quantity that is on the wrong side. To do this, you need certain rules, or axioms. Here is one of them: Axiom 1. If the same number is added to both members of an equation, the equation will still balance, or: If equals are added to equals, the results are equal. ( = s + =s are =.) Illustration 1. Solve the equation: x \u2014 3 = S. Remember we must get the x alone on the left. That means that we must get rid of the – 3. To do this, we add 3 to both sides. The given equation. Any quantity equals itself. If equals are added to equals, the results are equal. Check: Substitute 11, the answer, for x in the original equa- tion: 11 \u2014 3 1 8 A quantity may be substituted for an equal 8 = 8 one. Illustration 2. Solve: 4 ;r = 7 + 3 x We want the x on the left side of the equals only, so we must first get rid of the 3 % on the right side. We can do this by adding \u2014 3 % to both sides. i X = 7 + 3x \u2014 3 X = \u20143x X = 7 The given equation. Any quantity equals itself. If equals are added to equals, etc. Check: Substitute 7 for x in the original equation. 4×7 7-f3x7A quantity may be sub- 28 7-1-21 stituted for an equal one 28 = 28 (substitution). THE EQUATION IN PUZZLES 111 Class Exercises Solve these equations by adding the same quantity to both sides, and give the reason jor each step. Check your answers. 1. X = I 2. X – 7 = % 3. a: \u2014 2 = 5 4. ;c – 3 = – 3 6 . % \u2014 6 = \u2014 10 6. % -}- 4 == 9 7. % + 7 = 15 8. 3;c = 2% + 4 9. 4 X = 3 ;c + 11 10. 7 X = 6 X – 5 11. 13 X = 7 + 12 X 12. 5 X = 7 + 4 X 13. 9 X = 8 + 8 X 14. 2 X – 3 = X 15. 8 X – 9 = 7 X Optional Exercises 16. 5x-4=4x + 7 17. 3x + 5 = 2x + 8 18. 4x-3=3x + 7 19. 8x-7 = 7x + 5 20 . 2x \u2014 l=x + l 21. 9x + 7 = 8x-l 22. 7 X + 13 = 6 X + 5 23. lOx + 11 = 9x + 11 24. 5x\u2014 3=4x \u2014 5 25. 8x+2=7x-8 Honor Work 26. 4x+3\u2014 2x\u2014 7=x 27. 5x-9-4x + ll = 0 28. 2x+44-3x\u2014 6=4x 29. 3.4 X – 6.3 = 2.4 x 30. 1.7 X + 70 = .7 X Zl. X \u2014 a = b 32. 3x \u2014 c = 2x + & 33. 5x + 3\u00ab=4x \u2014 2a 34. Solve for C: C + 7. = A 35. Solve for g: c + g = s The Equation in Puzzles Many of the puzzles that you find in newspapers or that your friends spring on you may be solved by algebra. Puzzle. Charles said to George: \u201cThink of a num- ber. Now add 8. Now subtract 3. What answer do you get?\u201d George answered, \u201c11.\u201d Charles said, \u201cYou started with 6.\u201d Do you know how Charles got the answer? Well, he wrote: Let X = the number X ~-S – 3 = 11 x + 5= 11 – 5 = – 5 X =6 The equation made from the problem. Collecting like terms. Any quantity equals itself. If equals are added to equals, etc ^12 ALGEBRAIC ADDITION What is the number? 1. I am thinking of a number. If I subtract 5 from it and then add 2 to the result, my answer will be 9. 2. If I take 7 away from 6 times a number, I shall have 5 times the number left. 3. If 4 times a number is diminished by 14, the result is 3 times the number. 4. If I multiply a certain number by 4 and then add 8, I shall have 5 times the number. 5. A quart of milk without the bottle sells for 11^. Katherine returned 3 empty bottles and paid in addition to get 2 qts. of milk in 2 bottles. What amount did she get for each bottle that she returned? 6. If I double a certain number and add 3, the result is 11 more than the number. 7. If 3 times a number is increased by 10, the result is 6 more than twice the number. Thought Question Paul solved a problem that asked, \u201c In how many years will Ann be twice as old as Mary?\u201d His answer was \u2014 2. What does it mean? Review Exereises 1. Explain the meaning of these: the year \u2014 44, the latitude + 41\u00b0, a profit of \u2014 $400, the elevation of the Dead Sea above sea level is \u2014 1290 ft. 2. Count from \u2014 32 to + 43 by 3\u2019s. 3. Find the sum of 4 hw, \u2014 2 hw, \u2014 7 hw, 2 hw, and 6 hw. 4. Add: (\u00ab) +8 (h) \u2014 4 a’^bd (c) 38452 (d) 5 X 8.37 – 6 a^bc^ 29614 9 X 8.37 + 1 2 a^bc^ 7859 3 X 8.37 – 5 \u2014 4 a^bc^ 76428 7 X 8.37 + 3 3 a^bc^ 306 6 X 8.37 6. Express y yds. as feet. 6. If an angle grows larger, in what way does its supplement change? REVIEW 113 p = 340, and t = 6. 8. Find x: 3 x \u2014 4 = 19. 9. How many cents must Ed pay for s 1)^ stamps and t 3j^ stamps? 10. Using X and y for the numbers, express: {a) The sum of two numbers {h) The difference of two numbers (c) The square of the first number {d) Twice the product of the numbers (e) Three more than the second number (\/) The sum of the squares of two numbers 11. Add: {a) (+5) + (-8) (&) (_3) + (-7) (c) (-2) + (+5) {d) (+2) + (-5) + (+4) (^) (-7) + (+2) + (+4) (\/) (+4) + (-3) + (-l) 12. By substituting, find whether 3 or 5 is the correct root of the equation 3x+4=7x \u2014 16. 13. Solve and check: 7x \u2014 13 = ll+6x. 14. Write the equation and solve: If 5 times a certain num- ber is decreased by 13, the result is 4 times the number. 15. Anthony has 7 marbles less than Tom, and together they have 45 marbles. How many has Tom? 16. In 3 X -H y = 20, how does y change when x grows larger? 17. Using the formula C = f (F \u2014 32), find C when F = 50. 18. When two lines cross, what is the name of the pair of opposite angles? 19. Has the normal frequency curve an axis of symmetry? If so, in what direction does it run? 20. What is the average of + 15, \u2014 10, and -f 7? Of + 4, -7, and +3? 21. Explain why subscripts are sometimes employed in for- mulas. 22. Make a formula for the distance an automobile will go in t hrs. at 40 mi. an hr. 23. What does \u2014 2 in the net change of a stock-market report mean? 114 ALGEBRAIC ADDITION Make graphs suitable for these exercises: 24. Paul walks 3 mi. an hr. Make a graph showing the distance d he will go in t hrs. from ^ = 0 to \/ = 8. 25. The average temperature for June in these cities is: illbany, 68\u00b0; Atlanta, 76\u00b0; Bismarck, 64\u00b0; Dallas, 81\u00b0; Miami, 80\u00b0; and San Francisco, 58\u00b0. 26. The population of Michigan increased as follows: U810 1830 1850 1870 1890 1910 1930 cn O 8 32,000 398,000 1 , 184,000 2 , 094,000 2 , 810,000 4 , 842,000 27. 28. 29. Draw a small angle, and construct one 5 times as large. In this figure, read with 3 letters: {a) An acute angle (c) An obtuse angle {b) A right angle {d) A straight angle Is any angle supplementary to AEBl 30 . Is any angle vertical to AEBl 31. An angle has 78\u00b0. How many de- grees are there in its supplement? In its vertical angle? 32. Is a fish symmetrical? Is your image in a mirror symmetrical to you? 33 . Write a formula for the perimeter of: {a) A square whose side is 3 \u00ab + 4 {h) A rectangle whose width is w and whose length is 2 -f- 7 (c) A triangle whose sides are a, b, and c 34 . Using the formula T = 2 A he, find T when e = S and h = 5. 35 . Make a frequency polygon to illustrate these results: In a test 1 pupil received 40%; 3, 50%; 5, 60%; 8, 70%; 6, 80%; 4, 90%; and 2, 100%. 36. Find the average, median, and mode: In a class con- tribution 7 pupils gave 5^, 9 gave 10(2f, 12 gave 15^, and 2 gave 25^. 37. A watch ticked 80 times from the time a flash of lightning was seen until the thunder was heard. How many miles away was the lightning? TESTS 115 38. At what price must a clothier mark a suit so as to give a discount of 40% and sell it for $18.60? Formula: m = , ^ . 1 \u2014 a 39. Sarah sat 5 ft. from the fulcrum of a teeter board. Rob- ert, who weighs 90 ibs., found that he balanced her when he sat 6 ft. from the support. What was Sarah\u2019s weight? diWt Formula: Wi = d. 40. An elasticity formula from science is M = \u2014 \u2022 Find M if ea = 144,\/ = 25, e = .001, and a =3. Test on Addition 1. 5 6. 3 X 11. 3x2y3 16. 2Ak – 3 4 x^ys 3.1 k 2. – 6 7. – 4x 12. -3k^ 17. 5.8 3 \u2014 3 X – – 3.5 3. – 4 8. 3 ab 13. 3 18. – 5 \u2014 6 ab \u2014 mi\/Wa \u2014 m 4. – 6 9. – M 14. – 9pv^ 19. 2irs 9 -3M 10 pv^ – 3 r3 5. 3 10. -2h 16. – 10 hw 20. -3iB – 3 7h 10 hw 2k B Test on Equations with Axioms Solve these equations, and give a reason for each step. Check your answers. 1. a: – 5 = 4 6. X – 3.4 = 4.6 11. 3 – 4 = 2 % + 7 2. % -7 =2 7. :r + 1.3=8.3 12. 5 :r – 1 = 4 x + 9 3. – 9 = – 3 8. % – 5.6 = – 2.6 13. 4 x – 3 = S 3 x 4. ;c + l = -4 9.x + 2.9 = 2.9 14. 9 x + 6 = 10 + 8 x 5. X \u2014 5 = \u2014 5 10. X \u2014 5.3 = \u2014 8.3 15. 5 x + 8 = 4 x \u2014 2 Chapter 6 SUBTRACTION Review of arithmetic. Subtract: 1 . 38452 21965 4 . 54390 45077 7. 5142 3746 10 . 4.9852 4.2769 2 . 91002 73874 5 . 6452 5849 8 . 9780 9674 11 . 5.0000 2.8635 3 . 12764 875 6. 8763 1784 9 . 31.768 24.817 12 . 87 9.635 13. Beginning at 100, count down by 3\u2019s, by 4\u2019s, by 6\u2019s, by 7\u2019s, by 8\u2019s, by 9\u2019s. Making change. When you give a dollar bill in paying a debt of 32^, the merchant makes change by addition. He hands you 3 pennies, a nickel, a dime, and a half dollar, saying as he does so, \u201c32^, 35izi, 40^, $1.\u201d Make change from a dollar bill for each of these amounts: 14. 48^^ 16. 18. 60^ 20. 42)zi 15. 89^ 17. 12i 19. 18^ 21. 37?i ALGEBRAIC SUBTRACTION George Washington was born in 1732 and died in 1799. How many years did he live? What did you do to the numbers 1799 and 1732 to get your answer? Can you always find the number of years a person lived by sub- tracting the date of his birth from that of his death? Augustus Caesar was born in \u2014 63 and died in + 14. How many years did he live? Evidently he lived 63 yrs. before Christ and 14 yrs. afterwards, or 77 yrs. Can we get the answer 77 by subtracting \u2014 63 from + 14? 116 ALGEBRAIC SUBTRACTION 117 Subtraction as the opposite of addition. Example 1. Find the value of + 8 \u2014 ( + 3). Just as the clerk makes change by addition, so we can subtract 3 from 8 by asking ourselves what number we must add to 3 to make 8. If the numbers are laid off on a scale going in both directions from 0, we can count the number of spaces from 3 to 8. The answer is 5. We counted to the right. Is the answer positive or negative? Let us see. Add + 2 to + 5. Do you count to the left or to the right when you add a positive number to 5? Add \u2014 3 to + 5. Do you count to the left or to the right when you add a negative number to 5? ~9 -8-7-6~~5-4-3~2-l 012345678 I I ! I I I i I I I I I I I I I I I To add a positive number, count to the right; to add a nega- tive number, count to the left. Example 2. Find the value of \u2014 7 \u2014 ( + 4). This is the same as asking what number we must add to 4 to make \u2014 7. So starting at + 4, we find that \u2014 7 is 11 spaces to the left. The answer then is \u2014 11. Exercises What number must be added to the first number to give the second? 1. 1, 6 4. – 3, 5 7. 2, – 5 2. 2, 8 5. – 7, 1 8. 5, 2 3. 4, 0 6. – 5, 8 9. 4, – 7 10. -5,-3 11 . – 2,-8 12 . 0,-6 Investigating subtraction. 1. Subtract: 7 Think: What number must I add to 3 to 3 make 7? 2. Subtract: 6 Think: What number must I add to – 4 \u2014 4 to make 6? 118 SUBTRACTION 3. Subtract: \u2014 2 – 5 Think: What number must I add to \u2014 5 to make \u2014 2? 4 . Subtract: – 7 + 3 Think: What number must I add to + 3 to make \u2014 7? If you did these four exercises correctly, your answers ivere: 7 6 – 2 – 7 -h3 -4 – 5 + 3 4 10 3 – 10 Look at these addition exercises: 7 6 – 2 – 7 -3 +4 + 5 – 3 4 10 3 – 10 Compare the answers to these addition exercises with the answers to the subtraction exercises above them. Are they the same? Now compare the upper numbers (minu- end). Are they the same? Finally compare the numbers subtracted (subtrahend). Are they the same? How then can we change subtraction exercises into exercises in addition? Rule for subtraction. Change the sign of the number to be subtracted, the lower number, and proceed as in addi- tion. Of course, you will make this change mentally only, and not write it on your paper. Class Exercises Subtract. Then check by addition. 1. +6 + 5 – 7 – 4 -8 8 + 2 – 3 – 1 – 9 + 2 6 2 . 5 6 – 4 – 2 0 4 9 – 3 7 – 1 – 1 -3 3 . 5 – 7 – 8 0 – 6 – 7 – 1 2 – 3 5 0 5 3 . ALGEBRAIC SUBTRACTION 119 4. – 4 – 7 – 5 – 2 – 1 -0 – 6 – 2 ^ + 6 3 – 2 6. – 2 + 5 8 – 4 – 2 + 4 – 2 + 5 – 3 + 9 – 6 – 4 6. 3 0 – 5 ~ 6 + 0 -4 0 4 – 1 – 2 – 0 5 7. \u2014 3 \u00ab -2b 2c2 Sd 3c3 – 5\u00ab + 4 & 6c2 4^ – 3c3 8. IFG 4 w^x \u2014 3 0 -2FG 7 w^x \u2014 4 2 – 2^2 9. xyz 0 5 y^ 4 2 7rr2\/f xyz CO 1 2y3 Tir^h 10. +{Zx) – {+2x) – {-Sx) 11. +(-5\u00ab) -(-7a) + (-3\u00ab) 12. – (+2r) – (-3r) – (- r) 13. – (- m^) + (- 2 m^) – (+ 3 m^) 14. + (\u00a3”) – (2 \u00a33) – ( – 5E”) 16. \u2014 (4 WyW^) + (\u2014 2 \u201c (~ 16. 3.8 – 2.2 X Optional Exercises 7.34 y 5.4 c ab 3.29 j 2.77 c .2 ab .9 2 z 17. 3iM 2iM 2\/x \u2014 5(a + b) Vx \u2014 2{a + b) SVx \u2014 y \u2014 ^\/x \u2014 y 18. What must I add to 5 a; to make \u20143 a:? To make 0? To make 4 &? 19. How much greater is \u2014 5 \u00ab than \u2014 9 o? Than 8 m? a cy Honor Work ax bx^ M,{a + b) b_ dy X 2x^ M^ia + b) 2 amn 3 a:\u201c aVx + y abx aXx 2 bmn \u2014 2 x\u201c Vx + y acx bXx 22. If the value of x begins at \u2014 5 and grows larger, does it move toward 0 or away from 0? 120 SUBTRACTION 23. In subtraction, how does the answer change: {a) If the upper number increases, but the lower number remains un- changed? {b) If the lower number increases, but the upper number remains unchanged? (c) If both numbers grow larger by the same amount? Applied Problems 24. At 6 A.M. the temperature was \u2014 8\u00b0 and at noon it was + 18\u00b0. How many degrees did it rise? 25. A ship traveled north from latitude \u2014 42\u00b0 to latitude \u2014 14\u00b0. How many degrees north did it travel? 26. In a game Fred has \u2014 13 points and Robert has + 8. How many points must Fred win to overtake Robert? 27. Last August Katherine had \u2014 $3 and now she has + $7. How many dollars did she save? 28. How many years did each of these people live? Born Died Lincoln + 1809 + 1865 Charlemagne + 742 + 814 Alexander the Great \u2014 356 \u2014 323 Constantine + 272 + 337 Jesus Christ \u2014 4 + 29 Julius Caesar \u2014 100 \u2014 44 Ovid \u2014 43 + 18 Livy – 59 + 17 29. How many years elapsed from the founding of the Roman Empire in \u2014 753 to its end in -f 476? 30. How long did the Alexandrine age of literature last if it began about \u2014 323 and ended about 100? 31. Euclid lived about \u2014 300 and Descartes in + 1600. How many years passed from the time of Euclid to that of Descartes? 32. The boiling point of liquid hydrogen is \u2014 253\u00b0 C.; of liquid oxygen, \u2014 182\u00b0 C.; and of water, -|- 100\u00b0 C. Which is warmer and how much warmer when boiling: (a) hydrogen or oxygen? (b) Hydrogen or water? (c) Oxygen or water? 33. Water freezes at 0\u00b0 C. Is boiling oxygen warmer than ice? Note: For the subtraction of polynomials, see page 360. EQUATIONS 121 EQUATIONS We learned that we could change the form of an equa- tion by adding the same amount to both members. Some- times it was necessary to add a negative amount. Instead of this, we could have subtracted a positive amount and used the axiom: Axiom 2. If the same quantity is subtracted from both members of an equation, the equation will still balance, or: If equals are subtracted from equals, the results are equal. ( = s \u2014 =s are =0 Illustration. Solve forx: 3r-|-7 = 2x + 12 Here we must get rid of the 7 from the left member and the 2 a: from the right member. 2> X 1 = 2 X -j- 12 2x + 7 = 2x + 7 X = 5 Check: 3×54-7 I 2X5-1- 12 15 4- 7 I 10 4- 12 22 = 22 The original equation. Any quantity equals itself. If equals are subtracted from equals, the results are equal. Substitution. Exercises Solve these equations giving a reason for each step. Check. a: 4-3 = 7 6. 3i?4-5 = 2i?4-8 y 4- 10 = 15 7. 5\/z4-7 = 4\/?-3 z; 4- 8 = 4 8. 2s \u2014 5 =s4-5 ^4-5= -3 9. 4:W–S=3w–S mi 4- 21. = 41 10. SE + 1.6 = 7 E 3.6 Problems 11. One number is 3 x, and another is 12 \u2014 2 x. Their sum is 16, Find X and both numbers. , 12. One number is 2 w; a second number is 3 w; and a third is 10 \u2014 4 n. Their sum is 12. Find n and all three numbers. 122 SUBTRACTION 13. AB is 47 in. long and is cut into 3 parts, 3 42 \u2014 7 o, 3 a U 2 – 7 a 5 a _ ^ a. Find the value of a and ^ the length of each segment of AB. 14. Paul and John started from K, and both traveled east. Paul went 5 m mi., and John went 4 m mi. They were then 9 mi. apart. > ^ > How far did each travel? Thought Questions If X is growing larger, how is 15 \u2014 x changing? How large must X grow so that 15 \u2014 x will become 6? PARENTHESES Where have you already used the parenthesis? Of what use was it in multiplication? Have you used it in addition or subtraction? What is the value of 9 \u2014 (5 \u2014 2)? If this parenthesis were simply left out and we wrote it 9 \u2014 5 \u2014 2, would we get the same answer? Parentheses are used to indicate that a certain operation is to be performed on the enclosed expression as a whole. To remove the parenthesis, simply perform this operation. They are most often used to show that the enclosed ex- pression is to be multiplied by some number or is to be subtracted from some number. Other signs, the bracket [ ], the brace { }, and the vinculum are sometimes used in the same way as the parenthesis. The parenthesis in addition and subtraction. Illustration 1. 2a \u2014 hA{c \u2014 “id) means that c \u2014 3 is to be added to2a \u2014 h. Adding: 2a \u2014 b c – 3^ 2a \u2014 b c \u2014 2d Since in addition we do not change the sign of any term, the parenthesis is of no value here and may be dropped without changing the expression. Check by substituting a number for each letter in the original expression and in your answer. PARENTHESES 123 Illustration 2. 2 \u2014 {3 x \u2014 4) means that 3 x \u2014 4 k; to be subtracted from 2 \u2014 x”^. Subtracting: 2 x^ – x”^ 3x -4 2x^-x\u2018^-3x + 4 Since in subtraction we change the signs of all terms sub- tracted, the parenthesis may be omitted provided we change the signs of all terms that were inside it. Illustration 3. 2 x \u2014 3 \u2014 x \u2014 4) + {S \u2014 x). We change the signs of all terms that were in the parenthesis preceded by the minus sign, but do not change those in the parenthesis preceded by the plus sign. After the parentheses are removed, when there are like terms, we collect them into a single term. 2x – 3 – X – 4) + {3 – x) = 2x -3 + X + 4 + 5 – X = 2 AC + 6 Class Exercises Remove parentheses, and collect like terms: 1. 3a -b -4- {a -h) 2. X + 4 – {2 – X) 3. 5 & – 4 – (2 & + 3) 4. 3 TT -f (3 -f tt) 5. 3 T – (r – 4) 6. \u2014 2m \u2014 {\u20143m \u2014 4) 1. x^ – {2×4- 4) 8. m + (2 m \u2014 3) k- {-l-k) 10. 7y -1-3 2 -b (- 2y ^ 5 2) 11 . A \u2014 S \u2014 ( \u2014 A -B) 12. 5 – c – (c – 2) Optional Exercises 13. {2a -3c) 4- {2c -3a) 14. (\u00ab! 4- \u00ab2) – (2 fli -f \u00ab2) 15. – (3a; -4y) (5x – y) 16. – (- 2 – 2) – (-4 – 52 ) 17. (2r + S)-|-(-2r-S) 18. _(2\/ + 3g)-b(\/-3g) 19. 2 a: – (3 a:^ – 2 a: -f 4) 20. \u2014 4 -b (m^ \u2014 m 4- 4) 21. 3 – (a:^ -b – 1) – a: 124 SUBTRACTION 22. – + X – 3) – 2 -i- I 23. 3 W + (v -3 W) + 4 – V 24. 5m \u2014 (3 \u2014 2 m) \u2014 (4 m \u2014 2) Honor Work 25. a-^2b–[-5a-{-b-i3a-2b) – 2 b] + a 26. 3 – 4 – [2 a: – (;t:2 – x) + (3 % – 5) + 4 a:”] – 2 27. 3j^ – {2y+6) – [- (5 3^+3) + (- 4 + 23 ;)] 28. [- {3 – (2 +4%)} + 1] – [{-4 + (2x – 7)} – 6] 29. – {A – B,) – {[- (2 A – B) – (- 3 A – 5 B)] – B] 30. – 4 + [2 :)[: + 5 – 3 – 7 ;c + 4] – (a: + 1) + 3j Inserting parentheses. If we enclose terms in a paren- thesis and then remove the parenthesis again, we ought to get back the same expression that we had at first. Conse- quently when we enclose terms in a parenthesis, we change the signs of all the terms enclosed if the expression is preceded by a minus sign, but we do not change the signs if it is preceded by a plus sign. Exercises Inclose the last two terms in a parenthesis preceded by (a) a plus sign; (b) a minus sign: 1 . a 4r b c 6. 3 – 4 vy + 2 2. M – N – R 7. cd A- df \u2014 gh 3. 5x ~ y A-1 z 3. 5 + 3 a + 2 4. ar \u2014 bs \u2014 ct 9. \u2014 b”^ A- 6. mn + np \u2014 pq 10. Ax A- By \u2014 Cz EQUATIONS CONTAINING PARENTHESES Illustration. Solve and check: 5x – {4 X -3) =13 5x – {4x -3) – 18 Check: 5x \u2014 4:r-l-8 =18 50 – (40 – 8) 18 % + 8 =18 50 – 32 18 8 = 8 18 = = 18 X EQUATIONS CONTAINING PARENTHESES 125 Exercises Solve and check: 1. 3 % – (2 X – 1) 2. 9 y \u2014 (6 + 8 j) = 4 = 2 3. 4 yfe + (8 – 3 = 10 4. 5 2 – (2 + 4 z) = 8 5. 2 w + (3 – m) = 4 6. 7 r – (6 r – 5) = 9 7. (3 – %) + 2 X = 1 8. 5 3^ – (4 – 6) = 0 9. 4: a (3 \u2014 3 a) = \u2014 7 19. 9 – (2 X – 1) = – 20. (7 – 5 x) – 3 = 2 10. 4 X = 7 – (2 – 3 x) 11. (3 – 2 x) + (3 X – 1) = 0 12. 5 – (2 3^ – 3) = – (4 +33^) 13. 4 = 2 X – (x – 2) 14. 33^ + 1 + (5 – 23^) = – 8 15. (3 X – 4) + (2 – 2 X) = 4 16. 5 X – (4 – 3 x) = 5 + 7 X 17. 23′ + (3j + 9) -43^ =0 18. 3 = (5 X – 2) – (3 + 4 X) 3 – (4 + 3 x) + (1 – 6 X) Algebraic Expression 1. Which represents 4 more than ^:4^or^ + 4? 2. If 3; is a number, what does 3 y mean? What does y + 5 mean? 3. Write a number 4 larger than x; 3 times as large as x. 4. Two numbers differ by 5. The smaller is 7. What is the larger? 5. Two numbers differ by 5. The smaller is a. What is the larger? 6. Two numbers differ by d. The smaller is x. What is the larger? 7. By how much does 10 exceed nl y exceed 7? y exceed nl 8. What number exceeds 7 by x? Exceeds phy qi 9. Using n to represent a number, write: (a) A number increased by 4 (c) Twice a number (b) A number diminished by 7 (d) 6 less than a number 10. The difference of two numbers is y. The larger is 8. Find the smaller. 11. The sum of two numbers is 23. The smaller is 8. Find the larger. 12. The sum of two numbers is 12. The smaller is x. Find the larger. 13. The sum of two numbers is s. The smaller is x. Find the larger. 126 SUBTRACTION 14. John had m marbles. He lost 9. How many has he left? 16. The smaller of two numbers is x. The larger is 5 times the smaller. {a) Represent the larger. (&) Represent their sum, their difference, their product. (c) Write a number 8 smaller than the larger number, y smaller than the larger number, z larger than the smaller number. 16. Gladys had cp. She earns 20^ more. How much has she now? 17. If ^ is divided into 7 equal parts, what is one part? 18. If 20 is divided into n equal parts, what is one part? 19. If k is divided into n equal parts, what is one part? 20. Separate n into 2 parts so that the smaller is y. 21. X is one part of 15. X is increasing. How is the other part changing? 22. A number exceeds r by 7. If r is increasing, how is the other number changing? Algebraic Shorthand Translate the following sentences into algebra, and solve: 1. Six more than a number is 11. 2. When a number is increased by 8, the sum is 15. 3. In 5 yrs., Robert\u2019s age will be 16. 4. If Dorothy had 24jzf more, she would have $1. 5. If I add 8 to a number, the result is twice the number. 6. Twice a number increased by 12 equals the number in- creased by 17. 7. Five times a number is 4 more than 4 times the number. 8. If I subtract 7 from 3 times a number, the result will be 6 more than twice the number. 9. Three times a number exceeds 12 as much as twice the number exceeds 7. ALGEBRAIC NUMBERS IN MUSIC (Optional) Have you ever studied music? Do you remember what a nuisance it was to memorize all the keys? Can you tell ALGEBRAIC NUMBERS IN MUSIC 127 now what key has 3 sharps or what key has 4 flats? Would you like to learn an easy way by which you could figure it A#- G# F#’ Bb -Ab Gb Illustration. How many sharps has A? Since A is one tone higher than G, add 2 sharps. Then we have 1+2=3. A has 3 sharps. D#– jy –Eb Exercises C# 1. If G has 1 sharp, how many has B? C^? (# is the symbol for sharp, and b is the symbol for flat). 2. If B has 5 sharps, how many has A? G? F? Eb? Db? What do we mean by a negative sharp? out without memorizing? Here is the rule. Every time you raise the key a whole tone, add two sharps. For example, D is 1 tone higher than C. So the key of D should have 2 sharps more than the key of C has. Simi- larly the key of E has 2 sharps more than the key of D. What then should you do every time you lower the key a whole tone? When the number of sharps becomes greater than 7, decrease it by 7. This lowers the key | tone. C and G are good start- ing points, as C has no sharps and G has 1. JOHANN SEBASTIAN BACH ( 1685 – 1750 ) Famous German composer whose works influenced profoundly the course of modern music. 128 SUBTRACTION 3. If C has 0 sharps, how many has D? E? Or going down from the upper C, how many has Bb? Ab? Gb? On the ladder, notice that the intervals from E to F and from B to C are only half tones. Consequently, for example, one whole tone down from C gives us Bb, not just B. 4- How many whole tones is it from D to F;^? If D has 2 sharps, how many has F#? 5. How many whole tones is it from upper C down to Gb? How many flats (negative sharps) has Gb? Name each of these keys counting from C when the number of sharps or flats is even, and from G when that number is odd,. From C: 6 . 7 . 8 . 9 . 10 . From G: 11. 12. 13. 14. 16. 16. If F has 1 flat, what has A? 17. Using G and C as starting points, write in order the key having 1 sharp, 2 sharps, 3 sharps, 4 sharps, 5 sharps, 6 sharps, 1 flat, 2 flats, 3 flats, 4 flats, 5 flats, and 6 flats. When we lower the key ^ tone, we remove a sharp from each of the 7 notes in the \u201coctave\u201d; that is, we subtract 7 sharps. For example, if A has 3 sharps, Ab will have 3 \u2014 7 or \u2014 4 sharps or 4 flats. 18. How many flats has; (\u00ab) Db if D has 2 sharps? (c) F if F# has 6 sharps? ip) Eb if E has 4 sharps? {d) Cb if C has 0 sharps? 19. How many sharps has: {a) A if Ab has 4 flats? (6) B if Bb has 2 flats? REVIEW 129 Review Exercises 1. Make a circle graph to show these facts: Dry wood con- tains 50% carbon, 6% hydrogen, 41% oxygen, 3% nitrogen and ash. 2. From the sum of 8 and \u2014 11, subtract \u2014 5. 3. Last year Mr. Strong\u2019s balance was \u2014 $800. This year it is -f $1300. How much did he save? 4. Mt. Everest is -j- 29,141 ft. high. The deepest point in the Pacific Ocean is \u2014 35,400 ft. How far above this point is the top of Mt. Everest? 6. What number diminished by 5 equals \u2014 3? 6. How much greater is \u2014 2 than \u2014 9? 3 than \u2014 4? 7. Find the other number if the sum of two numbers is: ia) 15 and one of them is 7 (6) 12 and one of them is \u2014 4 (c) 0 and one of them is 5 {d) 8 and one of them is 13 (e) \u2014 8 and one of them is \u2014 12 (\/) 0 and one of them is \u2014 6 (g) \u2014 2 and one of them is 1 (h) \u2014 10 and one of them is \u2014 6 (i) 3 and one of them is 0 8. Solve for x and check: Zx –l –2x 9. Show by substituting whether % = 2 or a; = 3 is the cor- rect answer to x\u00ae \u2014 6 -f 10 a: = 3. 10. If an airplane travels m m. an hr., how many hours will it require to travel 500 mi.? 11. Sarah had 90jzf. She bought p pencils at each. How much had she left? 12. An angle has x \u2014 30 degrees. How many degrees are there in {a) Its supplement? {h) Its vertical angle? 13. An obtuse angle has d\u00b0. By how much does it exceed a right angle? 14. How many degrees are there in r right angles? In 5 straight angles? 15. The sum of the angles of a polygon (a figure made of straight lines enclosing a part of the plane) is found by using the formula S = {n \u2014 2)180\u00b0 where n is the number of sides. 130 SUBTRACTION Find the number of degrees in the sum of the angles of a polygon having: (a) 3 sides (c) 6 sides (e) 102 sides (b) 4 sides (d) 12 sides (\/) 2 sides (g) When the number of sides increases, does the sum of the angles increase or decrease? (h) When the number of sides is doubled, is the sum of the angles doubled? 16. Make a graph of the formula S = (n – 2)180 for values of n from 2 to 10. 17. Robert is y yrs. old. How old was he k yrs. ago? How old will he be in t yrs.? 18. Mrs. Heap bought 2 doz. buttons at a doz. and 5 yds. of cloth at 4 a yd. What was her total bill? 19. Translate into algebra: {a) Twice a number increased by 10 is 5 less than 6 times the number. ib) Three times a number is as much above 14 as the number is below 14. (c) If 4 times a number is decreased by 9, the result is still 1 more than the number. {d) If I multiply a number by 3 and then increase the product by 5, the result is 9 less than 5 times the number. 20. In a football game Georgia Tech gained 8 yds., \u2014 3 yds., \u2014 1 yd., and 7 yds. Did they make \u201cfirst down\u201d (10 yds.)? 21. Construct an angle of 22^\u00b0. 22. Subtract: {a) – 5 (6) 7 (c) – 13 (d) 3 (e) – 4v – 3 – 8^ – 11 y 23. Remove the parentheses, and collect terms: (<z) 3 X \u2014 (4 T \u2022^) T (3 X \u2014 6) {b) 2 \u00ab – 5 + (3 -\u00ab)-(- 4 \u00ab + 7) 24. A problem asked \u201cHow much heavier is Sarah than Katherine? \u201d Dorothy\u2019s answer was \u2014 10 lbs. What does it mean? 25. An airship with its passengers weighs \u2014 1500 lbs. After 10 passengers, averaging 150 lbs. each get out, what does it weigh? TESTS 131 Test in Removing Parentheses Remove the parentheses, and collect terms: 1 . 5 j + (x – 3) 7 . 4 c – (c – 3) + 3 c + 4 2. 7 m \u2014 (2 w? + 8) 3. 2 a – (fi – 4) 4. 3 & + (7 – &) 5. S h – (3 ~ 2 h) 6. 3 r – (~ 2r + 4) 8. h + (3 – h) – (2 h – 5) 9. 5 – (A + 1) – (1 – 10. – (x + 4) + 3 X – (2 X – 5) 11. p – q – ip – q) +3 12. 12 – (\u2122 7 + x) + (3x + 2) Test in Solving Equations Solve, giving a reason for each step, and check: 1 . X + 4 – 9 11. 7 m + 12 = 20 + 6 w 2. X + 7 – 13 3. X ~ 3 – 5 4 . X – 7 = 4 6. X + 5 = 2 6. X – 8 = – 4 7. X – 2 – – 5 8 . X + 6 – 6 9 . 4 X + 10 – 3 X 10. 6 X – 7 = 5 X 12. 3ife-5-2;fe + l 13. 3 ^ + 15 = 10 + 2 ^ 14. 5 – = 8 \u2122 2 15. 8\u2014 4w^8\u201c5\u00ab 16. 3 i ~ 2.7 -1.3 + 2 i 17. 5w + 8.72 – 4 m; + 9.22 18. 3.8 ^ – 7 = 4 + 2.8 t 19. 5.97 – .2 j – 8.97 + .8 y 20. .32 X – 1.2 – 8.8 – .68 x Chapter 7 MULTIPLICATION AND DIVISION MULTIPLICATION Review of arithmetic. 1 . Add the column of figures on the right. 2. Can you discover a short cut for this addition? Check your answer using this short cut. 3. How would you multiply a number by 10? By 100? By 1000? 4 . Find the product: {a) 3849 X 10; (b) 926 X 100; (c) 386 X 1000. Find these products: 6 . 382 X 8 7 . 5714 X 387 9 . 41567 X 3863 6 . 471 X 9 8 . 1497 X 429 10 . 86254 X 7421 11 . When you multiply a fraction by a whole number, do you multiply the numerator only, the denominator only, or both numerator and denominator? 12. If you multiply both numerator and denominator by the same number, what effect has it on the value of the fraction? Find these products: 13 . 7 X t"5 15 . 38 X f 17 . 2 X yV 19 – 83.4 X .58 14 . 4 X f 16 . 14 X A 18 . 6 X i 20 . .749 X .023 Find the cost of each of the following: 21 . 42 yds. at a yd. 23 . 31 gals, at 76^ a gal. 22 . 162 lbs. at 89^ a lb. 24 . 43 hrs. at 12(^ an hr. 25. By what number have you multiplied a number if you move the decimal point two places to the right, or if you add two O\u2019s? 26. If you move the decimal point two places to the right 132 48736 48736 48736 48736 48736 48736 48736 48736 48736 48736 MULTIPLICATION 133 and then divide the result by 4, by what have you multiplied the number? 27. Can you discover a short cqt for multiplying by 25? By 50? 28. Find the value of: 584 X 50; 3612 X 25; 8248 X 12^. 29. If you move the decimal point two places to the right and then divide by 3, by what have you multiplied the number? 30. Can you discover a short method of multiplying by 33i? By 66f ? By 16f ? 31. Find 38169 X 33^; 4578 X 66f ; 192114 X 16f. Find the per cent equal to: 32. 50% of 40% 34. 10% of 60% 36. 15% of 40% 33. 25% of 30% 35. 20% of 35% 37. 30% of 70% Aliquot parts of 100%. Numbers that are contained in 100% a whole number of times are called aliquot parts of 100%. Some of them and their multiples are very im- portant in business. Those used most often are: Fifths Sixths Eighths 20% = i 16f % = i 12i% = i 40% = f 33i% = i 25% 60% = f 50% =i m% = I 80% = f 66f % = f 62i% = f m% = f 75% =f m% = i Illustration. Find 16f % of 384. Instead of a long multipli- cation, divide 384 by 6. The answer is 64, Class Exercises Find the value of: 1. 33i% of 600 4. 66f % of 900 7. 16f%of600 2. 37i% of 800 6. 83i% of 1200 8. 87i% of 2400 3. 75% of 400 6. 621% of 1600 9. 50% of 1100 134 MULTIPLICATION AND DIVISION 10. I2i% of 600 11. 60% of 500 16. m% of 384 17. 66f % of 744 18. 62i% of 552 19. 37i% of 856 28. 40% of 389 29. 871% of 573 30. 121-% of 341 12. 20% of 700 13. 80% of 450 Optional Exercises 20. 40% of 380 21. 87i% of 944 22. 60% of 585 23. 83i% of 582 Honor Work 31. 50% of 81.92 32. 25% of 7.163 33. 621% of 84.3 14. 40% of 600 15. 25% of 200 24. 50% of 762 25. 33i% of 651 26. 80% of 475 27. 75% of 844 34. 331% of 61.7 35. 831% of .0052 36. 871% of 8.005 EXPONENTS IN MULTIPLICATION Thought problem. You have already learned that means a \u2022 a, that means a \u25a0 a \u2022 a and that means <i \u2022 a \u2022 a – a. See if you can discover for yourself what the answer would be if you multiplied by After you have found your answer, read the solution below. How many a\u2019s are multiplied together to make To make If we write them along on a line, how many c\u2019s shall we have multiplied together? How can we write the answer? What did you do to the exponents 3 and 4 to get your answer? flfS . ^4 {a – a ' d){a ' a – a \u2022 d) = \u00ab)(- 5\/.^) 28. (4M^)(5M) 33. {x^x^)^ ^ 1 ^ 2 ) 29. l5x^y’^)(2xy) 34. (-3kn){0) 35. 3a^^5ab-^) 36. \u2014 2mn(– 3m^n) 37. ~ Sw^w^i\u2014 38. 3h(-5h){0) 39. -4x^(-3x^) EXPONENTS IN ASTRONOMY 137 Optional Exercises 40 . (-2x)(-2x)i-2x) 46. – 5 \u2018( 5 \u2019) 60. (- 1) = 41. (-5a^b)(+2ab^)i-2a) 46. 51. (- I)’\u2019 42 . (7x){2 y)(z) 47. (- 3″)=’ 62. (- 1)* 43 . (-2 x^yz)(- 9 x\u2019\u2018yz\u2019\u2018){0) 48. (- 3*)=’ 63. (- 1)*”* 44. (- 4 x’\u2019yz'<)(2 xz^)(~ 4 y) 49. (. 3 )(- 2 ) 64. (.3 Honor Work 66 . UP \u2022 66. a\u00bb \u2022 c\u00ae 67. 68 . ~ ' a 69. ~ \u00ab) 60. -2"(-2) 61. – 2- \u2022 2\u00ab 67. What is the value of (\u2014 1)\u00ae when % is an odd number? When x is an even number? 68. If % is decreasing, will also decrease when X is {a) positive? (&) Negative? (c)When ^changes from positive to negative? 62. +3'-(~30 63. (-10)\u00ae 64. (- .3fl)(+ .02\u00ab")(- .Ifl) 65. w^w^) 66 . Exponents in Astronomy The distance to one of the nearest stars is about 26,000,000,000,- 000 mi., and to another star about 5,200^000,- 000,000,000 mi. How many times as far away is the second star as the first? Astronomers find it THE NEBULA OF ANDROMEDA Nebulas are other universes far beyond the stars of our universe. They are so far away that the stars in them appear as simply a cloud. Andromeda is about 900,000 light years away or about 5 X 10^\u00ae miles from us. Write the number in full. much easier to handle such extremely 138 MULTIPLICATION AND DIVISION large numbers when they are expressed as powers of 10. The power of 10 used equals the number of O\u2019s for: Power of Ten Exponent Number of Zeros 101 = 10 1 1 102 = 100 2 2 10\u00ae = 1000 3 3 101 = 10000 4 4 The first distance then is 26 X lO^^, the second 52 X 10^^ In this form it is easy to see that the second number is larger, and to determine how many times as large. Find the answer. Scientists often write very large or very small numbers with the decimal point after the first figure. In this case we have: 260 = 2 hundred and 60 = 2.6 X 100 = 2.6 X 10^ 2600 = 2 thousand and 600 = 2.6 X 1000 = 2.6 X 10\u00ae 26,000,000,000,000 = 2.6 X 10,000,000,000,000 = 2.6 X 101\u00ae 5,200,000,000,000,000 = 5.2 X 1,000,000,000,000,000 = 5.2 X 101\u00ae 1 . A light year, the distance light travels in a year, is about 6 X 1012 Express these distances in miles, using powers of 10: (a) 100 light years (c) 1,000,000 light years (b) 80,000 light years (d) 20,000,000 light years 2 . If there are about 10,000 light waves to 1 in., approx- imately how many light waves are on their way here from a star 1,000,000 light years from us? Polynomials What is a term? What signs tell you where one term ends and the next begins? How many terms are in the ex- pression 5 X? In 7x^y^z? In + 1? In 5 \u00ab \u2014 2 ft + 3? An expression having just one term is called a monomial. An expression having more than one term is called a polynomial. When the polynomial has just two terms, it is POLYNOMIALS 139 a binomial; when it has three terms, a trinomial. You will see that polynomials have some resemblance to num- bers containing more than one figure. Here we have three 3 + 4 + 6 = 13 whose sum is 13. How many of them ? 4 – ? 4 – ? = 26 must you multiply by 2 in order to make the sum twice as great, or 26? Here we ? X ? X ? = 60 three \/actors whose product is 30. How many of them must you multiply by 2 to make the product twice as great, or 60? If you multiply every factor by 2 and write 4 X 6 X 10, by what number have you multiplied the product? An expression is multiplied by a number when all of its terms are multiplied by that number or when one of its factors is multiplied by the number. How to multiply a polynomial by a monomial. Do you multiply every figure of the 342 by the 2? Why do A\/r If 1 q 4 9 begin at the right instead of at the Multiply: 3 4 3 get the same answer. Now multiply 342 by 7. Can you begin at the left in this example? Why not? When we multiply in algebra, we have no number to carry over from one term to the next, so we can, and by custom we do, begin at the left. Otherwise the multiplica- 3 4 2 3c + 4& + 2c tion is done just as in arith- 2 2 metic. 6 8 4 6a + 8& + 4c Beginning at the left, we mul- tiply 3 a by 2, then 4 ^ by 2, and finally 2 c by 2. In the same way: ~ 3 a'^b{2 – A a\u00a5 a\u00a5) ^ – Q a% A- 12 – lb Check by substituting any convenient numbers for the letters. Illustration. Multiply 3 \u2014 2 x \u2014 5 by \u20143x and check by substituting 2 for x. Note: x = 1 will check signs and coefficients, but will not check exponents, for = 1, = \/ 140 MULTIPLICATION AND DIVISION = 1. So a wrong exponent would give the same value as the right one. 3x2_2x -5 = 3-4-2-2-5= +3 = – 3 -2 = – 6 – 9 6 -{- 15 X =-9-8+6-4 + 15-2 – 18^. – 72 + 24 + 30 = – 18io 3 c 6 Optional Exercises 19. (-3 x^y^) (\u2014 xy) 22. (+ 18 m^n^) -i- (+ 6 m\u2018^n^) 20. (- 12\u00ab^&^) -4- i+’iab^) 23. (- Ux^^y’^^) ^ 21. (+ 14c\u00aefi?^) 4- (-2c^(\/^) 24. (+ 4- (+4i?^r\u00ab) 25. – %x,^x^^x.^ ^ {-2x.^x\u201e^x.^) 26. + 12\u00ab^6^c H- (- 6fl^c) 27. -lirr^h i-rh) 28. – 9 x^y^z^ 4 – 5 29. +2c’^x^y^^ 8 c^x^y^ 30. – 3^iM2M3″” -4 (- 3 Honor Work 31. n^+ i 34 ^ 37 2 x2 40. \u2014 3 \u00aby3\u00ab X\u201d a^~ ^ \u2014 \u00aby\u201c 32. yn + 5 36. X66 38. V 41. + 6 627 2n x^ \u2014 2 2\u00ab ALGEBRAIC DIVISION 151 How to divide a polynomial by a monomial. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Illustration. Divide: 6x^ \u2014 3 + 9 X\u00ae \u25a0 3^2 6x^ ^3 2 X, -3 = – 1, and 9 The answer is 2 a: \u2014 1 + 3 3 = 3 a;’ Exercises Divide: 3;r + 6 _ 3x^ \u2014 3x \u2014 ^ 5. g 2 7rr2 -f 2 wrh 3 X Trr 3a -13 Q 3x^ +3x in – a -yb – c 3 3x _ 1 -8^+4 \u201e 9x^ – 15 . . 2wh 2mh 4 ‘ -3 2h 6y \u2014 12 g \u2014 3 a^b 10 ab^ \u2014 14x3y2 _ 21 x2y4 – 3 \u25a0 3ab -7x^y^ \u2014 3 an + bn \u2014 3 cn 14 . – jx i X . _ 15 a^bx \u2014 10 ab^y – 3 ab Thought Questions How is a polynomial changed when you divide it by \u2014 1? Is the result the same as if you multiplied it by \u20141? The balance once more. What must you do to the weight in the right pan if you divide the amount of sugar in the left pan by 2? If you divide it by 3? By 5? Axiom 4. To keep the balance, both members of an equation must be divided by the same number, or: ‘ If equals are divided by equals, the results are equal. ( = s -i- =s are =.) Note: For the division of polynomials, see page 364. 152 MULTIPLICATION AND DIVISION By Ewing Galloway. THE BALANCE DIVISION IN EQUATIONS Illustration. Solve 5 x -1 = 2x + b 2x -1 =2x -1 3 X 3 X 12 ]2 3 4 7 = 2 a: 4- 5. The original equation. Any quantity equals itself. If equals are subtracted, etc. If equals are divided by equals, the results are equal. X DIVISION IN EQUATIONS Check: 5*4 \u2014 7 1 2*4 + 5 Substitution. 8 + 5 153 20-7 13 13 Class Exercises Solve, giving axioms, and check: 1. 2x = 10 6. 3 m A- m = = 20 11. 5 + + 8 = = 13 2. 5x = 35 7. 5 r -2 r : = 18 12. 9 m; = 12 A-5w 3. 3 X = 11 8. 2 k -7 k : = 15 13. Ay + 12 = 8 4. Ax = -8 9. 3 X A- Ax = = 10 14. A-7t = 11 5. -7x = – \u25a021 10. 4 n -7n = 6 15. s – 15 = 6 s 16. 5% + 7 = 3 X + 15 21. 2 – -Ax = 12 A- X 17. 4 + 3 h = 9 – 2h 22. 7 – – r = 12 – 6r 18. 32 + 10 = = 2 + 10 23. – 3+M = 4M-9 19. AR – \u25a0 7 = 13 – R 24. 14 -32 = 13+2 20. 3vA- 2 = 18 + 5v 25. 2t – 7 = = 4-^-1! Optional Exercises 26. 3 % – 2.1 = X + 5 34. .3 w – .82 = .01 w + .05 27. 4 + + .44 = + + .8 35. 4.3 – 2.5 % = 1.07 % – 4.625 28. 5 r – 7 – r + 4.84 36. 20% of x = 7 29. 3 yfe – 2.6 =8.5 – 2 k 37. 25% of = 18 30. 3-7:y = 2j; + 8.13 38. 12i% of % = 85 31. – 7 = .4 X + 5 39. 16f % of x = 8.3 32. 4 F + .18 = F + 6.6 40. ;c – 50% x = 15 33. N -4.2N = .8N –3 41. x + 33i% x = 28 Honor Work 47. 3x \u2014 a=x->rb 48. ax = bx 20 ^9. mx \u2014 X = k 50. cx \u2014 A = b + dx 51. ax + bx + dx = 0 Problems 52. Le Havre is in longitude \u2014 2\u00b0 and New York in + 73\u00b0. If the Normandie travels about 1\u00b0 every 2 hrs., in how many hours will it travel from Le Havre to New York? 42. Ax = b A3, mx + k = n 44. by \u2014 3 = A a 45. cx \u2014 a = ab 46. ax A- c = c 154 MULTIPLICATION AND DIVISION 63 . A balloon ascending into the stratosphere reported a temperature of \u2014 63\u00b0 at 12 mi. above the earth. If the tempera- ture at the ground was + 57\u00b0 that day, what was the average fall in temperature per mile rise? 64 . A boy on the edge of a cliff throws a stone upward with a velocity of 64 ft. a sec. Using the formula c? = 64 \/ \u2014 16 P, what distance up will it be at the end of (a) 3 sec.? (b) 5 sec.? (c) How do you interpret the answer to b? (d) Does d increase or decrease as t increases from 4 to 5? {e) Can you discover for what value of t d stops increasing? Consecutive Number Puzzles 1. Consecutive numbers are whole numbers (integers) taken in order, such as 13, 14, 15, 16. What do I add to 13 to get the next consecutive number? To get the second one? The third? 2. If w is the first of three consecutive numbers, what is the second? The third? 3. Numbers like 32, 34, 36 are called consecutive even num- bers. What must I add to 32 to get the second number? The third? 4. If w is an even number, what is the next larger even num- ber? The next larger than that one? What is the even number next smaller than w? 6. Numbers like 17, 19, 21 are called consecutive odd num- bers. What must I add to 17 to get the second odd number? The third? 6. If w is an odd number, what is the next consecutive odd number? The odd number next larger than that one? The consecutive odd number next smaller than nl 7 . If w is an even number, what are the next two larger odd numbers? 8. If w is an odd number, what are the next two larger even numbers? Illustration. Find three consecutive odd numbers whose sum is 81. Let n = the first Then w -f 2 = the second w + 4 = the third W + W + 2+ W + 4 =81 3 w + 6 = 81 w -}- 2 = 27 w -h 4 = 29 w = 25 SIMPLE EQUATIONS 155 Check: 25, 27, and 29 are consecutive odd numbers. 25 + 27 + 29 = 81 9. Find three consecutive numbers whose sum is 42. 10. Find four consecutive even numbers whose sum is 60. 11. Find three consecutive odd numbers whose sum is 45. 12. Find two consecutive even numbers such that 3 times the first added to the second equals 74. 13. Find three consecutive odd numbers such that twice the smallest plus the second and the third equals 58. 14 . Find three consecutive numbers such that the sum of the second and third decreased by the first is 11. 15 . The sum of the first two of three consecutive numbers is 23 more than the last number. Find the numbers. 16 . Paul said \u201cFind four consecutive whole numbers whose sum is 100.\u201d John said the problem was impossible. Show which was right. 17 . Can the sum of 3 consecutive numbers equal 100? SIMPLE EQUATIONS A short cut in solving equations \u2014 transposing. Illustration 1. Solve: ;t: \u2014 3 = 8 To get rid of the \u2014 3, we add + 3 to both members. Suppose we only indicate the addition. – 3 = 8 + 3 = +3 =8+3 ;c = 11 In the original equation we had \u2014 3 on the left side of the equals sign. In the new equation we have + 3 on the right side. The result is the same as if we moved the \u2014 3 past the equals sign and changed its sign from \u2014 to +. Illustration 2. Solve: 6 x = 5 a; + 4 Adding \u2014 5 x : \u2014 5 x = \u2014 5 x 6x =M :r = 4 156 MULTIPLICATION AND DIVISION In the original equation we had + 5 a: on the right side. In the new equation we have \u2014 5 x on the left side. The result is the same as if we moved the + 5 x past the equals sign and changed its sign from + to \u2014 . Rule. A term may be moved from one side of the equals sign to the other side provided its sign is changed. We say that the term is transposed from one side of the equation to the other. In solving an equation, all terms containing the unknown letter should be collected on one side of the equation, usually the left, and all other terms should be collected on the other side. Transpose all those terms that are not on the proper side. Terms that are not transposed, however, do not have their signs changed. Illustration. Solve: ^ x \u2014 1 \u2014 3x=2x-f4 \u2014 x Transposing: ^x \u2014 ?>x \u2014 2x–x = + 4 + 1 X = 5 Check: 25 – 1 – 15 | 10 + 4-5 9 = 9 The \u2014 1 is transposed to the right side and becomes + 1 ; the + 2 a: is transposed to the left side and becomes \u2014 2 X’, the \u2014 X is transposed to the left side and becomes + X. Exercises Solve the following equations, and check your answers: 1. 3x – 5 = X – 1 11. 4 X \u2014 X = :7+X 2. 4x + 14 = = 4 – X 12. 4 – 2y = + 5 3. 6y – 4 = 3 y + 8 13. 3 m \u2014 2 = = 7 m \u2014 6 m 4. 2z + 8 = 26-32 14. 3 – 2(x 4 – 4) = – 7 5. 5t – 14 = = 2 + \/ 15. 5(x – 2) : = 3x – 4 6. 7x + 6 = 2 X \u2014 4 16. \u20183{k + 1) \u25a0 -5 = k 7. 2 – – 5r = 2r + 16 17. 4(m \u2014 2) \u2014 7 = 2m 8. 3x + 4 = 9 +4x 18. 32 + 4(1 \u2014 z) =22 9. 8 s + 3 = 13 – 2s 19. 4 – 3(x – -l)+7 = 0 10. 9 – – 3k = k + 1 20. 2(p -3) = -3(p+ 4) REVIEW 157 Review Exereises Solve and check: 1 . 5x-3=2jt: + 12 2. 4% + 5= a: + 11 3. 73^ – 11 = 23 ; – 6 4. 2\u00ab-4 = ll-3fl 5. 8 \u2014 2w = 6m \u2014 12 6 . 4i7 + 9= – 9- 5\/\/ 7. \/?~13 = 5\/?-8 8 . + 3 = 5 \u2014 9. 4 m + 13 = 12 – m 10. w + 5= 4w + 5 11. 5%-3=% + 4 Multiply: 19. 3(5 X -8) 23. -y{x -y) 27. ( – 20. 5 x(l – n) 24. – 5(- 2)(4) 28. (2 m- r)^ 21 . 3 ajjC\u2014 2 a:) 25. \/>^r(\/?r) 29. 3 x{l.8 x^) 22. \u2014 4 m^(+ 5 m) 2%.Ax’^{2x^) 30. (- 1)^ Divide: 31. – _ 6 63 _ 9^4 4 – 12 62 Ww\u00ab 1 \u00ab a 3 62 32. ^ 3 ^ 14r2 -7r5 +7 h 7 33 . 1-” go 12 ^345 _ 9 ;c 23;4 _ 15 ;t 6 j ;2 3 ;cj\u20192 3r 34. 5 m2 \u2014 5 mw + 10 w 4c^ – 5 35. ^ (M 1 1 \u2014 m \u2014 a; Subtract: 41. \u2014 4 m3 42. – 5 r 43. k 44. \u2014 3 45. 0 + 3 m3 \u2014 5r 4^ \u2014 1 x’^y^ \u2014Za 12. X – = IZ – Ak 13. 2.3 A + .16 = 1.68 – 1.7 h 14. 2i + 3 = li + 18 P P 15. ^ + l=f + 7 16. ^ – I = e – 34 17. 2 r – ^ 14 ^ = 1 O 10 ^ ^ \u00ae ^ ^ ^ o 18. 4 3 ^ 158 MULTIPLICATION AND DIVISION Combine like terms: 46. – 3:t+7+4A:-2-3x+4+3;c-5 47. 5 m \u2014 (\u2014 3 w) + (\u2014 3 w) \u2014 (+ 5 m) 48. -2 xy^ + 5 xy^ – 4 Solve: 49. 5(x – 2) – 3(2 + a:) + 7 = 2 ;c + 4{x – 3) – 13 60. 3x(l -2 x) – 4(x – 3) = 6 x(5 – x) – 19 51. If = 2 a; \u2014 1, make a table of values of 3 ^ for a: = 0, 1, 2, 3, 4, and 5. 52. Two sides of a triangle are each x inches long and the third side is 3 more than their sum. What is the perimeter? 53. In = 2 a \u2014 5(w \u2014 1), find the value of K when \u00ab = 8 and w = 3. 54. A team won a games and lost h games. What fraction of its games did it win? 55. If m and n are two numbers, what is their average? 12 56. If 3 ^ = \u2014 , how does y change when x grows larger? 57. Remove the parentheses and collect terms: 3 m \u2014 2 a \u2014 3(a \u2014 3 m) + 2{m + a) ss 58. Find the value of r in r = \u2014 when g = 8 and s = 12. g +s ^ Historical Note on the Shorthand of Algebra: It is sur- prising how long it took the world to develop a good symbolism for algebra. Although geometry was well known to the Greeks, they never succeeded in working out even a satisfactory number sys- tem. All early writers except Diophantus used the full word or an abbreviation for it, so that their equations were more like rules than modem equations. Diophantus (275 a.d.) used many symbols but his work had little influence on later mathematics, for all through medieval times, no satisfactory symbols were used for even the operations of arithmetic. Plus (-f) and minus (\u2014 ) seem to have been first used in Widman\u2019s arithmetic (1489) in Germany. The multiplication sign (X) appeared in England about 1600. Our equality sign ( = ) was first used by an Englishman, Recorde in 1557. Before his time the whole word or an abbreviation of it had been used. Vieta (1590), a French mathematician, used letters to represent both known and un- known numbers. His is the earliest book to resemble a modem text in algebra. TESTS 159 69. A man earns e dollars a month and spends s dollars a month. How much does he save {a) in 1 mo.? (6) In 2 yrs.? 60 . Factor: {a) – 5y {d) 3b^ – 3P (b) am 2 an {e) ax \u2014 bx \u2014 cx (c) 6x^y -4 x^y^ (\/) 10 – 15 + 5 p 61 . Find the value of: (a) 79.34 X 834 – 79.34 X 833 (b) 27 X 7i + 5 X 7i 62 . What per cent of 852 is 355? 63. Make a graph olv = 12\u20142t from \/ = 0 to \/ = 6. 64 . A room is 15 ft. long, 12 ft. wide, and 9 ft. high. How many square feet of plaster are required? Formula: N = 2lh + 2 wh + n)l 66. The formula for finding the number of centimeters equal to a certain number of inches is c = 2.54 i. {a) Make a table of values of c for values of i from 0 to 10. (6) Make a graph from your table. (c) Find the number of centimeters corresponding to 7i in. (d) Find the number of inches corresponding to 8 cm. To 23 cm. (e) Does the number of centimeters increase or decrease as the number of inches increases? If you double the number of centimeters, is the number of inches doubled? Test in Multiplication Find the product of: 1 . \u2014 3 \u00ab(+ 4) 11 . – (xy)(- xz) 2 . (5 b){-2b) 12 . (+ 3 m)( \u2014 5 m^y) 3 . \u2014 3 m{ \u2014 2 m) 13 . (5 ab^c^){3 aHc^) ^ 4 . (+ 2 \u00ab)(+ 6 \u00ab) 14 . 2 x( \u2014 5 xy){+ 5 yz) 6 . 5b^{2 b^) 15 . 2x^(3x^){4 x^) 6 . -6r^(-4r^) 16 . – {-2p){~5p^) 7 . \u2014 2v{\u2014w) 17 . (_1)2(_2)3(+1)6 8 . a^{-a) 18 . (2fl”)(- 2fl)(2fl”) 9 . 4x^{4x^) 19 . 4863 (5976) (0) 10 . -3a^b(-2b^) 20 . CO 1 160 MULTIPLICATION AND DIVISION Test in Division Find the quotient: 1 . 7. 9 .g 9 \u2014 6 ab X 9 m3 \u2014 3a 2 . lb 8 . – 5x . 5 – 10 S3 7 5x -5 3. \u2014 2m 9. – 3x3j; 15 2 – 3 x2j\/ b 4. – 6 c 2 10 . 20×7>; 4 x 33 ; – 2 x 3\/2 3c^ \u2014 4 x^’ X 3 \/ 5. 4 11 . 3x – 3 3 m^n \u2014 w 4 a 3 \u2014 n 6 . – 4×4 12 . 8×2 _2x jg \u2014 5 x3 4- 3 \u00abx – 2 x 2 2x \u2014 X Test in Faetoring Factor: 1 . 6 m + 6 r 6 . ^hm + ^hn – ^hp 2 . 5h – lb k 7. P + Prt 3. \u2014 2m^ 8 . 72.96 X 813 – 72.96 X 613 4. – 8x^ +4x^ 9. 547 X 415 – 536 X 415 5. Trr^ + irrh 10. 34 X 55 X 2 – 34 X 45 X 2 Test in Solving Equations Solve: 1. ;c + 5 = 8 2. 4 = 20 3. 5 + 7 = 22 4. 3 & + 8 = 5 5. 7 – 2 c = 3 6. 2 a: = 5 \u2014 12 7. ? = 9 o 8 X 21 9.S_1=5 10. 4 x + 5 = 10 – a: 11. 3.4 a; = 10.2 12. 5 a: = 3.9 a: + 33 13. .3 a: – 7 = 2 14. 4(2 a: – 7) = x + 1 15. 8x – (- 3 + 5 a;) =4 ^ 4 – ^ 5 2 14 17 . 3^-5 = 24 18. 2a: + 1 = 2 5 Chapter 8 PROBLEMS O NE of the greatest values of algebra to you is the ability it gives you to solve problems. In life outside school, unless you are in certain lines of work, you will find very few equations ready-made for you to solve, but you will meet many problems that you must solve in some way, and usually the algebraic method is the easiest way. Then too, problems are a better test of your ability to think than any other part of algebra. In this chapter you will practise solving easy problems. Algebraic Shorthand Write the equations for these statements on a separate sheet as they are being dictated by the teacher. Then solve your equations: 1 . When 3 is added to 4 times a number, the result is 31. 2. If 5 times a number is diminished by 13, the result is the same as if twice the number were increased by 20. 3. Seven more than 3 times a number is 34. 4 . If a number is divided by 3 and then 4 is added, the result is 1 more than ^ the number. 6. If a number is divided by 4 and then 8 is subtracted, the result is 1. 6. Twice a number decreased by 9 equals 41. 7 . Six times a number exceeds twice the number by 24. 8. I am thinking of a number. If I multiply it by 4 and then add 3, the result will be 23. 9. Twice a number exceeds 10 as much as 17 exceeds the number. 10 . If 3 is subtracted from ^ of a number, the result is 4. 161 162 PROBLEMS 11. Twice a number increased by 52 is 20 more than 6 times the number, 12. If 3 is added to 7 times a certain number, the result is 18 more than 4 times the number. 13. Five times a number equals 18 diminished by the number. 14. Find a number whose double divided by 3 is 10 16. Four less than 5 times a number is 26. Problems Having More Than One Unknown Problem. One number is 7 more than twice another number, and their sum is 22. Find the numbers. Notice that there are two unknown numbers and two parts to the statement. Let some letter stand for one of the numbers, usually the smaller number. Use one of the parts of the statement for representing the other number. Then use the other part for making the equation. Solution. Let n Then 2 w + 7 \u00ab + 2 w + 7 3 w \u201ch 7 3 n n 2 w T 7 = the smaller number = the larger number (The first part) = 22 (The second part) = 22 = 15 Check: 17 is 7 more than = 5 twice 5. The sum of 5 and = 17 17 is 22. 1. Find two numbers if one is 5 times the other and their sum is 42, 2. One number is 3 times another, and their difference is 11. Find the numbers. 3. The greater number exceeds the smaller by 9, and their sum is 21. Find the numbers. 4. Find the dimensions of a lot if the length is 20 ft. more than the width and the fence around it is 280 ft. long. 5. One number exceeds another number by 6, and their sum is 40. 6. One number is 3 more than another, and their sum is 9. VERTICAL ANGLES 163 7 . One number is 3 times another, and their sum is 20. 8. Separate 33 into two parts so that the larger is twice the smaller. 9. Separate 15 into two parts so that the larger is 1 greater than the smaller. 10. One number is twice another. If the larger is diminished by 10, the result is 2 greater than the smaller. 11. The difference of two numbers is 8, and twice the smaller is 6 more than the larger. 12 . If one of two equal numbers is increased by 7 and the other is diminished by 2, their sum becomes 25. 13 . The length of a city lot is 75 ft. more than the width, and its perimeter is 250 ft. Find its dimensions. 14 . A book and a pencil cost $1.10, and the book cost $1 more than the pencil cost. What did the pencil cost? 15. A contractor ordered 460 lbs. of concrete containing 4 times as much sand as cement. How many pounds of cement will be needed? 16 . Find two numbers whose sum is 91, if the greater is 6 times the smaller. 17 . – If 3 is added to i of a number, the result is f of the num- ber. What is the number? 18 . The larger of two numbers exceeds twice the smaller by 6. The larger is also 1 more than 3 times the smaller. Find the numbers. VERTICAL ANGLES 1. An angle and its supplement together contain 180\u00b0. How many degrees has the supplement if the angle has 50\u00b0? 90\u00bb? 160\u00b0? ;\u00ab:\u00b0? 2. If Zx is growing larger, how is its supplement changing? 3. When Z x has become as large as its supplement, how many degrees has each? 4. There are x\u00b0 in LABC. How many degrees are there in its supplement Z CBDl 5. Since CBE is also a straight line, Z ABE is a supple- 164 PROBLEMS ment of A ABC. If Z ABC has x\u00b0, how many degrees are there in A ABE? 6. How does the number of degrees in Z ABE compare with that in Z CBD? U x grew larger, would these angles still remain equal? 7. But Zs CBD and ABE are vertical angles. Are vertical angles always equal? In these exercises we learned that Z s CBD and ABE are each 180 \u2014 x, so no matter what number x stands for, they are always equal. Vertical angles are equal. Historical Note: That vertical angles are equal was probably the first proposition in geometry ever proved. By proving it and a few other equally obvious propositions before anyone else thought to do so, Thales, about 560 B.C., became the founder of demonstra- tive geometry. The Equation in Geometry 1. Find X if one of two vertical angles has x\u00b0 and the other has 3 X \u2014 100 degrees. 2. Find x if one of two supplementary angles has 2 x + 10 degrees and the other has 5 x \u2014 40 degrees. Find x: 4 . Sx-sa PARALLEL LINES 165 Find x: 7. If \/? = a: + 10 and r = 2 x \u2014 20 8. If \/> = 5 X \u2014 80 and q = d>0 \u2014 x 9. i p = 2x and r = 200 \u2014 2 x 10. If g = I and s = X \u2014 60 11 . lip=^Rndq = ^ 12. If r = 2 ;c – 50 and q = 13. If s = ^ + 18 and ^ ^ + 40 4 X 14. If is a right angle and r == -g- \u2014 50 16. Zp is growing larger. What change is taking place in Zr? In Zq? In Zs? 16. As p grows from 0\u00b0 through 90\u00b0 to 180\u00b0, describe the change in q giving the number of degrees for the three sizes mentioned. PARALLEL LINES The lines AB and CD both lie on this sheet of paper and yet would never meet no matter how far they were ex- tended. They are crossed by a third line called a transversal. \u25a05 Do any of the angles look equal? Name the angles that appear equal to p. Which angles at the line CD are equal to r? To s? Is there an angle on CD equal to qi The pair of angles r and v and the pair s and w are called alternate interior angles. Can you discover why they are called interior angles? Why alternate angles? The pair of angles q and w, or p and v, or s and z, or r and y, are called corresponding angles. In what way do they correspond? 166 PROBLEMS Are corresponding angles on the same side or on opposite sides of the two lines? Are alternate interior angles on the same side or on opposite sides of these lines? Which kind are on the same side of the transversal and which on opposite sides? Two lines are parallel (|1) if they cross a transversal at the same angle, that is, if the corresponding angles are equal, li Ap equals Z v, then AB is parallel to CD. If a pair of corresponding angles are equal, do you think that a pair of alternate interior angles are necessarily equal? Suppose p = v, does r necessarily equal vl What is the name for the pair of angles p and r? Are they equal? i p = V and p \u2014 r, does v = r1 If lines are parallel, are alternate interior angles necessarily equal? If alternate interior angles are equal, are corresponding angles neces- sarily equal? Are the lines necessarily parallel? If two lines are parallel : 1. Alternate interior angles are equal. 2. Corresponding angles are equal. Exercises In these exercises AB is parallel to CD. Find the value of x, and check: 1. 2. 3. 4. If r = 70\u00b0, how many degrees are there in vl In s? In wl What is the sum of s and \u00ab;? Of r and wl (See page 165.) 5. If r = k\u00b0, how many degrees are there in vl In s? In w? What is the sum of s and vl Of r and wl Is the sum of a pair of interior angles on the same side of the transversal always the same, no matter what value k has? What is this sum? PARALLEL LINES 167 Find the value of x: 6. 7. Find the value of x: 9. If r = 4 – 30 and = ;c + 30 In these exercises show whether AB and CD are parallel: 15. r = 50\u00b0 and v == 50\u00b0 17. q = 110\u00b0 and v = 80″ 16. s = 140\u00b0 and z = 140\u00b0 18. q = 120\u00b0 and w = 60\u00b0 19. If line EF is turning around so that p is growing larger, what change is taking place in r? In ql In s? In v\u2019^ In w”? 20. If p becomes a right angle, what does w become? 21. If AB is not parallel to CD, {a) Does p \u2014 vl (b) Does p = r? (c) Does r ^ v7 168 PROBLEMS THE SUM OF THE ANGLES OF A TRIANGLE To find the distance to the moon A, an astronomer at B on the earth must find the number of degrees in ZA at the moon. Do you think he can determine this ^ from the two angles on the earth, Z B and Z C? Can you discover any relation between the angles ZL, B, and C? Do you think their sum is always the same? Would a small triangle have just as many degrees in the sum of its angles as a large triangle has? How great does that sum appear to be? 1. Draw a large triangle and a small triangle, and measure the angles with your protractor. What is the sum in the large triangle? In the small triangle? 2. Draw two triangles of very different shape, and measure their angles. Is the sum always the same? 3. Draw a large triangle, and cut it out. Now tear off the three angles, and fit them to- gether as shown here. Is the sum a straight angle? 4. In the triangle ABC the base BC is extended to Z>, and CE is drawn parallel to BA. What is the sum oit V A- If CE 1 1 BA, what kind of angles are v and r? w and s? How does the number of degrees in v compare with that in r? nw compare with that in s? How does the sum r + s + f compare with the sum t — v A- How many degrees then is the sum r -{- 5 + ^? THE SUM OF THE ANGLES OF A TRIANGLE 169 The sum of the angles of a triangle is 180\u00b0. Exercises Find the number of degrees in Z C: 1. It A ^ 38\u00b0 and B = 74\u00b0 2. It B = 48\u00b0 and A = C 3. If all three angles are equal 4. If Z, is a right angle, and B is twice as large as C 5. If is a right angle and B = C 6. ItC X, A = 2x, and j5 = + 20 7. If B is 10\u00b0 larger than C, and A \/ equals the sum of B and C \/ Nv 8. If 5 is 56\u00b0 larger than C, and ^ is \/ N. twice as large as C \u2014 9. If B is twice C, and is 3 times C 10. If B is twice C, and A equals the sum of B and C 11. If is a right angle and C is growing larger, how is B changing? 12. If triangle ABC grows larger, do the angles grow larger, grow smaller, or remain the same? Consider the cases {a) the triangle remains the same shape; {h) the triangle changes its shape. Thought Questions 1. If two angles of one triangle equal two angles of another triangle, what do you know about the third angles of these triangles? 2. (\u00ab) If Z^ = 80\u00b0 and Z5 = 70\u00b0, find the size of Z C. ib) If Z A’ – 80\u00b0 and Z5′ = 70\u00b0, find the size of Z C. 3. {a) If \/.A = x\u00b0 and AB = y\u00b0, find the size of Z C. (b) If Z A’ = and ZB’ = find the size of Z C’. (c) How do Z s C and C’ compare in size? Investigating the polygon. A figure made of straight lines enclosing a part of the plane is called a polygon. 1. How many sides has a triangle? How many straight angles are there in the sum of its angles? 170 PROBLEMS 2. How many straight angles are there in the sum of the angles of a four-sided polygon? Of a five-sided polygon? Of a six- sided polygon? 3. Complete this table, where s is the number of straight angles and n is the number of sides the polygon has: n 3 4 5 6 7’8 9 10 …. n 5 12 3 4. From this table make a formula beginning s = 6. Make a graph from the formula you made for Exercise 4. Algebraic Expression 1. Find 6% of $11, of $800, of P dollars. 2. Find 5% of $100 for 1 yr., for 3 yrs., for t yrs. 3. Find the proceeds (amount left) if a discount of 6% is deducted from $100, from $400, from P dollars. 4 . An agent receives a commission of 10% on sales. How much is the commission if the sales amounted to $800? To S dollars? When r is used to express the rate, it includes the per cent; that is, if the rate were 4%, r would be .04. 5. If the rate of interest is r, find the interest on $100 for 2 yrs., for t yrs. 6. Find the interest on P dollars at rate r for 7 yrs., for t yrs. Express these sentences in algebra shorthand: 7 . 60 is 4% of what number? 8. What per cent of 90 is 81? 9. What number is 8% of 320? Answer these questions: 10 . 15 exceeds 12 by what per cent of 12? 11. X exceeds y by what per cent of y? 12. What is the selling price if an article that cost $40 sells THE EQUATION IN BUSINESS 171 at a gain of 20% of the cost? That cost c dollars sells at a gain of 20% of the cost? 13. What is the selling price if an article that cost $24 sells at a loss of 25% of the cost? That cost c dollars sells at a loss of 25% of the cost? 14. What fraction of the cost does a dealer make: {a) If he buys chairs at $12 each and sells them at $16 each? {b) If he buys chairs at $12 each and sells them at s dollars each? (c) If he buys them at c dollars each and sells them at s dollars each? 15. What number increased by 33^% of itself equals 160? 16. What number decreased by 25% of itself equals 33? The Equation in Business How to solve percentage problems. Illustration. An author receives a royalty of 8% on the sale of his book. If the publishers sent him a check for $1624, what was the amount of the sales? Let X = amt. of the sales .08 X = no. of dollars in his royalty .08 X = 1624 Check: 8% of 20,300 = 1624 1624 ^ .08 = 20,300 1. Mr. Walsh sells goods at a profit of 20% of the cost. He uses the formula s = c + .20 c. He never uses the per cent sign (%) in his formulas. Find s if c = $1.40. 2. Mr. Houghton sells used cars at 40% below the cost. Make a formula for s. Find s when c = $680. 3. A merchant makes a profit of 15% of his sales. How much must he sell this month to make $375? 4. Nathan Goldberg, a clothier, wishes to put a suit on sale at 70% of the marked price. At what price must he mark it so that he can sell it for $35? 172 PROBLEMS 5. A grocer makes a profit of $420 on the sale of $4620 worth of groceries. What per cent of his sales does he gain? 6. A dealer bought shoes for $6.30 a pair. At what price must he sell them to make 25% of the selling price? 7. Mr. Campbell made a profit of $22.50 on the sale of a machine. He says that this was only 15% of the cost. What was the cost? 8. The Economy Jewelry Store wishes to sell a clock for $8 after giving the purchaser a discount of 33^% of the marked price for cash. At what price must they mark the clock? 9. Mr. Martin must borrow $343 from his bank for 4 mo. If the bank deducts a discount of 2% of the face of the note, for what amount must he write the note? 10 . The Harvey Marketing Company received a check for $3230 from their broker with the statement that he had deducted his commission of 5% of the amount of the sales. What was the amount of the sales? 11. For what amount must I write a promissory note if I need $2910, but must leave the bank 3% discount on the face of the note? 12. Mr. Hammond paid an income tax of $741, 3% of his income. What was his income? 13 . Mr. Haddon argues that the city should not object to spending $84,500 for the new school building as this is only 21% of the taxable value of the property in the city. If this is true, what must be the value of the taxable property? 14 . An importer bought goods in Europe, but had to pay an import duty of 30% ad valorem (of the cost). If the final cost of the goods was $2678, what did they cost in Europe? 15 . What change takes place in a merchant\u2019s profits if: (a) The cost of the goods increases, but the selling price remains unchanged (b) The cost increases, but the per cent of profit on the cost remains unchanged (c) The selling price increases, but the cost remains un- changed (d) The cost and selling price increase by the same amount PROBLEM ANALYSIS 173 PROBLEM ANALYSIS How to analyze a problem by the box method. Solving problems is a game. As in any game, you will succeed best if you know the rules and follow them. Rules of the game of solving problems. 1. First look at the question. Then put x in the box that represents the quantity, or one of the quantities, you are asked to find. 2. Fill the other boxes in the same column from data given in the problem. 3. If there are three columns, find data in the problem from which you can fill a second column. 4. Fill the remaining column of boxes (not always the right-hand column) from the quantities in the other boxes. Do not use information from the problem in filling these boxes. 5. Your equation will be made in terms of the subject of the last boxes filled. Read the problem again carefully to see what it tells about the subject of these last boxes. Make your equation using only this information and the expressions in the last column of boxes filled. Do not mix columns in your equation. Illustration. Widow Jones has been left $20,000. She wishes to invest it safely at 4%, but as she needs an income of $1040 a yr., she must put part of it in a less safe investment at 7%. How much can she invest at 4%? 1. The question is \u201cHow much can she invest at 4%?\u201d So let :i: be the principal at 4%. 2. To fill the box under x, note that she has 20,000 to invest, so the rest of it, 20,000 – x, is to be invested at 7%. 3. The rate column is filled from the 4% and 7% given in the problem. 4. The income column is filled from the product Principal Rate Income X .04 20,000 – X .07 Principal Rate Income X .04 .04 X 20,000 – X .07 .07(20,000 – x) 174 PROBLEMS of the principal and rate. Do not take anything from the word- ing of the problem for this column. 5. Now the equation must be made in terms of income, for the income boxes were the last filled. So look for income in the problem. It says, \u201cShe needs an income of $1040.” Therefore: .04 % + .07(20000 – x) = 1040 Solving : .04 X + 1400 – .07 X = 1040 4 X -t- 140000 – 7 X = 104000 4 X – 7 X = 104000 – 140000 – 3 X = – 36000 X = 12000 Check: 12000 X 4% = 480 8000 X 7 % = 20000 1040 The Equation in Investment Problems 1. I have $8000 from which I wish to receive an income of $380 a yr. If I invest as much as possible in 4% bonds and the rest in 6% stock, how much must I invest in the stock? 2. Henry Adams invested $3000, partly at 5% and partly at 6%. He says that he receives $18 a year more from the 5% investment than from the 6% one. How much has he at each rate? 3. Mr. Houghton has $4000 in a checking account that pays him only 2%. He decides to draw out and deposit in a savings bank that pays 4i% enough of it so that his interest will amount to $150. How much should he put in the savings bank? 4 . Mrs. Wilson has two investments, one paying 5%, and the other, $1000 larger than the first, paying 6%. Her income is $258. How much money has she at each rate? 5 . In his will a man directs that his daughter shall receive enough of his estate to furnish her with an income of $1400 a yr., f of her estate being in 5% bonds and the remainder in 4% bonds. How much of each should she receive? 6. If you invest $1000 at 4%, how much would you have to invest at 6% so that you would receive 5^% of your total investment? 7. Robert Smith left a hardware company because they paid him only 2% commission on his sales of $8000 and went to work for a farm equipment company that offered him a com- mission of 5% on his sales. How much must he sell there the INVESTMENT PROBLEMS 175 remainder of the year so that his total income may equal 3% of his total sales for the year? 8. What amount of money must a father invest at 6% simple interest so that his son will receive $2832 3 yrs. later for college expenses? 9. Mr. Arnold\u2019s mortgage for $924 will fall due in just 2 yrs. How much money should he put at interest at 5% so that it will amount to enough to pay the mortgage? 10. Chester Brown invested a sum of money at 5% and 3 times as much at 4%. The second investment pays $140 a yr. more than the first. How much has he at each rate? 11. Mr. Banker lent $3000 at 4%. How much additional must he lend at 6% so that the total income will be 4^-% of the total loan? Algebraic Expression 1. An automobile travels 35 mi. an hr. How far will it go in 6 hrs.? In 8 hrs.? In t hrs.? 2. How far can a boy row in t hrs. if he can row 2 mi. an hr.? r mi. an hr.? 3. Find a man\u2019s rate if he runs 120 yds. in 15 sec., in 12 sec., in t sec. 4. How long will it take to go 100 mi. at 8 mi. an hr,? At t mi. an hr.? 5. How long will it take to go m mi. at 10 mi. an hr.? At t mi. an hr.? 6. If you know the rate and the time, what would you do with them to find the distance? Write a formula for this relation using r, d, and t. 7. If you know the distance and the time, what would you do with them to find the rate? Write a formula for this relation. 8. If you know the distance and the rate, what would you do with them to find the time? Write a formula for this relation. 9. Can the formulas in Exercises 7\u2018 and 8 be derived from that in Exercise 6? Show this by solving rt = d for the proper letter. Illustration. A local passenger train starts east from Denver at the average rate of 30 mi. an hr. Two hrs. later a fast express leaves Denver over the same track at a speed of 50 mi. an hr. 176 PROBLEMS In how many hours after the express leaves must the train dispatcher arrange for the express to pass the local? 1. The question reads \u201cIn how many hours after the express leaves.\u201d Let x represent this number. 2. Now fill the box in the same column with the one filled. Here the local train started 2 hrs. earlier, so its time is ^ T 2. 3. To fill a second column of boxes, notice that the rates of the express and local are 50 4. Using rt = d, fill the last boxes without looking at the problem. 5. The equation is made in terms of distance. So read- ing the problem again, we see that the distances are equal. Then: Rate Time Distance Express. . 50 X Local. . . . 30 X \u201ct” 2 and 30 mi. an hr. Rate Time Distance Express. . 50 X 50 X Local. . . 30 X -|- 2 30(x -f- 2) 50 X = 30 (x -f 2) Express v ^ X = 3 Local ^ 30(x -f 2) The Equation in Motion Problems 1. George Brown leaves Atlanta at noon traveling south at the rate of 30 mi. an hr. At 2 p.m. another car leaves Atlanta with orders to overtake Brown by 5 p.m. At what rate must this car travel? 2. A slow train leaves Chicago for New York at 10:30 a.m. traveling at an average speed of 40 mi. an hr. Two hrs. later the Twentieth Century Limited leaves Chicago at an average speed of 60 mi. an hr. How far from Chicago will the Twentieth Century pass the other train? 3. An airplane leaves Oakland for the East traveling 180 mi. an hr. At the same time another airplane leaves Salt Lake City, 600 mi. farther east, and travels toward Oakland at the rate of 120 mi. an hr. In how many hours will they pass? 4. A train and a bus leave a town at the same time and travel in opposite directions. The train travels twice as fast as the bus, and in 4 hrs. they are 252 mi. apart. What are their rates? MOTION PROBLEMS 177 United Air Lines Photo. A MODERN PASSENGER PLANE With two 1150 horse-power motors, this twin-engined Douglas transport can take off in 980 ft. and climb to 8500 ft. with only one motor operating, and to 23,000 ft. with both motors running. These planes have a high speed of 220 mi. per hr. and fly from coast to coast in 15 hrs. 5 . Harold Green set out to walk home from college, a distance of 23 mi., at 4 mi. an hr., but a car picked him up after he had gone part way, and he finished his trip at the rate of 15 mi. an hr., arriving home in 3 hrs. from the time he started. How far did he walk? 6. Robert can run 12 ft. a sec. John can give him a start of 60 ft. and overtake him in 15 sec. How fast can John run? 7 . George rowed down the river at the rate of 5 mi. an hr. and returned at 2 mi. an hr. How far did he go if the trip took 3 hrs.? 8. As I have to wait in St. Louis 3 hrs. for a connecting train, I walk from the station at the rate of 4 mi. an hr. How far may I go if I can ride back at the rate of 12 mi. an hr.? If I had to wait 6 hrs. for the train instead of 3 hrs., would the distance I might go be doubled? 9 . A train leaves St. Albans for White River Junction aver- aging 40 mi. an hr. The railroad wishes to start a train from 178 PROBLEMS White River Junction for St. Albans at the same time and have them pass each other 75 mi. from St. Albans. If these towns are 120 mi. apart, what speed must the proposed train average? 10. If, in Exercise 9, the proposed train were to travel at a speed of 50 mi. an hr., how far from St. Albans would they meet? The Equation for the Merehant ; Mixture Problems Class Exercises Illustration. Mr. Rowe has 100 lbs. of 60^ coffee that does not sell easily. He wishes to mix it with 30^ coffee to make a mix- ture worth 40^ a lb. How many pounds of the 30^ coffee must he use? Kind Amount Value 30i X 30 X OOi, 100 6000 40i 100 4- X 40(100 + x) 30 ;c -f 6000 = 40(100 -f x) 30 X + 6000 = 4000 + 40 x X = 200 1 . The H and A Company has tea that should sell at 70^ a lb. With how many pounds of 30j^ tea should they mix 50 lbs. of it to make a mixture worth 40(zi a lb.? 2 . In order to make a mixture of nuts to sell at 60^ a lb., how many pounds of 40^ nuts should be mixed with 200 lbs. of 70^ nuts? 3. A druggist wishes to make up 200 lbs. of candy to sell for 70^ a lb. He has some candy worth SOj\/i a lb. and some worth 50j^ a lb. How much of each should he take? 4 . A paint store has white lead paint worth $4 a gal. and barytes paint worth $1.50 a gal. How much of each should be used to make 50 gal. of a paint worth $2.50 a gal.? 5. The Excelsior Feed Co. wishes to make 1200 lbs. of a poultry food to sell at $3 for 100 lbs. It has an oat food worth MIXTURE PROBLEMS 179 $4 for 100 lbs. and a scratch food worth $1 for 100 lbs. How much of each should be used? 6. H. G. Taylor has 60 lbs. of coffee worth 65ji^ a lb. How much coffee worth 40^ a lb. should he mix with it to make a mixture worth 50(!i a lb.? 7. A farmer has 300 lbs. of maple sugar worth 60jzi a lb. A city dealer will pay only 35^ a lb. How much granulated sugar worth 5^ a lb. must he mix with his maple sugar to make a product that he can sell for 35(zi a lb.? 8. Corn syrup is selling for 10^ a qt. and maple syrup for 30^ a qt. How much of each should a dealer use to make 40 qts. of a maple-flavored syrup to sell at 15^ a qt.? Optional Exercises Illustration. The druggist has a 40% solution of silver ni- trate, but the doctor\u2019s prescription calls for a 10% solution. How many ounces of distilled water must he add to 3 oz. of the solution to reduce it to the required strength? Let X = amt. of water added Amount of Solution % Amount of Silver Nitrate First solution. . . . 3 .40 .40(3) Material added . . X 0 0 Second solution. . 3 + X .10 .10(3 -h x) Since the amount of silver nitrate has not changed, the equation is: .40(3) = .10(3 + x) 40 X 3 = 10(3 -f x) 120 = 30 -1- 10 X X = 9 1. A druggist has a 40% solution of argyrol. How much water should he add to 10 oz. of it to make a 5% solution? 2. How many quarts of water are needed to reduce 4 qts. of an 8% solution of boric acid to a 3% solution? 3. The gold used for coins contains 10% alloy whereas that used for jewelry contains 40%. How much alloy should be added to 20 oz. of coin gold to reduce it to jewelry gold? 180 PROBLEMS 4. Pewter is made of tin with 15% of lead. How many pounds of tin must be added to 12 lbs. of pewter to reduce the lead to 10%? Honor Wqrk 6. A certain iron contains 5% chromium. How much chro- mium must be added to 850 lbs. of it to make a stainless steel that contains 15% chromium? 6. How many quarts of pure water must be added to 1 qt. of a 20% solution of ammonia to reduce it to a 5% solution? 7. A certain solution contains 20% sugar. How much water must be evaporated from 30 gal. of it to produce a 66f % sugar solution? 8. Milk contains 4% butter fat. If a city requires milk dealers to deliver a milk containing at least 3% butter fat, how much cream containing 30% butter fat may a milk dealer remove from 1080 qts. of milk? 9. A druggist has two solutions of iodine, one containing 5% and the other 20%. How much of each should he use to make 8 oz. of a solution to contain 10% iodine? 10. A milkman has milk testing 4% butter fat. If he wishes to make 465 qts. of a special grade- A milk to test 5% butter fat, how much cream testing 35% butter fat must he add to his milk? 11. How much pure water must be added to 3 qts. of a 40% solution of hydrochloric acid to reduce it to a 6% solution of the acid? The Formula in Medicine and Pharmacy Honor Work Modern physicians and druggists often use formulas instead of the methods you have just learned for changing the percentage of a mixture. They reduce n oz. of an x% solution to a y% solution by adding w = n{^ \u2014 1^ oz. of water. More often still the doctor writes ”Aqua q s ad ^ viii,\u201d which means \u201cAdd enough water to make 8 oz.,\u201d leaving it to the druggist to complete the work. REVIEW 181 Illustration 1. Reduce 3 oz. of a 40% solution of ether to a 10% solution. The druggist would add 9 oz. of water. To make up n c.c. (cubic centimeters) of a 3^% solution by- reducing an x% solution, take ^ c.c. of the solution and add w = ~ c.c. of water. X Illustration 2. To make 5 c.c. of a 24% solution from an 5 X 24 3 80% solution, take s ^ ^ c.c. of the solution and oU Z oO 2 Check: i -h i \u2014 5 80% X f C.C. = 1.20 c.c. 24% X 5 c.c. = 1.20 c.c. 1. How many cubic centimeters of oil should be added to 100 c.c. of a 50% solution of camphor to make a 20% solution? 2. How many ounces of alcohol should be added to 8 oz. of a 50% solution of chloroform to make a 30% chloroform lini- ment? 3. How many ounces of a 15% solution of silver nitrate, and how many ounces of water should be used to make 12 oz. of a 2% solution of silver nitrate to be used in the eyes? 4. How much water should be added to a 50% solution of phenol to make 500 c.c. of a 5% solution for disinfecting? 5. Find the amounts of water and of a 20% solution of boric acid needed to make 1000 c.c. of a 5% solution. Review Problems 1. One number is twice a second number, and the second number is 3 times a third number. Their sum is 70. Find the numbers. 2. A man lends part of $7200 at 4% interest and the remainder at 5%. The income is the same for both loans. How much did he lend at each rate? 3. Albany and New York are 150 mi. apart. A car leaves Albany for New York and averages 40 mi. an hr. Another car 182 PROBLEMS leaves New York for Albany at the same time and averages 35 mi. an hr. How far from Albany will they meet? 4. The difference between two numbers is 72 and i of the larger equals ^ of the smaller. Find the numbers. 6. A man earns 4 times as much a day as his son. One week the man worked 5 days and the son 6 days. Their combined wages were $52. How much did each receive a day? 6. John and Henry started from the same place and rode their bicycles in opposite directions. John started at 9 a.m. and rode 8 mi. an hr. Henry started 1 hr. later and rode 10 mi. an hr. How long had Henry traveled when they were 35 mi. apart? 7 . Grace paid $2.30 for stamps. She bought 3 times as many 3(zf stamps as 5^ stamps and twice as many l^j^ stamps as 3f!i stamps. How many of each did she buy? 8. Tom had $2000 invested at 7% and $5000 at 4%. How much additional money must he invest at 6% to make his total income 5% of his total investment? 9 . Paul received 73 in English, 80 in French, and 89 in geom- etry. What mark must he get in history to make an average of 85? 10. Katherine invested $500, part at 5% and the remainder at 4%. The interest amounted to $22 annually. How much did she invest at each rate? 11. A merchant wishes to mix candy worth 55^ a lb. with candy worth 35^ a lb. How many pounds of each must he use to make a mixture of 20 lbs. worth 40^ a lb.? 12. How many ounces of water must Mrs. Richardson add to 8 oz. of a 12% solution of argyrol to reduce it to a 5% solution? 13. To make up 200 lbs. of nuts to sell at 50j!i a lb., how many pounds of 60^ nuts should be mixed with 35jzi nuts? 14. Our baseball team has won 3 games out of 6 played. How many must it win out of the next 9 games to make its average 66f %? 15. Carl must be back in school in 2^ hrs. How far can he ride his bicycle up hill at 3 mi. an hr., if he can ride back at 12 mi. an hr.? 16. How many tablespoons of water should be added to REVIEW 183 9 tablespoons of a 60% solution of soda to reduce it to a 40% solution? 17. Eighteen k. gold contains f gold and i alloy. How much alloy must be added to 16 oz. of 18 k. gold to reduce it to 14 k. gold which is xV gold? 18. Two trains start at the same time from cities that are 140 mi. apart and travel toward each other at rates of 15 mi. an hr. and 20 mi. an hr. What distance will the slower train have traveled when they pass? 19. How many pounds of 30^ coffee must be mixed with 45^ coffee to make a mixture of 300 lbs. worth 35^ a lb.? 20. How many pounds of peanuts worth 15j^ a lb. should be mixed with 40 lbs. of walnuts worth 50^ a lb. to make a mixture worth 30^ a lb.? 21. A baseball player has a batting average of .200 in 80 times at bat, but is now batting .400. If he keeps up this rate, how many times must he bat to bring his average up to .240? 22. A baseball team won .375 of its first 24 games, but will now meet easier opponents from whom it ought to win .750 of its games. How many games must it still play to bring its record up to .525? 23. One angle of a triangle is 50\u00b0. A second angle is 30\u00b0 larger than the third. Find the number of degrees in each angle. 24. An isosceles triangle has two equal angles. The third angle is 20\u00b0 less than their sum. Find the number of degrees in each angle. 25. Find the acute angles of a right triangle if one of them is 18\u00b0 more than the other. 26. The second angle of a triangle is twice the first, and the third angle is 20\u00b0 more than the first. Find the number of degrees in each angle. 27. One angle of a triangle is 3 times another, and the third equals their sum. Find the number of degrees in each angle of the triangle. 28. Two lines are parallel. One of a pair of alternate interior angles contains 38\u00b0 more than half the other. How many degrees are there in each? 29. Two lines are parallel. One of the two interior angles 184 PROBLEMS on the same side of the transversal contains 40\u00b0 more than 3 times the other. Find the number of degrees in each. 30. Two lines are parallel. One of a pair of corresponding angles equals 3 % \u2014 20 and the other equals x + 100. Find the size of each. 31. Find the size of two supplementary angles if one of them is 30\u00b0 less than twice the other. 32. How large is an angle if it is 56\u00b0 larger than its supple- ment? 33. A merchant bought suits at $12 each. At what price must he sell them to make 33^% of the selling price? 34. Mr. Haddon invests $12,000, part at 5% and part at 6%. He receives $60 a yr. more from that at 6% than from that at 5%. How much has he at each rate? 35. In a park, 300 benches around the grand stand have a seating capacity of 1700 people. If some of them seat 6 people each and the rest seat 5 each, how many of each kind are there? 36. Find two numbers whose sum is 38 and whose difference is 4. 37. The numerator of a fraction is 3 less than the denom- inator, and the fraction can be reduced to f. What is the fraction? 38. The denominator of a fraction is twice the numerator, and if I add 8 to the numerator the fraction becomes 1. What is the original fraction? 39. In building a through road, the state agrees to pay twice as much as the town, and the Federal government will pay as much as both together. How much must a town pay if the road through it costs $21,000? 40. Emily bought a coat at a i off sale for $16. What was its former price? 41. A profit of 20% of the selling price is what per cent of the cost? Review Exercises 1. Add: 6 8\u00ab2 A he \u2014 6 \u2014 3 The 10 -5 x2 \u2014 2m 2 m REVIEW 185 2. Subtract: 15 a: 8y 4 c \u20146a – 5 62 9a: 11 y -7 c -6a – 8 62 3. Multiply: (5a:5)(2a:2) (-7 ab)(-3b^) (-2&)(8 c) (x^)(x)(x^) (3:r3)(- 4xy) (-5)(-2)(-3) 4. Divide: 6 a:6 15 m3\u00ab2 8c3 3a:3 \u2014 3 m^n^ 8c3 – 8 fl 2 – 20 b^ Sab^c^ 4 a – 4 b^ 2a^b^c^ 6, Remove parentheses, and combine like terms: 5\u00ab + (6& \u2014 2) m \u2014 {\u20142m 3 c – (5 + 4) 2k -2{k -Q) 6a – (3a -7) x – 2(4 – x) – 3 6. Solve and check: + 5 = 8 – 3 = 9 3:r = 12 2 a: – 7 = 17 7. Find the value of: ^ _ a + b c ^ ~ 2 a = – 2)180 n n _ 3 pm 2ia)\u00a5 3 ^ _ 4) =14 2(9 – 3 2) + 3(4 – 2) = 21 if \u00ab = 17, & = 12, and c = 13 if w = 12 i p = 8000, m = 200, w = 6, and h = 8 8 . Find the value of: 38 X 51 – 28 X 51 489 X 372 – 479 X 372 8432 X 3769 + 8432 X 6231 9. Does y increase or decrease as x increases in: y = 3×4-4? 6^ 2;c+3^ = 20? ^ y = 10 \u2014 X? ^ x’ = 18? ^ x’ 186 PROBLEMS 10. If 3 apples cost x cents, what does 1 apple cost? How many hours will it take to go m mi. at x mi. an hr.? If 5 books cost d dollars, what will 7 books cost? A and B were m mi. apart but have traveled toward each other for h hrs. at x and y mi. an hr. respectively. How far apart are they now? 11. Charles made this graph to show how his father\u2019s estate was divided among 4 children and their mother, the mother receiving the largest share. If the estate was worth $54,000, find the share of each. 12. The telephone company charges a business house $12 a mo. for a telephone exchange and 30^ for each phone. Make a formula for the cost of \/ phones. Find c when \/ = 20, when \/ = 40. When the number of phones is doubled, is the cost doubled? Test in Making Equations from Problems Write equations for these problems: 1. If 3 times a number is increased by 12, the result is 30. 2. Six times a number exceeds 42 as much as 22 exceeds twice the number. 3. One number is 7 more than 4 times another number. Twice the larger number exceeds 8 times the smaller number by 14. 4. The second angle of a triangle is 10\u00b0 more than twice the first angle, and the third angle is 10\u00b0 more than the sum of the first two. 5. Mr. Washburn sold merchandise for $9 making a profit of 80% of the cost. Find the cost. 6. Two lines are parallel. If one of a pair of alternate interior angles were decreased 25\u00b0 and the other were increased 15\u00b0, the larger angle would lack 5\u00b0 of being twice the smaller. 7. The B and Q Company have tea worth 60jii a lb. and tea worth 30^ a lb. How many pounds of each should they use to make 180 lbs. worth 40fi a lb.? 8. Mr. Arnold has $8000 invested, part at 5% and part at 6%. The part at 5% produces $70 a yr. more income than that at 6%. How much has he at each rate? TEST 187 9. A sets out on a trip at 20 mi. an hr. Three hours later B follows from the same place at 35 mi. an hr. In how many hours will B overtake A? 10. Dr. Martin has 15 oz. of a solution containing 20% iodine. How much alcohol should he add to reduce it to a 6% iodine solution? Chapter 9 SETS OF EQUATIONS THE GRAPH IN MOTION PROBLEMS M any problems can be solved graphically without making an equation. Then from the graph, many other questions can be answered that could not be answered from the equation. Whereas one set of equations solve a single problem, the graph solves a group of related problems. Illustration 1. Robert started for the country on his bicycle at noon, traveling 9 mi. an hr. John will get out of school at 2 p.M. and can ride 12 mi. an hr. He wants to know how quickly he can over- take Robert and how far they will then have gone. John made this graph. Since Robert goes 9 mi. an hr. and started at noon, how far will he have traveled at 1 p.m.? At 2 P.M.? At 3 P.M.? If John starts at 2 p.m. \/ [p Y Hours and travels 12 mi. an hr. At 4 P.M.? At 5 P.M.? how far will he have gone at 3 p.m.? Can you tell from John\u2019s graph at what time he will overtake Robert and how far they will then have gone? When were they 8 mi. apart? If Robert had stopped traveling at 4 p.m., at what time would John have overtaken him? How far apart were they at 7 p.m.? If they continue riding, how far ahead will John be at 9 p.m.? 188 THE GRAPH IN MOTION PROBLEMS 189 Illustration 2. A local train leaves Chicago for the west at 1 p.M. traveling 40 mi. an hr. at a station, then travels at the same speed until 3 P.M. when it makes a 20-min. stop. It then goes on at the same rate. A through ex- press leaves Chicago for the west at 2 P.M. averaging 50 mi. an hr. At what time and how many miles from Chicago should the train dispatcher arrange to have the ex- press pass the local? What does the horizontal part of the graph mean? At what time does the express overtake the local? How far from Chicago are they then? How far apart are they at 2:30 p.m.? How far does the express travel while the local is standing at the first station? At what time are they 30 mi. apart? Illustration 3. A and B are 300 mi. apart. A train leaves A at noon for B averaging 40 mi. an hr. At 1 p.m. a train leaves B for A going 60 mi. an hr. At what time and how far from A will they pass? How far apart are the trains at 1 p.m.? At 2 p.m.? At 4 p.m.? At what time are they 100 mi. apart? By extending the graph, find at what time the train from B will arrive at A. How far from A will the train that left A then be? Exercises Solve these problems by graphical methods: 1. At noon a train leaves Kansas City for St. Louis at the rate of 40 mi. an hr. At 1 p.m. it is forced to slow down to 25 mi. an hr. At 2 p.m. another train follows on the same track at 50 mi. an hr. At what time and how far from Kansas City must the train dispatcher arrange to have them pass? \u25a0p 1 1 T\u201c 1 1 T~ 1 1 2 3 4 5 .Time It stops from 2 p.m. until 2:30 p.m. X’ 1 2 3 4 Time 190 SETS OF EQUATIONS Courtesy of Union Pacific Railroad. A MODERN STREAMLINED TRAIN Much mathematical research plays a part in reducing the wind resistance of trains, airplanes, and automobiles of today. 2. A patrol boat that can travel 35 mi. an hr, sights a smuggler 15 mi. away and traveling away from it at 28 mi. an hr. How far must the patrol boat go to overtake the smug- gler? 3. A train leaves New York for Philadelphia traveling 45 mi. an hr. At the same time another train leaves Philadelphia, 90 mi. away, and travels toward New York at 30 mi. an hr. In how many hours will they meet and how many miles from New York? 4. An automobile leaves town at 25 mi. an hr. Three hrs. later a second automobile pursues it. At what rate must it travel to overtake the first in 5 hrs.? 6. At noon a car leaves town going 20 mi. an hr. At what rate must a second car travel if it leaves the same place at 1 p.m. and plans to overtake the first when it has gone 70 mi.? 6. Dorothy has $5 and saves $1 a week. If Robert saves $1.50 a week, in how many weeks will he have as much money as Dorothy? How much more money has Dorothy than Robert at the end of 5 weeks? THE GRAPH IN MOTION PROBLEMS 191 7. Paul sets out at 9 a.m. by bus going 10 mi, an hr. At 11 a.m, John follows by car at 18 mi. an hr. How far apart are they at noon? At what time will John overtake Paul? At what time will John be 12 mi. ahead of Paul? 8. It costs a publisher $2000 to prepare a book and make plates and $600 extra for every 1000 books sold. If the book sells for $1 a copy, how many thousand copies must be sold before they begin to make money? How much will they make if they sell 10,000 copies? 9. A and B start toward each other at 8 a.m. from two towns 240 mi. apart. A travels 15 mi. an hr. and rests from 12 noon until 2 p.M. on the way, while B travels 25 mi. an hr. and rests from 11 A.M. until 2 p.m. At what time will they meet? 10. A starts from a town at noon and travels east 30 mi. an hr. If B starts at 2 p.m. from another town 20 mi. farther west, at what speed must he travel to overtake A at 6 p.m.? 11. Two men, 60 mi. apart, travel toward each other. One travels 4 mi. an hr. The other starts 1 hr. later and travels 3 mi. an hr. In how many hours will they meet and how far will each have traveled? The graph as a ready reckoner. 12. This graph shows the relation between pounds and kilograms. For example, 5 kilograms (Kg.) equal 11 lbs. Can you find this on the graph? How many pounds equal 3 Kg.? 8 Kg.? 10 Kg.? How many kilo- grams equal 4 lbs.? 9 lbs.? 15 lbs.? 2 4 6 8 10 12 14 16 18 20 22 Pounds Historical Note on Sets of Equations; The use of two unknowns seems to have been discovered by the Chinese long before the time of Christ but they had no suitable symbolism for them. Later the Hindus used colors, such as red and blue, to stand for two unknowns. About the sixteenth century the letters A, B, and C began to be used. These equations were then, solved by the addition-subtraction method. 192 SETS OF EQUATIONS GRAPHS OF FUNCTIONS How to map the plane. You have already made graphs of formulas in which you found points that corresponded to two numbers. This same method may be used for locat- ing any point of the plane. We shall call the horizontal axis, XX’, the :r-axis; the vertical axis, YY’, the y-axis; and the point where they cross, the origin. Where is 0 placed? Is 0 the same on both axes? Does positive x run to the right or to the left? Negative a:? Which way does positive y run? Negative y? Can you find a point that is 2 in the a: direction and also 3 in the y direction? Charles says that this point is P on the graph. Is he right? Can you find a point that is \u2014 3 in the x direction and + 2 in the y direction? Sarah says that S is the required point because it is 3 units to the left and 2 units up. GRAPHS OF FUNCTIONS 193 The X distance of a point is called the abscissa, and the y distance is called the ordinate of a point. Both together are the coordinates of the point. Here is a short way to record the coor- dinates of a point. Write the x value first, then a comma, then the y value, and finally enclose all in a parenthesis. For example P is (2, 3), Q is (4, 2^), R is (4, – 2), and 5 is (- 3, 2). Exercises Make a graph like that shown here, and locate the following points on it: RENE DESCARTES (1596-1650) Famous French philosopher and mathematician. By plotting the graph of algebraic functions, he established a connection between algebra and geometry. In addition to his work in mathe- matics, he is considered one of the founders of modern philosophy. 1 . ( 1 , 2 ) 5 . (-4,3) 9 . (-5, – 1 ) 13 . (4,0) 2 . (4, 7) 6 . (- 2, 2) 10 . (2i, 3) 14 . (0, – 3) 3 . (5, – 1) 7 . (3, – 6) 11 . (3i, – 4) 15 . (- 2, 0) 4 . (3, – 2) 8 . (- 4, – 7) 12 . (- 1, – 4i) 16 . (0, 0) 17 . Write the coordinates of points A, B, C, . . . N. Thought Question In what direction would point P move if its x value grew larger but its y value remained unchanged? How to make the graph of an equation. Find two num- bers whose sum is 10. If x and y are the numbers, the equation is % -h y = 10. 194 SETS OF EQUATIONS Gladys says that x = 5 and y = 5. She put point A on the graph. Is she right? Katherine says that x = 7 and y = 3. She put point B on the graph. Is she right? Is there any one solution to this problem? Can you find other solutions to this equation and put other points on the graph? Can one of the letters be nega- tive? The class made this table of values for x and y : 1 ^ 1 2 I 3 4 5 6 7 8 9 10 11 12 – 1 -2 – 3 9 8 1 7 6 5 4 3 2 1 0 – 1 – 2 11 12 13 Are there other points that satisfy this equation between these points? For instance, could x = 3^? 5f ? Then the class plotted this graph. They called it the graph of the equation x + y = 10. What do you notice about the location of these points? GRAPHS OF FUNCTIONS 195 Could we have made the graph without plotting so many points? Does every point on that line satisfy the equation? Are there any points not on the line that satisfy the equation? Exercises Draw the graphs of these equations: 1. X + y = 6 5. y – 2 X \u2014 3 9. 2x + 3y = 9 2. X – y -= 2 6. y = 2 X \u201c 1 10. 3 X ~ 2 y – 4 CO I !! ! CO eo 7. y = 2 X + 4 11. y -ix 4. y = X + 4 8. y – 3 X + 4 12. X + y – 0 How to solve equations by graphs. Illustration. Solve the equations: 3% + 2y = 12 3 X ~ j = 3 First, let us make a table of values for each equation. Since two points are needed for locating a line and one other point for a check on our work, we shall make a table of three sets of values for x and y. Of course we shall choose the easiest points to find, those for which one letter is 0, as this will save us the trouble of transposing. So we let X = 0 and find the value of y, then let y = 0 and find x. Then any other value of x will do for the checking point. T able for: T able for: 3x + 2y = 12 3x-y=3 X 0 1 3 y . – 3 0 6 X 0 4 1 y 6 0 4i Making the graphs of both equations on the same axes, we have the figure on page 196. What are the codrdinates of the point where the lines cross? 196 SETS OF EQUATIONS Does every point on the graph of3A + 2y = 12 sat- isfy the equation? Does (2, 3) satisfy the equation 3 x 2 y = 12 ? Does every point on the graph of 3 A \u2014 y = 3 satisfy that equa- tion? Does (2, 3) satisfy it? Is there any other point that satisfies both equations? There is only one solution to the pair of equations \u2014 the values of A and y at the point where the graphs cross. Exercises Solve graphically: 1. x+y ^3 X \u2014 y = 1 2. 2 X –y = 7 a; \u2014 3 y = 0 3 . a: \u2014 2 y = 2 a: + 2 y =6 4 . 2A;-3y = 7 a: +y = 1 5. X \u2014 5 y = 2 2 X \u2014 y = \u2014 5 6. 5 a: + y = 3 3a: -2y = -6 7 . a: – y = 0 2 X + y = 15 8. y = 3 X – 7 y = – X -1- 5 9. X -f y = 0 X \u2014 y = 0 10 . 3 X \u2014 5 y = 2 2 X + 3 y – 14 11 . 3x+4y=-4 2x-5y = -18 12 . y = X -f 2 y = 3 X -f 2 13 . X 4- y = 5 3 X 4- y = 11 14 . X \u2014 y = 5 X \u2014 4 y = \u2014 5 15. X \u2014 y = \u2014 4 3 a: – y = – 12 16 . 7 X -h 5 y = 2 X -h2y = – 1 17. X – y = 0 2 X \u2014 3 y = 6 18 . 6x – 7y = 1 7x – 8y = 2 Try to solve these equations graphically: 19. 2 X – y = 5 20. 2 X 4- 4 y = 10 4x-2y = 3 x4-2y = 5 ADDITION-SUBTRACTION METHOD 197 21. What did you discover about the graphs of Exercise 19? Do they intersect? Can we find the answers from the graph? Equations that have no common solution are called inconsistent equations. The graphs of inconsistent equations are parallel lines, because two lines that have no common point are parallel. 22. What did you discover about the graphs of Exercise 20? Does more than one point satisfy both equations? Many? Is it true that any point that satisfies one equation will satisfy the other? How many answers then have these equations? Two equations are dependent if every point on the graph of either is also on the graph of the other. You can change one equation into the other by multiplying every term by some number. We shall learn more about inconsistent and dependent equations later. ALGEBRAIC SOLUTION: THE ADDITION- SUBTRACTION METHOD In solving equations by the graphical method, it is necessary to estimate the roots, and often, as when the lines are nearly parallel, it is difficult to get an accurate result. We shall now learn a method that is generally simpler and more accurate. Illustration 1. Solve: 2 % \u2014 y = 4 a: + y = 5 Do you remember how to solve an equation when there is only one letter in it? If you could get rid of one of the letters here, could you find the value of the other? Look at these two equations. Is there anything that you can do to them that will get rid of y? 198 SETS OF EQUATIONS If you knew the value of x, could you find the value of y? Solution: We can get rid of y by adding the two equations: 2x – y = 4: X + y = 5 3x =9 X =3 Now we can find the value of y by substituting 3 for x in either equation. The second is the easier. 3 +y = 5 y = 2 Check your answers by substituting in both of the original equations. 2x \u2014 y = 4 X + y = 5 2*3-2 14 3 + 2 I 5 4=4 5=5 Illustration 2. Solve: 3 x + 2y =4 2x + 3y = 1 Here we cannot get rid of either letter by adding or sub- tracting, because the coefficients of neither x nor y are the same in both equations. Is there anything that we can do to make them the same? Can we multiply all terms of an equation by anything we please? What is the smallest number of y that 2 y and 3 y will each go into? By what must you multiply the first equation to obtain 6 y? The second equation? Solution. If we multiply the first equation by 3 and the second by 2, we shall have 6 y in both equations. Multiplying by 3: 9% + 6y = 12 Multiplying by 2: 4x + 6y = 2 Subtracting; 5 x =10 = 2 ADDITION-SUBTRACTION METHOD 199 Substituting in the first: 3x + 2y =4 & ^2y =4 y =-l Check by substituting in both of the original equations. Class Exercises Solve and check: 1. 3 X –y = 7 9. 2 a: – 3 y = 1 X \u2014 y = 1 3% + 2y = 21 2. X + 2 y = 4: 10. 3x \u2014 4y=\u2014 6 3 X + 2 y = 8 4 % + 2 y = 14 3. x + 2y = 9 11. 5;c + 6y = – 1 X \u2014 2y =3 3 X 4- 2y = 1 4:. 5x\u2014 2y = 4 12. 4 + 3 m; = 24 3 % + 2 y = 12 3 + 2 m; = 17 5. 2p + Sq = 5 13. 2 \/? + 5 yfe = 1 4p -3q = 1 3h -4k = -10 Q. 2a + 3b = 8 14. + 3 = y 2a – 51? = -8 2 y \u2014 10 = a: 7. 3r-2s = 6 15. m \u2014 w = 0 3r -7s = -9 n = 3 m \u2014 4 8. X \u2014 5y = 3 16. 3 + 2 & – 29 = 0 X -i-2y = – 4 b = 2a -3 Optional Exercises 17. 2 ;ri + 3 %2 = 13 22. .4;r + 1.2y = 7.6 3 \u2014 4 a:2 = \u2014 6 1.5 X – .08y = 5.6 18. 5 m;i \u2014 7 m;2 = 12 3wi \u2014 5w2 = 8 19. i\u2018X + xy = 2f- T X ^ y = 4^ 20. 2ix \u2014 l^y = 0 24.^+^ = 7 lix + 2^y = 22 21. .3x 4- .7y = 1 X \u2014 .06 y = .94 ^ ^ = 12f 200 SETS OF EQUATIONS 26. 1.2 X – My = 2.4 3 X + 1.5y = 6 27. 10 TT – 4e = 20.5428 3 TT + 10 e = 36.6078 2 4 2(x+3′) -3(3;c-3′) = 30. 3(x-l)+2(2>’ + l) = X –5 y -{-3 _ 2 Honor Work 31. 3x – Ay = – fl + 76 2a: +33^ = 5a – 6 _ . 32. ax \u2014 by = ah bx -A- ay = 2 6^ -|- ^2 33. ax by = c dx -{- ey = f 18 34. mx \u2014 ny = w* + 3 nx my = + w\u00ae 36. ax -A- by = a bx \u2014 ay =6 + 4 SUBSTITUTION METHOD Another method of solving sets of two equations is called the substitution method. We first solve one equation for a letter. Then we substitute the value found in the other equation. Illustration 1. Solve: 2 a: + 3^^ = 21 (1) 4 a: + 3^ = 17 ,(2) Solving (2) for 3^ = 17 \u2014 4 a: (3) Substituting this value of j in (1): 2 a: + 3(17 \u2014 4 a:) = 21 2 X + 51 – 12 a: = 21 – 10 X = – 30 X = 3 3; = 17 – 12 3^ = 5 4 X + 3; = 17 12 + 5 I 17 17 = 17 3 a – 5 6 = 22 (1) 5a +36 = 14 (2) Substituting x = 3 in (3) : Check: 2x + 33 ^ = 21 6 + 15 I 21 21 = 21 Illustration 2. Solve: SUBSTITUTION METHOD 201 Solving (1) for a Substituting (3) in (2); Substituting h 2 in (3) Check: 3 fl – 5 6 = 22 12 + 10 I 22 22 = 22 3 fl = 22 + 5 & 22 + 56 )+3 6 (3) 14 no + 25 6 + 9 6 = 42 34 6 = – 68 b = -2 22 – 10 5 c + 3 6 = 14 20 – 6 I 14 14 = 14 Exercises Solve and check: 1. 4 a: + j = 13 5 a: + 3 3^ = 18 2. 2a:+33’=8 X \u2014 Qy = \u2014 11 3. 2 a: + 3^ = 8 5x -2y = 11 4. a: \u2014 2 3^ = 2 4 a; + 33; = – 3 5. 3 X \u2014 y = 2 2 a; + 33^ = 5 6. a; = 3 3′ + 5 3^ = 2 a: \u2014 20 7. 3a:-43;=2 5 a: + 2 3^ = 25 8. 2a: + 33’=-12 5a: – 73^ = – 1 9. 2 a; \u2014 5 3^ = .5 3a: – 73^ = 1 10. 1.5 ;c + 2.8 3^ = 17 .5 a: \u2014 3^ = \u2014 4 11. 2.4 a: – 3 3^ = 7.8 .85 a: + .01 3^ = 1.69 12. 3.25 a: + .45 3^ = 4.15 2.75 a: – 3^ = .75 13. In the formula D = dq + r, find d and r if Z> = 103 when q = 3, and D = 115 when ^ = 9. 14. A merchant, using the formula C = S + Z,, finds his loss, L, to be $8. If the selling price, S, had been 50% greater, L would have been $2. Find the cost and selling price. 15. Find a and h if the graph of ax by = 10 passes through the points (1, \u2014 2) and (4, 2). 16. In the formula d = vt ^ aV-, d = 104 when t = 2, and d = 336 when t = 4. Find v and a. 202 SETS OF EQUATIONS Sets of Equations in Geometry Is AB parallel to CD if: 1. r + v = 112\u00b0 3 r – 2 ?; = 56\u00b0 2. r + v = 210\u00b0 r \u2014 V = 0\u00b0 Z. r + v = 100\u00b0 Algebraie Expression If X is the larger of two numbers and y the smaller, repre- sent: 1. The sum of two numbers is 17. 2. The difference of two numbers is 12. 3 . One number exceeds the other by 20. 4. Twice the larger number exceeds 3 times the smaller by 15. 5 . The sum of two numbers is 5 times their difference. 6. The quotient of two numbers is 3. Problems involving equations with two letters. It is often much easier to write the equations for a problem by using two letters than it is by using only one letter. But it is necessary to form as many equations as there are letters. When you use only one letter, you will need only one equation, but when you use two letters, you must make two equations. Illustration. The sum of two numbers is 15, and twice the larger exceeds the smaller by 9. Find the numbers. Let X = the larger y = the smaller X A y = 2x \u2014 y = 9 Solving: x = 8 y = 7 Check : The sum of 8 and 7 is 15. Twice 8, or 16, exceeds 7 by 9. 203 PROBLEMS IN TWO UNKNOWNS Algebraic Shorthand Express the following problems as equations in two letters. Then solve your equations. 1. The sum of two numbers is 21, and the difference of these same numbers is 11. 2 . Find two num- bers whose quotient is 3 and whose sum is 24. 3. Find two num- bers whose sum is 11, if twice the first ex- ceeds 5 times the sec- ond by 1. 4 . Find two num- bers whose difference is 6 and whose average is 7. 5 . Find two numbers whose average is 8, if the first is 4 more than 3 times the second. 6. The larger of two numbers is 3 more than twice the smaller, and twice the larger is 1 isaac newton more than 5 times the Isaac Newton (1642-1727) was the first to solve smaller. equations by the substitution method. He dis- -j.” . covered the laws of gravity, of moving bodies, 7. It twice a number and of the propagation of light. His work laid is added to 3 times a foundation for modern mechanics and as- smaller number, the \u00b0 ^ sum is 17. But if twice the smaller is added to 3 times the larger, the sum is 18, 8 . Sam Ling charged me 66(zf for laundering 3 shirts and 10 collars one week. The next week he charged me 45^ for laundering 2 shirts and 7 collars. What is his price for each article? 9 . John said to Robert, \u201cI have 6 marbles more than you have.\u201d Robert replied, \u201cTomorrow I shall bring twice as many 204 SETS OF EQUATIONS as I have today, and then I shall have 8 more than you have.\u201d How many marbles has each boy today? 10. At a special sale the King Grocery Company gave 3 lbs. of coffee and 5 lbs. of sugar for $1, or 2 lbs. of coffee and 3 lbs. of sugar for 65^. For what price a pound were they selling each? 11. Mr. Farrell saves $4 a day when he works, but it costs him $3 a day to live when he is idle. If he must save $198 during the next 60 days, how many days must he work? 12 . For a football game the tickets cost 25?i if bought at the school, but 50^ if bought at the field. The manager says that $270 was taken in for the 960 tickets sold. How many were sold at the school? 13 . A club allows its secretary $1 for sending notices of a meeting. If he must notify 58 persons, to how many may he send 2^ letters and to how many must he send post-cards? 14 . A contractor says that he will furnish me with 5 men and 3 boys for $44 a day, or with 4 men and 5 boys for $43 a day. What rate is he charging for a man and for a boy? 15 . In sending a telegram a certain charge is made for the first 10 words and a fixed amount for each additional word. Dorothy paid 61^ for a 22- word message and 94)zi for a 33- word message. What are the two rates? 16 . Two investments total $9000. Find their amounts if one is at 6% and the other at 5%, and the total income per year is $504. 17 . Three times one number added to 5 times a second num- ber gives 49. But if I subtract 3 times the second from twice the first, the remainder is 1. Find the numbers. AREA PROBLEMS Many area problems can be expressed easily in terms of a single letter, but there are many others that either re- quire two letters or are much more easily stated in terms of two letters. Area Problems Requiring but One Letter 1. The width of a rectangle is w, and its length exceeds its width by 5. What is its length? Its perimeter? Its area? AREA PROBLEMS 205 (c) id) {e) (\/) 2 . The width of a rectangle is w, and its area is + 4 w. Write an expression for its length. 3. The length of a rectangle is 7 more than the side of a square, and the width is 3 less than the side of the square. Letting s represent the side of the square: {a) Represent the dimensions of the rectangle. (&) Represent the area of the rectangle. Represent the perimeter of the rectangle. State algebraically that the area of the rectangle is 200. State that the length is twice the width. State that the area of the rectangle exceeds that of the square by 40. (g) State that the sum of the perimeters is 90. {h) If the length of the rectangle were decreased 3 and the width increased 4, express the new dimensions, the new area. (f) State that the area of the new rectangle exceeds that of the original rectangle by 50. Illustration. The length of a rectangle is twice the width. If the length is increased 3 and the width diminished 1, the area is increased 3. Find the dimensions. In solving geometry problems, it is best to draw the figures and write the dimensions and areas on them. Since the area is increased by 3, the second area is 3 more than the first, or {2w + 3) {w 2w^ w\u20141 (2w+3) (w-1) 2w+3 1 ) = 2 3 . w = 6, the width 2 = 12, the length 4 . The length of a rectangle is 10 ft. more than the width. If each dimension is increased 5 ft., the area will be increased 475 sq. ft. Find its dimensions. 5. The length of a rectangle is 3 times the width. If the length is decreased 4 and the width is increased 3, the area will be increased 8. Find its dimensions. 6. Find the side of a square whose area is increased 32 when its dimensions are each increased 2. 7 . If the base of a square is increased 3 and the altitude de- creased 2, the area remains unchanged. Find its side. 8 . Mrs. Hart has the choice of two rugs, both having the same area for her square dining room. One is the length of the 206 SETS OF EQUATIONS room but 4 ft. narrower, whereas the other is 2 ft. shorter than the room and 2^ ft. narrower. Find the dimensions of the room. Area Problems in Two Letters 1. The width of a rectangle is w and its length is 1. Express its perimeter, its area. 2. Using \/ and w for the length and width of a rectangle, write an equation for: (fl) Its perimeter is 34. (&) Its area is 90. (c) The length is 3 less than twice the width. id) Twice the length exceeds 3 times the width by 5. ie) If the length were increased by 3, the area would be 50. (\/) If the width were decreased by 8, the area would be decreased by 70. (g) If the length and width are each increased by 4, the area is doubled. 3. If \/ is the length and w the width of a rectangle, express in words; (a) \/ + 3 and w \u2014 2 {c) Iw + 64 {b) (I + 3){w – 2) ‘ (d) (1 + 3)(w – 2) = \/m; + 64 4 . If the length of a rectangle is increasing, but its width re- mains unchanged, what change is taking place in its area? In its perimeter? 5. Twice the length of a rectangle exceeds 3 times the width by 1. If each dimension is decreased by 2, the area is decreased by 22. Find the width. i 6. If the length of a rectangle is increasing, but its area re- mains unchanged, what change is taking place in its width? 7. If the length of a rectangle is increased by 3 and the width increased by 2, the area is increased by 40. But if the length is increased by 2 and the width by 3, the area is increased by 42. Find the original length and width. 8. If the length of a rectangle is increased by 1 and the width decreased by 1, the area is decreased by 3. But if the length is decreased by 1 and the width increased by 2, the area is in- creased by 9. Find the length and width. 9. If the length of a rectangle is decreased by 3 and the width increased by 2, the area remains unchanged. Also if the length OUR NUMBER SYSTEM 207 is decreased by 4 and the width increased by 3, the area is unchanged. Find the dimensions. 10. If I cut a strip 1 in. wide from all sides of a sheet of paper, I reduce its area by 40 sq. in. If, instead, I cut a strip the same width from the two ends only, the sheet becomes a square. Find its dimensions. OUR NUMBER SYSTEM The number system that we use today was invented by the Hindus and came to us through the Arabs. Its great advantage over all older systems will be evident to you if you attempt to perform any of the ordinary opera- tions with, for instance, the Roman numerals. For example, try to multiply LXXIX by XIV. Even a simple addition is difficult. TryXLII + XXVIIL The figures 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 are called digits. When two digits are placed together, the number formed is the sum of the right-hand digit and 10 times the left- hand digit. For example, 37 = 10 X 3 + 7. When the digits are reversed, a new number is formed, for now it is the other digit that is multiplied by 10. 73 = 10 X 7 + 3. If the tens\u2019 digit is t and the units\u2019 digit is u, the number is 10 ^ -f w. Questions on Digits 1. How many units are there in a number if: (c) Its tens\u2019 digit is 4 and its units\u2019 digit is 7? (6) Its tens\u2019 digit is x and its units\u2019 digit is y? 2. Write a number whose tens\u2019 digit is 5 and whose units\u2019 digit is n, one whose tens\u2019 digit is n and whose units\u2019 digit is 5. 3. If the tens\u2019 digit of a two-digit number is t and the units\u2019 digit is u: {a) Write the number. ih) Write the number having these digits reversed. (c) Write the sum of the digits. (d) If i = 6 and u = 9, what is the number? What is the number with the digits reversed? What is the sum of the digits? 208 SETS OF EQUATIONS 1234667890 Year NUMBER SYSTEMS Our present number system is the culmination of the experiments of many people for centuries. The Arabs, Persians, Egyptians, and Hindus all proba- bly had a hand in originating our numerals. Some of the numerals are found in inscriptions made in the 3rd century B.c. Numerals are thought first to have appeared in Europe in 976 A.D. From Hill, The Development of Arabic Numerals in Europe by permission of Oxford University Press, publishers. OUR NUMBER SYSTEM 209 Illustration. The tens\u2019 digit of a two-digit number exceeds twice the units\u2019 digit by 2. The number is 27 more than the new number formed by reversing the digits. Find the number. Tens\u2019 Digit Units\u2019 Digit Number Original number t u lot –u Reversed number 1 u t 10 M -f- i t – 2u = 2 t u ~ 27 -)- 10 M -f- ^ Check: 4 exceeds 2 X 1 by 2. M = 1 t = 4 10 f -f M = 41 41 is 27 more than 14. Exercises on Digits 1 . The units\u2019 digit of a certain two-digit number is twice the tens\u2019 digit. If 18 is added to the number, the digits will be reversed. Find the number. 2. The sum of the digits of a two-digit number is 6, and if its digits are reversed, the new number exceeds the original by 18. Find the number. 3. The sum of the digits of a two-digit number is 7. If the digits are reversed, the new number is 2 more than twice the original number. Find the original number. 4. A two-digit number is 3 times the sum of its digits, but if the digits are reversed, the new number is 45 more than 3 times the sum of its digits. What is the number? 6. A certain two-digit number exceeds the sum of its digits by 18 and exceeds the number formed by reversing the digits by 9. Find the number. 6. If a two-digit number is divided by the sum of its digits, the quotient is 4. If it is divided by 1 more than the sum of its digits, the quotient is 3. Find the number. 7 . A two-digit number is 6 times its units\u2019 digit. If the digits are reversed, the sum of the new number and the original number is 66. Find the original number. 8. The sum of the digits of a two-digit number is 8. If the digits are reversed, the number is not changed. Find the number. 210 SETS OF EQUATIONS USING TWO UNKNOWNS Many other problems of types that you have already solved in this course using only one letter, can be stated more easily in terms of two letters. Illustration 1. Find two numbers whose sum is 26 and whose difference is 10. Using X and y: x y = 26 X – y = 10 Illustration 2. Mr. Head invested $10,000, partly at 5% and partly at 6%, If his annual income was $570, how much had he at each rate? Using X and y: Principal Rate Income X .05 .05 X y .06 .06 y 10,000 570 X -hy = 10000 .05 X + .06 y = 570 Illustration 3. Mr. Merchant has some coffee worth 40j^f a lb. and some v\/orth 25^ a lb. How much of each should he take to make 150 lbs. worth 35^ a lb.? Using X and y: Kind No. of Lbs. Value 40 X 40 X 25 y 25 35 150 35(150) X -h y = 150 40x + 253^ = 35(150) If you use initial letters for the unknowns, you can often write your equations as shorthand without filling the boxes. USING TWO UNKNOWNS Problems 211 Solve these problems using two letters. 1 . To make 200 lbs. of tea worth 40^ a lb., how many pounds of 50^ and of 35)Z^ tea must I mix? 2. From an investment of $8000, part at 4% and part at 5%, Mrs. Dyke has an annual income of $350. How much has she at each rate? 3 . On a trip of 40 mi. Paul walked part way at 4 mi. an hr. and then got a ride the rest of the way at 12 mi. an hr. He arrived in 6 hrs. How far did he walk, and how far did he ride? 4 . Mr. Broker sold 50 bonds for $40,000. Some were $500, and some $1000 bonds. How many of each did he sell? 5. In making up 500 lbs. of mixed nuts to sell for 40(zi a lb. Allan used some nuts worth 50^ a lb. and some worth 25jzi a lb.. How much of each did he use? 6. Emily and Sarah together weigh 200 lbs. On a teeter board they balance when Emily is 6 ft. from the support and Sarah is 4 ft. from it. What are their weights. 7 . Jones lends part of $7200 at 4% and the rest at 5%. The incomes from the two parts are equal. How much has he at each rate? 8. For a high-school entertainment 186 tickets were sold. Adults were charged 50jzi and children 25(2^. If the receipts amounted to $75, how many tickets of each kind were sold? 9 . The difference between two numbers is 32, and ^ of the larger equals of the smaller. What are the numbers? 10 . A man\u2019s daily wages were 4 times his son\u2019s. One week he worked 5 days and his son worked 6 days. Their combined pay was $39. How much did each earn a day? 11. Last week, Mr. Bartley, a salesman, included 6 dinners and 4 lunches in his expense account for which he asked $5.10. This week he charged the company $5.90 for 7 lunches and 5 dinners. What price does he pay for each meal? 12. Smith invested $5000, part at 5% and part at 4%. His yearly income was $220. How much did he invest at each rate? 13. The units\u2019 digit of a two-digit number exceeds the tens\u2019 212 SETS OF EQUATIONS digit by 5. The number increased by 63 equals 10 times the sum of its digits. Find the number. 14 . If the width of a rectangle were doubled and the length were decreased by 3, the figure would be a square whose area would be 60 less than double the area of the rectangle. What are the dimensions of the rectangle? Review Problems 1 . Mr. Stearns has shares of stock A and of stock B. Last week stock A went up $3 a share, and stock B went up $2 a share. Consequently Mr. Stearns made $230. But this week, although stock A went up $1 a share, stock B went down $4 a share, and he lost $110. How many shares of each stock has he? 2. Robert sat 8 ft. from the fulcrum of a seesaw and balanced John who sat 6 ft. from the fulcrum. Then Robert moved back to a distance of 9 ft. from the fulcrum and held his dog weighing 9 lbs. in his arms, and it was necessary for John to move back to a point 1 ft. from the fulcrum to balance him. What are their weights? 3. In mathematics Charles got 72, 90, 70, and 80 in 4 tests. What must he get in the next test to make his average 80 for the 5 tests? 4 . The train dispatcher arranged to send out a local train at 3 A.M. and an express at 6 a.m. When he had decided on the average rates of these trains, he computed that the express must pass the local at 10 a.m. This was too early so he decided to increase the speed of the local by 6 mi. an hr. and have the express overtake it at 1 :30 p.m. What rate had he planned at first for each train? 5. A man invested one sum at 4% and another at 6% which gave him an income of $900 a year. The next year he was able to invest the first sum at 4^% and the second at 7% which increased his income to $1030 a year. How much money had he in each investment? 6. A grocer mixed 60(;i coffee with 35^ coffee and computed that the mixture was worth 50(zi a lb. To reduce it to a value of 45(zi a lb., he then added 50 more lbs. of the cheaper coffee. How many pounds of each did he use at first? REVIEW PROBLEMS 213 7 . The sum of the digits of a two-digit number is 10. If the digits are reversed, the number is increased by 36. Find the number. 8. The sum of the digits of a two-digit number is 13. If 27 is subtracted from the number, the digits will be reversed. What is the number? Solve these problems graphically: 9. A leaves town at 20 mi. an hr. Three hours later B follows at 35 mi. an hr. In how many hours will B overtake A, and how far will they then have gone? 10. An express leaves New Orleans for Atlanta at 9 a.m. traveling 45 mi. an hr. At 11 a.m. the flyer leaves Atlanta for New Orleans and travels 60 mi. an hr. If the distance from Atlanta to New Orleans is 493 mi., how far from Atlanta must the train dispatcher arrange to have them pass? 11. The leading baseball team is now 10 games ahead of the second-place team and is winning just i of its games. The sec- ond team is winning f of its games. After how many more games will the second team overtake the first? 12. Show on graph paper how the point {x, y) moves, if: (a) X y = 8 (b) y = 3 X (c) x \u2014 y = 2 (d) X varies but y always remains 4 (e) y varies but ai; always remains \u2014 2 (\/) X and y both vary, but their sum is always 10 (g) X and y both vary, but y is always 3 more than x Test in Geometry Define: (a) Right angle id) A perpendicular (b) Supplementary an- (e) Perimeter gles (\/) Alternate interior angles (c) Vertical angles 2. What is the sum of the angles of a triangle? 3. If 2 lines are parallel, what fact is true about their corre- sponding angles? 4 . Write the formula for the area of: (a) A parallelogram (c) A trapezoid (&) A triangle (d) A circle ‘214 SETS OF EQUATIONS Test in Solving Equations Solve: 1 . X + y = 1 X \u2014 y = ?> 2. ?,x + 2y = ll X + 2y = ^ 3. 7 X – 4 = 18 3 X -j- 2 y = 30 4. 3x+8>’ = l 2x – = 17 6 . .06 X + .045 3 ; = 90 X – 3; = 100 6 . 5 X \u2014 9 3^ = 54 3 X + 63 ‘ = 21 7. .3 X + 2 3 ; = 13 2x + 1 . 23 ; =26 8 . X + 3^ = 7 9. 2 X – 3 ; = 4 X + 2 3^ \u2014 3 4 5 10. 2 X + 3 ^ \u2014 (x \u2014 3 ^) X + y X – y 22 \u2014 Matching Test in Graphs Wn’le the numbers 1 to 10 in a column, and after each write the letter of the corresponding graph. 1. y = X + 2 6 . 3^=3 2j=0 7. x+J=5 2 . X 3 . X + 2 3; = 4. X = 3 5. X – 3^ = 0 8. X + 2 3; = 0 9. 2x\u2014 33’=6 10. 3 X + 4 3^ = 12 Y^\u2014 -4 4 \/ ^z! \/ “?3 4- -tx -T -7^- -K r “z’ 0 ‘ z – \u201c 7 )’ 1 h 7 c r ^ C 1 1 ^ Y -f- Y Y * s ‘> 3 . s : s s , ”s. T f \u25a0 ‘ w , -3 1 X 01 0 112 3 4’^ -3 01 A A 1 01 2 3 1 2 — f _ i – 1 4 Chapter 10 THE FORMULA APPLIED LITERAL EQUATIONS M r. Long is an electrical engineer and every day he encounters a large number of problems in which he must use the formula C \u2014 R nr to find n. Of course Mr. Long could substitute the values of E, R, r, and C in this formula for each problem and then solve his equation, but Mr. Long is too wise to do all that unnecessary work. He first solves the formula for n, once and for all, and then he has only to substitute to get the answers immediately. Whenever anyone has to use the same formula many times, he always saves labor by first solving it for the unknown letter. It will be easier for us, if before we attempt to solve this formula, we review what we have learned about solving equations. What we have already learned about solving equations. To solve the equation | \u2014 3 = | + 4 means to obtain the answer x = 21. That is, we must get x alone on the left side of the equals sign and the number on the right side. Our example has a 2 and a 3 on the left side that should not be there and an ^ on the right side that should not be on that side. Solving an equation then consists principally 215 216 THE FORMULA APPLIED of getting rid of those quantities that are where we do not want them. To get rid of a quantity, we perform an operation exactly opposite to that which connects the quantity with the un- known letter. Addition and subtraction are opposites. If I add 2 to 7 and then subtract 2, I get back the original 7. If I sub- tract 3 from 5 and then add 3, the answer is again the orig- inal 5. To get rid of a quantity that is added, subtract that amount from both members. X + 4 = 13 . Subtract 4: 4=4 % =9 Courtesy of New York Telephone Co. DIAL TELEPHONE CENTRAL OFFICE What a mathematics problem this electrical engineer had to solve! Think of designing a machine that can automatically select any one of a million telephones for you as you turn the dial at home! LITERAL EQUATIONS 217 To get rid of a quantity that is subtracted, add that amount to both members. Add 7: 1 11 II II to Multiplication and division are opposites. If I multiply 8 by 5 and then divide the result by 5, I get back the 8. If I divide 15 by 3 and then multiply the result by 3, I get back the 15. To get rid of a multiplier, divide both members by that amount. 3 a; = 18 Divide by 3: 3;c 18 3 3 a: = 6 To get rid of a divisor, multiply both members by that amount. Multiply by 5: ^.5.4 a: = 20 Illustration. Solve: ^ ~ 3 = ^ + 4 A D It is generally best to get rid of denominators first. Since one x is divided by 2 and the other by 6, we must multiply by a number that will contain each of these. 6 will do this. Multiply by6:6-|-6*3=6-| + 6- 4 3 X – 18 = + 24 To get rid of 18 from the left side which is subtracted, add 18 to both members. Add 18: 3 a: – 18 = a: + 24 18 = 18 Zx = a: + 42 218 THE FORMULA APPLIED To get rid of the x on the right side which is added, we must subtract x from both members. 3 X = X + 42 Subtract x: x = x 42 We have still to get rid of the 2 that multiplies the x. To do this we must divide both members by 2. T^. . , o 2x 42 Dividing by 2: ^ Y X =21 The literal equation. In a literal equation, some of the letters represent known quantities whose values have not yet been substituted. Except in formulas, where initial letters are used for both known and unknown quantities, it is customary to let the last part of the alphabet represent unknown quantities and the first part known quantities. For example, in ax + dy = c, a, b, and c are supposed to be known, whereas x and y stand for unknown quantities. How to solve literal equations. Actually we solve literal equations in exactly the same way that we have solved other equations. But we must be careful to keep in mind which letter is the unknown quantity. Illustration 1. Solve: ax + & = c. Here x is the unknown letter. The 6 is added. To get rid of it, subtract: ax = c \u2014 b The a multiplies the x. To get rid of it, divide by it. a Check; + & c a c \u2014 h –h c c = c Illustration 2. Solve mx \u2014 n = m \u2014 nx. x is the unknown. We must get all terms containing x in the left member and all terms not containing x in the right member, so n and nx are on the wrong sides. Both are subtracted, so add : mx nx = m n LITERAL EQUATIONS 219 Since x occurs in two places, to find what x is multiplied by, factor. {m n) X ^ m n Check: m ^ n m(l) \u2014 n m \u2014 n{l) m Divide hy m n m n = 1 n = m \u2014 n Class Exercises Solve for the last letter of the alphabet occurring in the equation, and check your answers: 1. X a = h Z. 2x-{-m = lm 2. 5y=6w4-3y 7. ct\u2014 Ad = 2d\u2014 6d Z. mx — n = 2 m n 8. 7x\u2014 b=5b+4:X hv \u2014 k = ha \u2014 k d. mx + k = k am 6. cy\u2014 5=3a \u2014 5 10. to + 4 c = 4 c Optional Exercises 11. ax \u2014 2 b = 2 a \u2014 bx 14. mz \u2014 m = md 12. ax \u2014 ah = a”^ 15. 8 ^ – 15 = 5 fl + 3 ^ IZ. hv \u2014 kv = k \u2014 h 16. miv \u20142mi = mm^ + 2 mj Honor Work 17. az a =\u25a0 a 20. ay a \u2014 \u2014 a^ \u2014 y IZ. TTX + 2 k = A: k 21. ax + 9 b^ -= a”^ -2 bx Id. kx \u2014 1 = X \u2014 k 22. ax b = cx d Using the answer to Exercise 22 as a formula, solve: 23. 7x+5=3x + l 24. hx k = mx + n How to solve the formula for a given letter. Now we are able to solve Mr. Long\u2019s formula for n. C = En R nr Since one term is divided by i? + nr, we shall multiply by that quantity first. C{R + nr) En{R + nr) ~ R nr 220 THE FORMULA APPLIED Getting rid of the common factor from numerator and denominator, we have: C(R + nr) = En Multiplied out: CR + Cnr = En Now remember that n is the unknown quantity, so get n on the left side only. Also get rid of CR from the left side. CR + Cnr = En CR En = En CR Cnr \u2014 En = \u2014 CR To find n, we must divide by the number that multiplies it, so first factor out n to get its multiplier together. n{Cr – E) = – CR Dividing by Cr \u2014 E: This is a satisfactory answer, but as Mr. Long never has problems that give a negative answer, it would be more convenient for him to have the CR positive. So he multi- plies both numerator and denominator by \u2014 1. This gives him: To check, substitute any numbers for E, n, R, and r in the original and find C. Then substitute this in your answer and get back the same value of n you used first. Class Exercises Solve each formula for the letter following it: 1. pv = k V 1. s = ^ at^ 2. p = kvt V Z. L = wh + Ih 3. S = 2Trh r 9. A = i h(B + b) 4:. A = ^bh b 10. 5. A = p + prt t 11. S = 2 Trr\/i + 2 irr^ 6. C = f (F – 32) F 12. S = i n(a + 1) h I a h THE FORMULA IN SCIENCE 221 Find the value of the unknown letters in: 13. \/ = c + (\u00ab – l)d, if \/ = 15 when n = A, and I = 27 when w = 7 14. \/I = i h{B + b), i h = ^ when B = 10, and ^ = 8 when 5 = 4 1^. y = mx + h, if 3^ = 5 when x = 2, and y = 7 when X = & 16. ax^ + by”^ = 33, if 3^ = 5 when x = 2, and 3^ = 1 when a; = 4 Optional Exercises Solve: fl+&+Cr rl \u2014 a r 17. s = \u2014 \u2014 for c 20. S = \u2014 \u2014 for r 2 r \u2014 1 18. i = – + ^ for p 21. F = for m fPP’ gr 19. C = \u2014 for i? 22. r = for s 5 + wr g + s Honor Work These formulas are used in engineering and in science. Solve each for the letter following it: 23. P = t{a \u2014 d)s d 28. C = \u00bb(| – z 24. L _Mt – g 1 29. ^ M n t (1 – ck)Ad 25. Zi _^2 Pi 30. t{a – <\/)S _ ^ d ^ 2 Pi ad a 26. K _ irr^E E 31. M = \/2 180 Cl C 2 27. P = s-\u00ab C 32. j, _ + ^2\u00bb2 mi A r Wi + m 2 The Formula in Science and Engineering 1. To find two unknown electrical resistances, R and r, an 2 n engineer uses the formula 5 + \u00abr = He finds that when he 222 THE FORMULA APPLIED Courtesy of Radio Corporation of America. IN A TELEVISION STUDIO Designing television apparatus is today one of the most interesting problems for the electrical engineer. uses 20 cells {n = 20), the current (C) is .2; and when he uses 60 cells, the current is .3. What were the unknown resistances? 2. An optician can find the focal length (\/) of a lens by measuring the distance {d) from some object to the lens and the distance of the image from the lens (f), and using the formula j = ^ + If he wishes to select a lens having a focal length of 10 in., and holds the lens 30 in. from an object, at what distance should he expect the image to appear? 3. Mr. Waters is building a dam and must determine the pressure it will have to stand. He finds that at a depth {d) of 10 ft. the pressure (P) is 58 lbs. and at a depth of 20 ft. it is 101 lbs. If he uses the formula P \u2022= dw — a, find the value of w and a. 4. A heavy storm has broken down the telephone wires some- where in the woods. Mr. Burns, the company electrician, knows THE GENERAL SOLUTION 223 that the wire has a resistance (R) of 1 for every 100 ft. of wire, so he measures the current through the wire for a voltage (F) of 110, and finds that C = 2. Solve the formula C = ^ for R, and from it find how far from the station Mr. Burns should look for the break. 6. In the formula S = 2 m;\/ + 2^\/H-2 wh, S is the number of square inches of tin used in making a box whose dimensions are I, w, and h. {a) Solve for h. (b) John wants to make a box having a base 7 in. by 5 in. What depth should he make the box if his father will allow him to use only 166 sq. in. of tin? 6. The formula w = is used in radio for changing kilo- cycles to wave length, (a) Solve for k. (b) Dorothy finds that the newspaper gives the wave length of KDKA as 306 M. If her radio dial reads in kilocycles, where should she set the pointer to tune in KDKA? (c) WGY operates on a wave length of 380 M. How many kilocycles does this represent? 7 . The normal weight of a person over 60 in. tall is given by the formula w = \u2014 40), where i is the height in inches. Find w when i is (\u00ab) 64 in. {b) 68 in. (c) 72 in. THE GENERAL SOLUTION Many times throughout this book you had to solve several problems of the same kind but containing different numbers. For such groups of exercises you could have made a formula yourself, and then it would have been necessary only to substitute the numbers of each exercise to get the answer. How to make your own formulas. All you have to do to make a formula is to use letters in place of all numbers in the exercise, and solve. This gives you a general solution in which the numbers of any exercise can be substituted. It is customary to use the first letters of the alphabet, a, b, c, d, etc., in place of known numbers. Then the last letters of the alphabet are the unknowns for which you must solve. 224 THE FORMULA APPLIED Illustration 1. Make a formula for solving exercises like: 7;c+2=4;r + ll Make an equation: ax + b = cx d Transpose: ax \u2014 cx = d \u2014 h Factor: (a \u2014 c)x = d \u2014 b Divide by a \u2014 c: x = a \u2014 c Now to solve any number problem, just substitute back the numbers in your answer. In 7 + 2 = 4 X + 11, fl = 7, 6 = 2, c = 4, and d = 11. Then: ;c = ^ ~l = 3 Illustration 2. How shall I invest $20,000, part at 4% and part at 7%, so as to have an income of $1040? Change to: How shall I invest p dollars, part at a and part at b, so as to have an income of z? Principal Rate Income X a ax p – X b b(p – x) ax + b{p \u2014 x) ax -r bp \u2014 bx ax \u2014 bx {a \u2014 b)x X i i i \u2014 bp i \u2014 bp i \u2014 bp a \u2014 b In this problem a Then x = .04, b = .07, p .04 – .07 .03 20,000, and \/ = 1040. 12,000 Illustration 3. Solve for x First solve for x: Multiply the first by e: Multiply the second by b: Subtract: Divide by ae \u2014 bd: In our exercise a = 3, b = \/ = 6. 13 X (-3) – 3X (-3) – 3x + 2y = 13 Ax \u2014 3 y = 6 ax by = c dx + ey = f aex + bey = ce bdx + bey = bf (ae \u2014 bd)x = ce \u2014 bf ^ ce – bf ae – bd 2, c = 13, d = A, e = – 3, and 2 X 6 _ – 51 _ 2X4 – 17 X THE GENERAL SOLUTION 225 Exercises Solve these problems by first making a formula for all problems of the same type, and then substituting the numbers of the problem given here in your formula. 1. At what price a pound must a dealer sell steak that cost him 24jz^ a lb., to make 331% of the selling price? 2. Solve 5 \u2014 (a: \u2014 3) = 19 for x by first solving the equation ax \u2014 (x \u2014 b) = c. 3. One number is 3 more than another and their sum is 17. 4 . Separate 27 into 2 parts so that the larger is twice the smaller. 6. Solve p – ^ = 3 by first solving – \u2014 ^ = c. 5 10 a b 6. The supplement of an angle is 10\u00b0 more than twice the angle. , 7 . Find x if one of two alternate interior angles of parallel lines is3x \u2014 10 and the other is ;r + 70. 8. Two automobiles, headed in opposite directions, leave a town at the same time, one traveling 30 mi. an hr. and the other 35 mi. an hr. In how many hours will they be 455 mi. apart? 9. A grocer has 60 lbs. of candy worth 80^ a lb. How much candy worth 30jz^ a lb. must he put with it to make a mixture worth a lb.? 10. Find the side of a square whose area is increased 63 when its side is increased 3. The Formula in Business If you studied business arithmetic, you learned three cases of percentage depending on whether you had to find the base, the rate, or the percentage. In algebra, however, it is not necessary to know those separate cases. One formula br = p which you can easily solve for any letter, can be used for all cases. 226 THE FORMULA APPLIED The base is the number on which the percentage is to be computed. i The rate is the per cent or hundredths to be taken. The percentage is the product of the base and rate. In 6% of 300 is 18, .06 is the rate, 300 is the base, and 18 is the percentage. Note that the rate is .06 and not 6. Illustration. Mr. Pell has an income of $1800 a yr. He spends $360 a yr. for rent. What per cent of his income is his rent? Here h = 1800, p = 360, and r is unknown. Solving the formula for r: ^ ^ 1800 = .20 or 20% 1 . Mr. Jones invested $3400 at 4%. \u2022 What interest does he receive? 2. A house worth $12,000 is insured for $7500. For what per cent of its value is it insured? 3. A manufacturing plant valued at $60,000 depreciates at the rate of 5% a yr. Find the amount of depreciation and the value of the property at the end of 1 yr. 4. Mr. Black receives a 2% discount on all bills because he pays them within 10 days. How much does he save in 1 mo. if his bills amount to $5300 for that time? 6. A bankrupt has $18,000 with which to settle a debt of $45,000. What per cent can he pay on claims? 6. The dividend paid by ABC stock is 5% of its present value. If Henry Brown received a dividend check for $650 from this company, how much is his investment worth? 7 . A family could save 10% of their food bill by buying for cash. How much could they save in 1 yr. on an account aver- aging $60 a mo.? 8. Rapid Railway stock pays a dividend of $4 a share. If I can buy this stock for $60 a share, what per cent will I realize on my investment? 9. What per cent can I save by buying 6 cakes of soap for 25jzi if it sells for 5?^ a cake? BUSINESS FORMULAS 227 10. Mr. Reed pays a property tax of $180. If the tax rate is 4%, what is the assessed value of his house? Simple Interest The amount ^ of P dollars invested at rate r simple interest for t yrs. is given by the formula: i4 = jP + Prf 1. Solve this formula for t. 2. In how many years will $120 amount to $148.80 at 6% simple interest? 3. In how many years will $200 double itself at 5% simple interest? 4. Solve for r. 6. What rate must I receive on $1800 so that it will amount to $2286 in 6 yrs.? 6. Solve for P. 7. What sum of money must Robert deposit at 6% simple interest so that he will have $1232 with which to start college 9 yrs. from now? Accounting Capital is the value invested in a business. Assets include all property that belongs to the company and all money owed it. Liabilities are the debts of the company. Capital, assets, and liabilities are related according to the formula: C = A – L 1 . If the accountant does not use negative quantities, can you show why he puts the assets on one side of his book and the liabilities and capital on the other? 2. Find the capital of a corporation if its assets amount to $654,000 and its liabilities amount to $432,000. 3. A corporation, whose capital is $128,000, has assets of $287,000. Find the amount of its liabilities. 228 THE FORMULA APPLIED 4. The liabilities of a corporation are $46,000 and the cap- ital is $78,000. What are its assets? The Formula in Economics 1. The average quantity of money in circulation, M, mul- tiplied by its velocity of circulation, V, plus the average quan- tity of bank checking deposits, M', multiplied by their rate of turnover, V', equals the average unit price paid for goods, P, multiplied by the total volume of trade, T. MV + M'V' = PT {a) Solve this formula for P. (b) Find P if M = $5,000,000,000, V = 20, M' = $40,- 000,000,000, V' = 20, and T = $450,000,000,000. (c) If the amount of money in circulation, M, were in- creased, other things being equal, what effect would it have on the price of goods, P? {d) How would inflation, a great increase in the amount of money in circulation, affect prices? 2. The purchasing power of the farmer, P, equals the price he receives for farm products, P\/, divided by the price he pays for industrial products. Pi. What effect will it have on the purchasing power of the farmer if: (a) P\/ is doubled, but Pi remains unchanged? (h) Pi is doubled, but P\/ remains unchanged? (c) Both Pf and Pi are doubled? d) P\/ is doubled, but Pi is multiplied by 3? The Formula in Installment Payments When you buy goods on the installment plan, you pay interest on the unpaid installments and the cost of handling them, in addition to the cash price of the goods. It is an expensive way of buying goods. The interest that is added to the cash price can be found by using the formula i = 1^ .. where n is the number of monthly install- ments and p is the amount unpaid. Before writing the INSTALLMENT PAYMENTS 229 value of p, a payment down at the beginning should be subtracted. Illustration. A radio dealer sells a certain type of radio for $58 cash. How much interest at 6% should he add if the terms are $10 down and the remainder in 15 installments? Deducting the cash payment, p = 4S, r = .06, and w = 15 _ pr(n + 1) ' 24 _ 48 X .06 X 16 24 = 1.92 1. A real estate dealer sells houses for $7000 cash. How much interest at 6% should he add if the buyer pays $1000 down and the rest in 59 monthly installments? 2. Mr. Farrell borrowed $600 which he is to pay back in 12 monthly installments with interest at 5%. How much interest will he pay? 3. The Realty Building and Loan Corporation lent $5400 on a mortgage to be paid in 40 monthly installments with interest at 6%. What is the total amount of it? Illustration. An automobile dealer offers a car for $200 down and $50 a mo. for 1 yr., or for $735 cash. What rate of interest is he asking on his money? You can find the rate by using the formula: ^ ^ 24 ip + ni – c) n{n l)f where r is the rate, p the first payment, n the number of in- stallments, i the amount of each installment, and c the cash price. Then: _ 24(200 + 12 X 50 – 735) 12 X 13 X 50 = .20 He is asking 20% interest on his money. 4. Mary Coyle bought a fur coat for $30 down and $10 a mo. for 8 mo. If she could have bought the coat for $105.50 tash, what rate of interest is she paying? 230 THE FORMULA APPLIED 5. Mrs. Eldredge bought an electric refrigerator for $25 down and 24 monthly installments of $5 each. The cash price was $125. What rate of interest is she paying? 6. The Real Estate Promotion Company offers a house for $1000 down and $50 a mo. for 10 yrs. If the house would sell for $2432 cash, and the rent is worth $30 a mo., what rate of interest is the company asking? 7. The Royal Radio Company sells a certain radio for $20 down and 12 monthly installments of $5 each. If their cash price is $73.50, what rate are they charging on their money? How much would I save if I borrowed the $53.50 for a year at 6% and paid cash? Interest by the 60-Day Method Mr. Peabody loaned Mr. Yates $300 for 40 days at 6% interest. To find the interest due him, Mr. Peabody uses the formula i = Prt P = 300, r = .06, and t is the time in years. He could express the time as of a year, but this would be in- convenient, so in agreement with other bankers he uses 360 days as 1 yr. For 360 days, i = .06 P. For 60 days (i of 360), i = .01 P. Now to multiply by .01 we move the decimal point two places to the left. This is most conveniently done by draw- ing a line down to represent the decimal point in all the numbers in a column. To find the interest for 40 days, divide 40 into numbers that are factors of 60, as 30 and 10. As 30 days is half of 60 days, divide 3.00 by 2 and as 10 days is i of 60 days, divide 3.00 by 6. Then the sum of the interest for 30 days and for 10 days is the interest for 40 days. 00 60 days 50 30 days 50 10 days 00 40 days SIMPLE INTEREST 231 Illustration. Find the interest on $480 for 77 days at 6%. Factors of 60 that make 77 are 60, 15, divide by 30. To do this, divide by 3, and move the decimal point one place to the right. 4 80 60 days 1 20 15 days 16 2 days 6 16 77 days Class Exercises Find the interest at 6% on: Principal No. of Days 1 . $200 30 2 . 500 45 3 . 450 50 4 . 600 22 6 . 720 36 6 . 870 42 7 . 64 70 8 . 954 52 9 . 840 47 10 . 420 28 Principal No. of Days 11 . $ 740.20 20 12 . 86.50 24 13 . 4362.40 80 14 . 1145 12 15 . 28.67 90 16 . 2463.80 77 17 . 624.60 130 18 . 583.10 14 19 . 82.13 17 20 . 1691.43 243 Optional Exercises When the rate is not 6%, find the interest first at 6%, and then correct it for the proper rate. Illustration. Find the interest on $643.55 for 28 days at 4i%. 6 4355 60 days at 6% 3 2178 2145 30 days 2 days To find the interest for 28 days, find it for 30 and 2, and subtract. 3 0033 7508 28 days at 6% at 1|% Divide by 4, since l|-% is -|- of 6%. Then subtract. 2 2525 at 4|% Keep extra decimal places to the end, and then adjust to the nearest cent. $2.25 is the answer. 232 THE FORMULA APPLIED Principal Time Rate Principal Time Rate 21 . $ 420 40 d. ^% 26 . $ 166.40 2 mo. 7% 22 . 335 70 d. ^% 27 . 486.12 3 mo. 8% 23 . 251 38 d. 5% 28 . 2100.40 5 mo. 5% 24 . 6540 50 d. 3% 29 . 58.25 7 mo. 4% 26 . 63 46 d. 4% 30 . 554.36 1 mo. 2% BANK DISCOUNT Business men, whose credit is good, often give a promis- sory note in payment of a bill. This is a promise to pay a sum of money in a certain number of days, usually 30, 60, or 90, or in a certain number of months. If the man who receives this note needs the money before the note will be paid, he takes it to a bank. The banker deducts interest in advance for the time he must wait for the money and gives the man the remainder. ^300.\u201c^ ^ Jcu \u20221 \u2014 ^ \u2014 * ot S\u2019 cAtJ~ \/CO M , – *Due Q \u2022 COtioev A PROMISSORY NOTE The date on which he takes the note to the bank is the date of discount; the date when the note must be paid by the man who wrote it is the date of maturity; and the time between these two dates is the term of discount. The date on which the note was written is the date of the note, and the amount of money it promises to pay is the face of the note. How to find the date of maturity. If the time is given in months, you simply count the required number of months BANK DISCOUNT 233 ahead and take the same day of that month. For example, a note dated April 18 promising to pay in 3 mo. would mature July 18. If the time is stated in days, you must count the exact number of days from the date of the note. For example, a 60-day note dated June 3 would be due August 2 (27 days left in June, 31 in July, and 2 needed from August to make 60). How to find the term of discount. Count the exact num- ber of days from the date of discount to the date of matu- rity. For example, if a note due August 2 was discounted June 11, the term would be 52 days (19 left in June, 31 in July, and 2 in August). Bankers often use a table for finding the time between dates. Bankers\u2019 Time Table From ANY To THE Same Day of the Next Day OF Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sept. Oct. Nov. Dec. Jan. 365 31 59 90 120 151 181 212 243 273 304 334 Feb. 334 365 28 59 89 120 150 181 212 242 273 303 Mar. 306 337 365 31 61 92 122 153 184 214 245 275 Apr. 275 306 334 365 30 61 91 122 153 183 214 244 May 245 276 304 335 365 31 61 92 123 153 184 214 Jun. 214 245 273 304 334 365 30 61 92 122 153 183 Jul. 184 215 243 274 304 335 365 31 62 92 123 153 Aug. 153 184 212 243 273 304 334 365 31 61 92 122 Sept. 122 153 181 212 242 273 303 334 365 30 61 91 Oct. 92 123 151 182 212 243 273 304 335 365 31 61 Nov. 61 92 120 151 181 212 242 273 304 334 365 30 Dec. 31 62 90 121 151 182 212 243 274 304 335 365 This table gives the time from any day of a month on the left to the same day of the month at the top of a column. To find the time from August 5 to December 5, find August on the left side of the table. Then look across on the same line to the column headed December. The time is 122 days. 234 THE FORMULA APPLIED To find the time from May 8 to September 20, find May on the left and look across to the column headed Septem- ber. The 123 days given there is the time from May 8 to September 8. As there are 12 days more from September 8 to September 20, add 12 to the 123. The time then is 135 days. To find the time from October 29 to March 18, we find 151 days from October 29 to March 29. As March 18 is 11 days earlier, subtract 11. The time is 140 days. Exercises Find the date of maturity and the term of discount for these notes: Date of Note Time Date of Discount 1 . Apr. 27 4 mo. May 10 2 . Jan. 25 2 mo. Feb. 1 3 . Sept. 5 3 mo. Sept. 19 4 . Jun. 24 3 mo. Aug. 27 5 . Jan. 2 4 mo. Mar. 30 6 . Jun. 3 60 d. Jul. 10 7 . Dec. 14 90 d. Jan. 4 8 . Oct. 29 30 d. Nov. 8 9 . May 20 60 d. May 28 10 . Jul. 2 90 d. Jul. 19 How to find the proceeds of a note. The amount of money left after the discount is deducted from the face of the note is called the proceeds. To find the proceeds, we must first find the discount or interest on the face of the note for the term of discount. Then we subtract this dis- count from the face of the note. p=f-fdt where p is the proceeds, \/ the face of the note, d the rate of discount, and t the time in years. COMPOUND INTEREST 235 Illustration. Find the proceeds of a 60-day note for $240 dated April 23 and discounted May 5 at 6%. Date of maturity (60 days after April 23) is June 22. Term of discount (from May 5 to June 22) is 48 days. Discount on $240 for 48 days at 6%. Proceeds = face – discount 2 40 60 d. Face 240.00 48 12 d. Discount 1.92 1 92 48 d. Proceeds 238.08 Exercises Find the proceeds of these notes: Face Date Discounted Rate Time 1 . $ 400 Jul. 2 Jul. 20 6% 60 d. 2 . 350 Aug. 27 Sept. 5 6% 30 d. 3 . 420 Jan. 2 Feb. 1 6% 90 d. 4 . 140 Mar. 4 Mar. 4 6% 30 d. 5 . 3460 May 12 Jun. 24 6% 3 mo. 6 . 240.65 Oct. 29 Nov. 20 H% 2 mo. 7 . 65.49 Jul. 19 Aug. 4 5% 4 mo. 8 . 343.44 Feb. 20 Mar. 1 4% 1 mo. 9 . 1243.66 Apr. 27 May 6 3% 2 mo. 15 d. 10 . 6543.21 Mar. 30 May 1 5i% 5 mo. 20 d. COMPOUND INTEREST In a savings bank interest is computed to certain dates, usually January 1 and July 1. If you leave the interest in the bank, it is added to the principal, and the interest for the next period is computed on this larger principal, and so on. When interest is added once a year, it is said to be compounded annually. When it is added every half year, it is compounded semi-annually; and when it is added every three months, it is compounded quarterly. When interest is compounded semi-annually, the rate for one period is half the annual rate. 236 THE FORMULA APPLIED Illustration. Find the amount of $420 for 3 yrs. at 6% com- pounded annually. Principal at beginning $420. Interest first yr. 25,20 Amount at end of 1 yr. 445.20 Interest second yr. 26.712 Amount at end of 2 yrs, 471.912 Interest third yr. 28.3147 Amount at end of 3 yrs. 500.2267 Answer to the nearest cent $500.23 Exercises Find the amount of: 1. $300 for 2 yrs. at 6% compounded annually 2. $540 for 3 yrs. at 6% compounded annually 3. $200 for 2 yrs. at 6% compounded semi-annually 4 . $784 for 2^ yrs. at 6% compounded semi-annually 6. $100 for 2 yrs. at 4% compounded quarterly How to make the compound-interest formula. You can see that the work of finding the compound interest for a long period of time would be very great. Here is a short cut that reduces the labor to a single multiplication. Later you will learn another short method that can be used for any rate and any time. You have already learned that you can make a general formula by using letters in place of all numbers and solving the problem. Let us then find the amount of P dollars at rate r for n yrs. compounded annually. Principal at beginning P Interest first yr. Pr Amount at end of 1 yr. P -f Pr Factoring P(1 -f r) Notice that P(1 + r) shows that we would have gotten the same result if, instead of finding the interest and adding COMPOUND INTEREST 237 it to the principal, we had multiplied the principal by 1 + r. In that case since P(1 + r) be- comes the new prin- cipal on which the next year\u2019s interest is computed, we ought to be able to get the amount at the end of the second year by multiplying by 1 + r again, and so on. That would give us: Principal at beginning P Amount at end of 1 yr. P(1 4- r) Amount at end of 2 yrs. P(1 4- r)^ Amount at end of 3 yrs. P(1 4- r)\u00ae What would be the amount at the end of 4 yrs.? Of 5 yrs.? Of 10 yrs.? Of yrs.? The formula for the amount of P dollars at rate r for n yrs. compounded annually is: A = P(1 + r)” How to use a compound-interest table. A compound- interest table gives the values of (1 4- r)\u201d, so it is only necessary to multiply by P to find the amount of P dollars. To find (1 4- r)\u201d, look down the column on the left, headed n, until you come to the value you need. Then look across that line to the column under the proper value of r. \u00a9 G. A. Douglas from Gendreau, N.Y. THE DOOR OF A BANK VAULT 238 THE FORMULA APPLIED Illustration. Find the amount of $750 for 8 yrs. at 5% com- pounded annually. Here P = 750, n = 8, and r = 5%. Look down the column headed n until you come to 8. Then fol- low that line across to the column headed 5%. You will find 1.477455. This is (1 + r)\u201d. Multiplying by F = 750, we have A = 1108.09125 or $1108.09. Compound Interest Values of (1 + r)’^’ n 1% 2% 5% 4% 5% 1. 1.010000 1.020000 1.030000 1.040000 1.050000 1.060000 2. 1.020100 1.040400 1.060900 1.081600 1.102500 1.123600 3. 1.030301 1.061208 1.092727 1.124864 1.157625 1.191016 4. 1.040604 1.082432 1.125509 1.169859 1.215506 1.262477 5. 1.051010 1.104081 1.159274 1.216653 1.276282 1.338226 6. 1.061520 1.126162 1.194052 1.265319 1.340096 1.418519 7. 1.072135 1.148686 1.229874 1.315932 1.407100 1.503630 8. 1.082857 1.171659 1.266770 1.368569 1 477455 1.593848 9. 1.093685 1.195093 1.304773 1.423312 1.551328 1.689479 10. 1.104622 1.218994 1.343916 1.480244 1.628895 1.790848 Class Exercises \/\/ interest is compounded annually, find the amount of: P r n F r n 1. $100 6% 5 8. $8421.50 4% 4 2. 452 6% 9 9. 3175.68 5% 10 3. 300 4% 8 10. 7436.54 3% 8 4. 560 3% 6 11. 2107.09 6% 6 5. 287 5% 7 12. 6342.50 2% 10 6. 194 5% 10 13. 5891.65 6% 7 7. 891 2% 8 14. 3230.17 5% 9 Optional Exercises When interest is compounded semi-annually, let n be the number of interest periods {twice the number of years), and let r be the rate for one period the annual rate). If interest is compounded semi-annually, find the amount of: COMPOUND INTEREST 239 Principal Rate Time Principal Rate Time 15. $100 6% 3 yrs. 20. $673.42 6% 3 yrs. 6 mo. 16. 300 4% 4 yrs. 21. 211.54 4% 2 yrs. 6 mo. 17. 685 2% 5 yrs. 22. 583.90 6% 4 yrs. 6 mo. 18. 492 6% 3 yrs. 23. 87.64 4% 2 yrs. 6 mo. 19. 147 4% 2 yrs. 24. 786.59 2% 4 yrs. 6 mo. Honor Work When the time is not an exact number of periods, find compound interest for the greatest number of full periods, and then find simple interest on this amount for the part period. 25. $100 4% 3 yrs. 8 mo. Semi-annually 26. 584 5% 8 yrs. 9 mo. Annually 27. 643 4% 2 yrs. 7 mo. Quarterly 28. 967 6% 4 yrs. 10 mo. Semi-annually 29. In how many years will $100 at 5% compounded annu- ally amount to $134? Review Exercises 1. Find the interest on $842 at 6% for 42 days, 2. Find the amount of $900 at 5% compounded annually for 7 yrs. 3. A note dated May 12 and payable in 90 days is discounted June 18. (\u00ab) Find the date of maturity and the term of discount. {h) If the face of the note is $320 and the rate of dis- count 6%, find the proceeds. 4. The fundamental equation in accounting is A \u2014 L = C. What change takes place in the: {a) Capital, if the liabilities increase, but the assets remain unchanged? {h) Capital, if the assets increase, but the liabilities re- main unchanged? (c) Liabilities, if the capital increases, but the assets remain unchanged? {d) Assets, if the capital increases, but the liabilities re- main unchanged? 240 THE FORMULA APPLIED 6. Use the percentage formula p = br. (a) 18 is what per cent of 90? (b) 48 is 12% of what number? (c) Find 15% of 80. 6. How much money must I place at 6% interest com- pounded annually so that I shall have $1000 in 8 yrs.? 7. Solve for x and y: ax + by = b dx + y = 1 8. (a) Solve for x: mx + k = x \u2014 5 (b) By using your answer as a formula, solve: 5x + 3= x\u2014 5 9. (a) Solve for x: ax b = nx r (b) Using your answer as a formula, solve: 7x + 5 = 5:r + ll 10. (a) Two automobiles d mi, apart are traveling toward each other at rates of ri and ra mi. an hr. In how many hours will they meet? {b) Using your answer as a formula, find the time if d = 260, Ti = 25, and ra = 40. (c) What is the effect on the time: If d only is increased? If Ti only is increased? If d, ri, and ra are all doubled? If d is doubled, but ri and ra are each multiplied by 3? 11. (\u00ab) Henry Jones invested P dollars, part at rate b and part at rate c. His income is d dollars a yr. How much has he at each rate? {b) Using your answer to part \u00ab as a formula, find the amounts at each rate if Mr. Jones invested $6000, part at 5% and part at 6%, and his income was $310. 12. (\u00ab) How much water must Sam add to 2 oz. of a \/?% solution of alcohol to reduce it to an w% solution? (&) Using the answer to part a, find how much water he must add to 16 oz. of a 40% solution to reduce it to a 15% solution. 13. How much must I invest at 5% to yield an income of $1200 annually? $2400 annually? To double the income must you double the investment? 14. The population of S increased 15% in the last 10 yrs. and is now 43,700. What was the population of the town 10 yrs. ago? REVIEW EXERCISES 241 15. Robert Brown sold his car for $580 at a loss of 33^% of the cost. How much did he pay for it? 16. At what price must Cohen Brothers sell a suit bought for $16.80 to make 40% of the selling price? 17. In the compound-interest table, when n increases, does (1 + r)\u201d increase or decrease? 18. If you subtract 1, which is the principal, from each value of (1 -f r)\u201d, the remainder is the interest. When n = 10, is the interest just 10 times what it was when w = 1? 19. As r increases, does (1 -f r)” increase or decrease? When r = 6, is the interest just 6 times the interest when r = 1 (a) if ^ – 1? (6) If w = 10? Solm these formulas for ifte letter following the formula: 20. fd = w f 27. – 20 + 32 \/ t 21. C = 2 irr r 28. D dq + r 22. p=2l + 2w p 23. C – 5 w E 29. Wi _ d^ w 2 d-j^ W 2 R 24. F – ^ h 30. P 3 25. = 2 gs s 31. Q MCl ^ \/x+w\/2 M 26. S – 2 irrh r Solve for x: 32. kx – hk 41. 33. TTX = 11 TT b 34. acx ~ abc 42. 35. – = 3 & 43. s a nx = 36. \u2014 r == 1 a + b 44. 45. a = bx h c = – 37 . ^ = 5 46. x ax -f- &x = a + & 38. – = s 47. 4x-|-8a = 12& X 48. 3x-fm = X AT m 39. ax \u2014 a === 0 49. 2px \u2014 a\u2014px-j-^a 40. nx an = 0 50. 2(x + 2 6) – 10 6 242 THE FORMULA APPLIED 61. 3(x – a) = 2{x + a) 62. – = – a c X \u2014 n 63. 66. ^ + – = 1 P Q 67. = r r s 1 2 a 64. – + \u00ab = \u2014 a: X 68 . 2a 69. ? + I _ ^ a b c 66. a{a \u2014 x) = b{b \u2014 x) Test in Literal Equations: One Unknown Solve Jor x: 1. 3x = b X + a = 7 2 . ax = b 7 . 2 X \u2014 b = r 3 . 5 kx = m 8. 5x\u2014 3n = k 9. ax + 5 m = 3 c X 12. – + ^ = 10 a 2 13. . X 4. p \u2014 ^ 10. i +\/ = S \u2019 d ^ 11. ax -{-b = 3cx 16. 5 c nx 2 kx n 2 k Test in Literal Equations: Two Unknowns Solve for x and y: 1 . X y = a 3 . 3x–Ay = k = h X \u2014 y = 5 4 X \u2014 3 y = m 2 . ax -i- 2y = b 4 . mx + py = q X + y = c 3 X ry = 7 6. ax + by = c X = dy 6. (a + l);t + 2^; = c {a \u2014 )x \u2014 3y = d Test in Solving Formulas Solve for the letter following the formula: 3. F 1. L = 4 as 2. \/?=2a + 26 mv^ r 4 7?= ^ P^ 2wh^ 6. P = a + dgh 6.^ = ^^ 7. C = 8. s = 9. c = 10. a = R + r rl \u2014 a r – 1 h. h{l^ \/j) (^2 – mf)g Chapter 1 1 SQUARE ROOT AND RADICALS J OHN and Robert found the height of a cliff by dropping a stone and noting the time it took to reach the bottom. They used the formula d = substituting the number of seconds for t and getting the height d in feet. Robert then asked, \u201cHow long would it take a stone to fall a mile?\u201d \u201cThat is easy,\u201d said John. \u201cWe shall substitute 5280 ft. for d and solve for \/.\u201d He got this result: = d 16 = 5280 P = 3 ^ t = V330 John wrote t = V330, meaning the square root of 330, for finding the square root is the opposite of squaring, and he knew that to get rid of a square he should do its oppo- site. Then he found the square root of 330. Could you have done it? Let us see how it is done. Investigating square root. How many digits are there in the square of a one-digit number? Of i a two-digit number? Of a three-digit num- 9^ – 81 ber? ^ 10^ = roo If we group the digits in the square by loo^ – I’OO’OO two\u2019s beginning at the decimal point, 999^ ^ 99’80’01 how does the number of groups compare with the num- ber of digits in the square root? Notice that if we separate the square into groups of two digits each, beginning at the decimal point, each group will represent a digit in the square root. 243 244 square root and radicals How to find the square root of a number. Illustration. Find the square root of 3969. First begin at the decimal point, and separate the number into groups of two digits each. The square root of 39 ’69 has two digits. Since 39 is between 36 and 49 (the squares of 6 and 7) the square root of 39 ’69 is between 60 and 70. Call it 60 + x. Look at the square. It is made up of a large square, two rectangles, and a small square. If we subtract the area of the large square, 60^ = 3600 from 39 ’69, we have 369 left for the two rectangles and the small square. Laying the three pieces out to form a single rectangle, we note that its area is 369 and that its length is a little \u2014 \u2018 more than twice the side of the square or 120. 60 (120->^)x 120+x- So we can approxi- mate the width by dividing the area remaining by the length which is twice the answer already found. 369 120 = 3 -f. Then we find the exact length of the rectangle by adding this 3 to 120. Since 123 X 3 = 369, x is exactly 3, and the side of the square, 60 + a:, is 63. g We can arrange the work as follows: Subtract 60^ = 3600 from 3969 : 36 00 The trial divisor is twice 60: 120 | 3 69 Since 120 goes into 369 3 times, add 3 to the answer making 6 3 the answer 63: Add 3 to the trial divisor 36 00 making it 123: 123 | 3 69 Multiply the divisor by the new answer, 3, 3 69 and subtract: Since there is no remainder, the square root is exactly 63. As in long division, unnecessary O\u2019s, such as those after the 36 and the 0 of the 120 may be omitted. Thought Question Is the square root of 1690 exactly 10 times the square root of 169? Explain. APPROXIMATE MEASUREMENT 245 Exercises Find the square root of: 1. 1936 2. 9409 3. 3136 4. 1024 6. 6724 6. 29.16 7. 5.76 8. .8649 9. 12.96 10. .000841 11. 701.7201 12. 26.2144 13. 466.56 14. .351649 15. 1176.49 When a number is not a perfect square, its square root will not come out even, but you can find it to as many decimal places as needed by adding enough O\u2019s after the decimal point to make a group of two figures for each decimal place in the number. Find these square roots to three decimal places: 16. 2 19. 11 22. 38.47 26. 6.4 17. 3 20. 39 23. 4.912 26. 5.8 18. 5 21. 53 24. .617 27. .58 Thought Questions Can the same number have more than one square root? What number is the square root of 49? Is there any number other than 7 that, when multiplied by itself, gives 49? How many square roots then has 49? However, to avoid confusion, we shall agree that when we write the form 4a, we shall mean the positive square root only: that is, V25 is + 5. APPROXIMATE MEASUREMENT How accurately can you measure? Measure the length and width of the picture showing the transit on page 310, and read your result to just as small a fraction of an inch as you are able. Now find the area of the picture. Do not let anyone know your answer until all members of the class have finished. Now, do all of you get the same result? If not, which of you is right, or are all wrong? On how many figures of your answer do most of you agree? What 246 SQUARE ROOT AND RADICALS then is the use of all the other figures if they are probably wrong? Would not the figures of which you are sure, followed by O\u2019s, be just as accurate? You see from this that measurement is not exact. We can think of a line segment as exactly 8 in. long or as exactly equal to another segment, but because we are unable to make instruments that are absolutely accurate, and to read them beyond a certain degree of exactness, the real length of a segment can never be knovi^n. All physical measurements are only approximately true. With an ordinary ruler we cannot measure the length of this page more accurately than to the hundredth of an inch. We read the length as 7.25 in. if, in our judgment, it is more than 7.245 and less than 7.255, that is, if it is nearer to 7.25 than to 7.24 or 7.26. But we cannot read accurately enough to tell if it is 7.249 or 7.252. Therefore, we say that the length is 7.25 in. to the nearest hundredth of an inch. If we used a more accurate instrument than the ruler, we might determine that the length was 7.252 and not 7.251 or 7.253, but we would still be unable to tell whether it differed from 7.252 by 1 or 2 ten-thousandths of an inch. All that our more accurate instrument can do is to move the uncertainty to a different decimal place. Significant figures. Figures obtained by actual meas- urement which indicate the degree of accuracy of the measurement are called significant figures. Sometimes 0 is significant. When we measured with the more accurate in- strument, if we had found that the result was nearer to 7.25 than to 7.251 or to 7.249, we would have written the result 7.250 in. This 0 does not change the result, but it shows that we have measured the length more accurately than to the hundredth of an inch. Consequently it is sig- nificant. In the first measurement, the 7, 2, and 5 are all sig- nificant figures, for the result is accurate to the hundredth of an inch, but if O\u2019s were written after the 5, they would not be significant. APPROXIMATE MEASUREMENT 247 More often, however, O\u2019s are not significant figures, but are used only to determine the decimal point. When a man states, for example, that he expects to save $1000 next year, he does not mean that he will not save $1,143.57, but only that he does not expect to differ from $1000 by many hundred dollars. In this case, the O\u2019s merely determine the decimal point; they show that he does not expect his savings to be around $10,000 or $100. So the number 1000 has only one significant figure. Similarly, the distance from the earth to the moon, 238,000 mi., has only three significant figures, for it may differ from the exact distance by 300 or 400 mi. A number that has O\u2019s to determine the decimal point is called a round number. All figures except O\u2019s are usually significant. We are accustomed to say that a distance is about 200 mi. when we know it is 198 mi., but we do not say it is about 198 mi. when we know it is 200 mi. Computation with approximate numbers. If we find by measurement that the side of a square is 5.4 in., correct to the nearest tenth of an inch, then the diagonal will be 5.4 V2. Now V2 == 1.41421 correct to five decimal places, which gives us 7.636734 as the length of the diagonal. However, 5.4 in. to the nearest tenth of an inch tells us only that the side of the square is between 5.35 and 5.45 in., and may differ from the exact length as much as .04 in. The length of the diagonal then, computed from this number, may differ from the true length by more than .04 in. Consequently it would be not only useless but misleading to give the result 7.636734 when even the 3 in the hundred\u2019s place may be incorrect. Since 5.4 is accurate to only two significant figures, the length of the diagonal cannot be depended on to more than two significant figures. As a rule, in working with approximate measurements, we carry the result to only as many significant figures as the least accurate of the measurements used in obtaining it. 248 SQUARE ROOT AND RADICALS How to round off a number. When we express a number to less significant figures than are given, we call it rounding off the number. When the figure to be omitted is less than 5, we simply drop it, but leave the other figures as they were (or if on the left of the decimal point, we replace it by a 0). When, however, the figure to be omitted is 5 or more, we increase the last figure retained by 1. For example, the value of TT to 7 significant figures is 3.141593. To six significant figures it is 3.14159, for since the 3 is less than 5, we drop it. To five significant figures the value is 3.1416, for since the 9 is more than 5, we increase the 5 to a 6. Exercises Express these numbers to three significant figures: 1 . 25.83 4 . 487.8 7 . 5932 10 . .7699 2 . 3.164 5 . 631.3 8 . 80471 11 . 49999 3 . 59.66 6 . .4885 9 . 24333 12 . 62.99 In these exercises draw a line under O’s that are significant: 13 . 1020 15 . 0.034 17 . 1000 19 . .050 14 . .001 16 . 5.003 18 . 24.0 20 . 30.0 21. Is 0 significant in the date 1940? 22 . Is 0 significant in 5 X 4 = 20? 23 . John measured the length of his room and found it to be 12.84 ft. to the hundredth of a foot. Robert found the width to the nearest foot to be 11 ft. What was the area of their room? 24 . On a map, the distance from A to R is 2.4 in. and from B to C 4.8 in. If the distance from .4 to 5 is 267.6 mi., what is the distance from B to C? Thought Questions 1. The value of tt is 3.14159265. How many significant figures of it would be reasonable in finding the circumference of a circle whose radius is: {a) 5.41? {b) 103.7? (c) 204.0? {d) 124.537? 2. Would you round off the answer in finding the cost of 3847 books at $.93 each? SQUARE ROOT AND THE FORMULA 249 SQUARE ROOT AND THE FORMULA Class Exercises 1. (a) Solve A = for s. (b) Find s to the nearest hundredth when A = 43.4. 2. The pressure on a dam is p = 32 wd^ where w ;s the width of the river and d its depth. {a) Solve for d. (b) Find d to three significant figures if w = 350 and p = 105,000. 3. The velocity of a falling body is given by = 2 gh, where g = 32 and h is the height fallen. {a) Solve for v. (b) Find v to three significant figures when h = 1200 and g = 32. V\u2018i 4. The number of watts of electricity used is w = where V is the voltage and r the resistance. (\u00ab) Solve for v. (b) Find v to two significant figures when w = 61 and r = 370. (c) If V is constant, how does w change as r increases? 6. Another electricity formula is m; = ch. (a) Solve for c. (b) Find c to tenths when w = 330 and r = 800. (c) If c is constant, how does w change as r increases? \u25a0 6. (a) Solve d = ^ aP for t. (b) Find t to the nearest unit when a = 32.4 and d = 10,- 560. 7. (a) Solve A = 6 for e. (b) Find e to as many significant figures as A has, if A = 7.340. 8. The volume of a square prism is F = s%. (a) Solve for s. (b) Find s to tenths if F = 55 and h = 5. Optional Exercises Find the answers to two significant figures: 9. In A =^Trr find r when A = 616 and tt = -y-. 10. In F = Trr% find r when tt = 3.14, h = 11, and F = 628. 250 SQUARE ROOT AND RADICALS 11. In c 2 = -j- ^ 2 ^ find a when c = 38 and h = 23. 12. In = 22, find y when {a) x = 0, (b) x \u2014 1, (c) X = 2, (d) X = 3, {e) x = 4. (\/) Plot these points on a graph. How does the point (x, y) seem to move as x increases? 13. In E = i mv\u2019^, find v when E = 24 and m = 16. 14. The slant height s of a cone is 4- r^. {a) Solve for h. (b) Find k when r = 9.0 and s = 14. Honor Work 16. Any two bodies in the universe attract each other with a force\/ = where k = .00 000 000 000 014, wi and are SATURN AND ITS RINGS These rings are probably made of small particles held in their couise around the planet by the force of gravitation. the weights of the bodies, and d is the distance between their centers. (a) Solve for d. {b) Find d when Wi = 2,000,000, W 2 = 30,000,000, and \/ = .004 (c) Look at the original formula. As d increases, does \/ increase or decrease? THE PYTHAGOREAN THEOREM 251 {d) Is there any body so far from the earth that the force exerted on it by the earth is exactly 0? 16. One answer of a quadratic equation, an equation con- taining x’^, can be found by using the formula: _ \u2014 ft -f. yjb^ \u2014 4 ac \u201c 2^ Substitute the following values in the formula, and find x, to the nearest tenth when it does not come out an even number: (a) a = 2, b = 5, c = 2 (c) a = 1, b = 11, c =- 5 (ft) \u00ab = 3, ft = 8, c = 2 (d) a = 3, b = – 2, c = – 6 If in this formula, a is the coefficient of x% b the coefficient of X, and c the number, solve: (e) + 2 =:0 (g) 2;r2 -6:r – 5 = 0 (f) x^ + 2x – 3 =0 (h) 3 x^ -Sx + 4 = 0 17. When money is placed at compound interest, the amount in 2 yrs. is A = P(1 -f r)^. , (a) Solve for r. (Do not multiply out the parenthesis.) (b) Find r if P = 350 and A = 378.56. 18. A formula used to find IT is X = Find% to 2 significant figures if s = 1 and r = 1. THE PYTHAGOREAN THEOREM Triangle ABC is a right triangle. It has a right angle at C. In this par- ticular triangle, the sides are 3 in., 4 in., and 5 in. What is the area of the . square on 3? On 4? On 5? If you add the squares on sides 3 and 4, | how does the sum com- pare with the square on 5? A triangle is called a right triangle when one of 252 SQUARE ROOT AND RADICALS its angles is a right angle. The side opposite the right angle, the longest side of the triangle, is called the hypotenuse, and the other two sides are the legs. In triangle ABC, AB or c is the hypote- nuse. AC and BC, or b and a, are the legs. The relationship that you found true for the 3, 4, 5 tri- angle, is true for all right triangles. It was known to the an- cient Egyptians, but was first proved by a Greek mathemati- cian Pythagoras about the year -550. In a right triangle, Pythagoras (580-501 b.c.) was a famous Greek Square of the hy- mathematician. He formed a secret society in potenuse equals the southern Italy which worked out many of the theorems of geometry long before Euclid wrote SUUl 01 the squares his epochal book. Although special cases of r loac the theorem that bears his name were used in practical measurement long before his time, he ^ 2 K 2 was the first to prove that it was true for all cases. ^ \u2014 0-^-0 Illustration 1; Find the hypotenuse of a right triangle whose legs are 8 and 15. c = ^289 = 82 -f 152 = 17 = 64 -f- 225 = 289 Illustration 2: Find the leg of a right triangle whose hypot- enuse and other leg are 82 and 80. = c2 – &2 = 822 _ 802 PYTHAGORAS a = V324 = 18 THE PYTHAGOREAN THEOREM 253 Exercises Find the hypotenuse of a right triangle whose legs are: 1. 6 and 8 4. 7 and 24 7. 38 and 43 2. 5 and 12 5. 5 and 7 8. 63 and 57 3. 21 and 28 6. 10 and 10 9. 184 and 362 10. Solve the formula for a, for b. Find the other leg of a right triangle if the hypotenuse and one leg are: 11. 15 and 9 14. 184 and 138 17. 3.96 and 3.12 12. 26 and 10 15. Ill and 94 18. .083 and .075 13. 41 and 9 16. 342 and 341 19. 76.0 and 63.4 The table of square roots. Mr. Brown is a machinist, and uses squares and square roots very often in his busi- ness. It would not pay him to stop to work out a square root every time he needed one, so he uses a table like that on page 254. To find the square root of 43, he looks down the column headed No. until he comes to 43. Then he looks across to the column headed Square Roots. Here he finds that the square root of 43 is 6.557 to four significant figures. Exercises By using the table, find the square root of: 1. 5 3. 57 5. 50 7. 31 2. 8 4. 72 6. 83 8. 95 9. Can Mr. Brown find the square root of .24 from this table? If the first group of two figures is just right of the decimal point, where would you place the decimal point in its square root? 10. To find the square root of .8, would you look opposite 8 or 80? Why? What is the square root of .8? 11. Using the table, find the square root of 4100. 12. How does the square root of 8 compare with that of 2? Can you give a reason for your answer. 254 SQUARE ROOT AND RADICALS Table of Squares and Square Roots No. Squares Square Roots No. Squares Square Roots No. Squares Square Roots 1 1 1.000 34 1,156 5.831 67 4,489 8.185 2 4 1.414 35 1,225 5.916 68 4,624 8.246 3 9 1.732 36 1,296 6.000 69 4,761 8.307 4 16 2.000 37 1,369 6.083 70 4,900 8.367 5 25 2.236 38 1,444 6.164 71 5,041 8.426 6 36 2.449 39 1,521 6.245 72 5,184 8.485 7 49 2.646 40 1,600 6.325 73 5,329 8.544 8 64 2.828 41 1,681 6.403 74 5,476 8.602 9 81 3.000 42 1,764 6.481 75 5,625 8.660 10 100 3.162 43 1,849 6.557 76 5,776 8.718 11 121 3.317 44 1,936 6.633 77 5,929 8.775 12 144 3.464 45 2,025 6.708 78 6,084 8.832 13 169 3.606 46 2,116 6.782 79 6,241 8.888 14 196 3.742 47 2,209 6.856 80 6,400 8.944 16 225 3.873 48 2,304 6.928 81 6,561 9.000 16 256 4.000 49 2,401 7.000 82 6,724 9.055 17 289 4.123 50 2,500 7.071 83 6,889 9.110 18 324 4.243 51 2,601 7.141 84 7,056 9.165 19 361 4.359 52 2,704 7.211 85 7,225 9.220 20 400 4.472 53 2,809 7.280 86 7,396 9.274 21 441 4.583 54 2,916 7.348 87 7,569 9.327 22 484 4.690 55 3,025 7.416 88 7,744 9.381 23 529 4.796 56 3,136 7.483 89 7,921 9.434 24 576 4.899 57 3,249 7.550 90 8,100 9.487 25 625 5.000 58 3,364 7.616 91 8,281 9.539 26 676 5.099 59 3,481 7.681 92 8,464 9.592 27 729 5.196 60 3,600 7.746 93 8,649 9.644 28 784 5.292 61 3,721 7.810 94 8,836 9.695 29 841 5.385 62 3,844 7.874 95 9,025 9.747 30 900 5.477 63 3,969 7.937 96 9,216 9.798 31 961 5.568 64 4,096 8.000 97 9,409 9.849 32 1,024 5.657 65 4,225 8.062 98 9,604 9.899 33 1,089 5.745 66 4,356 8.124 99 9,801 9.950 Square Root in Geometry 1. Find the side of a square equal in area to a rectangle whose length is 342 ft. and whose width is 195 ft. 2. Find the diagonal of a square whose side is 10. 3 . Find the diagonal of a rectangle whose sides are 12 and 18. SQUARE ROOT IN GEOMETRY 255 4. Find to three significant figures the side of a square whose area is 8 sq. ft. 6. ABC is an equilateral triangle whose side is 8. What is the length of the altitude AD if it cuts the base in halves and makes a right triangle ABD? 6. The diagonal of a box is given by the formula d = V\/2 + 10’^ + h-^ Find d when \/ = 12, w = 8, and h = 14. 7. In the square pyramid VABCD, the altitude VH makes a right angle with HK, and H is the center of the square. Find the slant height VK if the altitude VH = 12 and the sides of the square base are each 18. How to find the distance between points on a graph. To find the dis- tance from A to B, complete the right triangle ABC. Then = AC^ -f BC^. But AC = X2 \u2014 Xi and BC = y2 ^ yi. So by the Pythagorean Theorem: = {X2 – XiY + {y2 – yiY Find the distance between these points: 8. (5, 6) and (2, 2) 12. (0, 5) and (0, 0) 9. (8, 5) and (6, 2) 13. (0, 0) and (-3,-4) 10. (7, 6) and (- 5, 1) 14. (2, 0) and (0, 2) 11. (-3, – 5) and (4, – 1) 16. (- 7, 0) and (0, – 6) Y B (a 2\/2 L V’. y, A c \u25a1 1 X 256 SQUARE ROOT AND RADICALS Square Root Applied 1 . A baseball diamond is a square 90 ft. on a side. Find the distance from home plate to second base. 2. Can Dorothy pack her umbrella which is 27 in. long in the bottom of her suitcase which is 23 in. long and 15 in. wide? 3. A vacant lot is 100 ft. by 80 ft. What is the length of a path diagonally across it? 4 . A carpenter is making a gate 4 ft. wide and 3 ft. high. How long should he cut the diagonal piece used to strengthen the gate? 5 . How long a wire will be needed to reach from the top of a pole 16 ft. high to a point on the ground 12 ft. from the foot of the pole? 6. A vacant corner lot is 120 ft. long and 50 ft. wide. What is the length of a straight path running diagonally across the lot joining two opposite corners? What distance is saved by taking the path? 7. A ship sails 30 mi. east and then 40 mi. north. How far is it then from its starting point? Square Root in Science and Engineering 1 . The diameter of a pipe that will carry as much water as 2 pipes whose diameters are di and d^. is calculated from the formula d = Find to the nearest inch the diameter of a pipe that will supply 2 pipes whose diameters are 5 in. and 7 in. Will two 3-in. pipes carry as much water as a 6-in. pipe? 2. The pressure of the air on the wing of an airplane depends on the velocity of the airplane and is given by the formula p = .0005 v^. What value of v is needed to make p equal to 12? 3. The time it takes a pendulum to swmg depends on its length and is given by the formula t = tt How long will it take a pendulum whose length is 42 in. to make a single swing? 4 . When a stone is dropped from a height, the distance fallen at the end of t seconds is d = 16 How long will it take for it to fall 800 ft.? 6. The velocity of water flowing under a head is v = c V64 h. SQUARE ROOT IN ART 257 What is the velocity to three significant figures when c = .500 and h = 14.6? 6. A carpenter must order rafters for the gable of a house. What length rafters will he need if the height of the gable is 12 ft. and the span is 20 ft.? 7. The distance between rivets in a steam boiler is found by means of the formula P + 8 r)(\/? + _ 8r) ^ Find when 10 p = 3 and r = ^. 8. A boat travels 40 ft. across a river while the current carries it 30 ft. down-stream. Draw the path of the boat, and find the distance it has moved. Span Ex. 6 9. A man walks at the rate of 5 mi. an hr. across the deck of a boat that is traveling 12 mi. an hr. What is his actual speed? 10. A force of 60 lbs. is pulling directly north and a force of 80 lbs. directly east. If the resultant force is represented by the diagonal of the rectangle whose sides are as many inches long as the forces are pounds, find the amount of this force. Square Root in Art Artists think that a picture is more beautiful if its length is V2 or V3 or V5 times as long as the width. These shapes are called root 2, root 3, and root 5 rectangles. 1. A landscape painter wishes to make a root 2 picture, so he starts with a square ABCD whose sides AD and CD are each 10 in. Next he takes AC as a radius and C as center and makes an arc cutting CG at E. Finally he draws the rectangl^^ FBCE whose base CE equals the diagonal CA- 258 SQUARE ROOT AND RADICALS {a) Does this give him a root 2 rectangle? Explain. {h) Find the length of CE to the nearest tenth. Is this length V2 times the width 10? 2. When an artist wants a root 3 rectangle, he first m.akes a root 2 rectangle FBCE as explained above. Then he takes its diagonal CF as radius and C as center and draws the arc FG. If FE = 1.00 and CE = 1.41, find CF and CG to the nearest hundredth. Now find the square root of 3 to the nearest hundredth. Does CG equal the V3? RADICAL EQUATIONS Paul was repairing an old Grandfather\u2019s dock. He wanted to make the pendulum just long enough so it would beat seconds. In a book he found the formula t where I is the length of the pendulum. An equation such as this containing a root sign is called a radical equation. Paul had to solve this equation for \/. How could he do it? To get rid of a square root, he must perform its opposite. But what is the opposite of square root? Here is Paul\u2019s work: t Dividing by tt: – TT P Squaring both sides: \u2014 TT^ Multiplying by 32 : TT^ Substituting t = 1 and tt = 3.14, he found I = 3.25 ft. Rule. To get rid of a square-root sign: 1. Get the radical alone on one side of the equation. 2. Square both sides of the equation. 32 259 RADICAL EQUATIONS Thought Question How would you get rid of a cube-root sign (^)? One peculiarity of radical equations is that sometimes they have no answer. Consequently, it is necessary to check, for even though you make no mistake in your work, your result may not be an answer. Illustration. Solve : Squaring: Transposing: Check: Solve and check: 1. Va: = 5 2. V3^ = 6 3. Va: -b 4 = 3 4. V x-2 = 4 6. \u2022yJ2i a: -f- 3 = 6. V3x – 2 – 7 V% – 3 – 4 ;r – 3 = 16 X = 19 V19 I 4 V16 i 4 4=4 Class Exercises 7. 2 8. ^|Ax = S 9. ^% = 4 10 . VxT2 = 3 11. ^|x – 3 =2 12. Vx + 5 = 7 13. V3^ + 4 = 1 14. Vx -f- 4 \u2014 5 = 0 15. -f 2 = 0 16. – 7 = 2 _ 17. ^Jx_= 8 – Vx_ 18. 2 Vx \u2014 9 = ^|x Optional Exercises 19. V2 X + .15 = .5 20. 2 V3 X + .1 =2 21. V.7x – .17 = .5 22. 8 – V3 X – 1 = 5 23. 3 + ^Vx + 4 = 5 24. 7 – 3 Vx + 4 = 2 + 2 Vx -b 4 In these formulas find the value of the unknown letter: 25. t = \/ = 5 and g = 32 . r = yj~ V = 942, TT = 3.14, and r = 10 27- ^ r = 7 and TT = V c = 13 and a = 5 28. c = V \u00ab2 + 62 260 SQUARE ROOT AND RADICALS The Radical Equation in Science 1. The voltage needed to produce a certain amount of heat IWr in an electric iron is F = -W-y- Find the amount of heat pro- duced in an iron if F = 110 volts, r = .05 ohms, and t = 60 sec. 2. The radius of the pump on a fire engine necessary for pumping g gals, of water a sec. is r = where L is the length of the stroke and N is the number of strokes a sec. How many gallons a second will be pumped by an engine if r = 7, L = 10, and N = 21? 3. The diameter of a motor-boat engine\u2019s cylinders to pro- fit duce a certain horse-power is d = What horse-power h has an engine if the diameter d is 10 and the number of cylinders n is 8? 4 . A formula used in magnetism is T when T =22, M = 100, and H = 240. Find I 5. The amount of illumination on your book from a light depends on the distance of the light from your book. If the distance becomes twice as great, the area the light spreads over becomes 4 times as great, and therefore the intensity of illumination becomes only i as great. {a) How many times as much light will fall on the pages of your book if you move up from a distance of 10 ft. from the light to a distance of 2 ft. from it? (b) Ifd 4 find I when d = 5 and k = 50. 6. The radius of a cylindrical vegetable can is given by the formula – \/I What value must r have if h = 3 in.. that F may be 44 cu. in.? So that F may be 30 cu. in.? 7. In the hydraulic press, used for exerting great pressure, the diameter D of the large cylinder is given by the formula REVIEW EXERCISES 261 D = Find the force F if the force\/ = 100 lbs., the diam- eter of the small cylinder d = I m., and the diameter D of the large cylinder = 20 in. 8. A silo for storing feed for cattle is usually a cylinder stand- ing on end. The radius of a silo 10 ft. high that will hold t tons of feed is r tons will it hold? If the silo has a radius of 6 ft., how many Review Exercises 1 . Find the amount of $5840 at 5% compounded annually for 7 yrs. 2. Solve for r; h = \u2014 ^ 12 wv^ 3. The U.S. Marine rule for the diameter of rivets in building ships is 6? = .78 X 12000 n and w is the weight supported. Solve for w. 4 . Solve for x and y: , where n is the number of rivets 3 a: \u2014 5 y = 6 2 a: + 3 y =23 5 . At what price should a furniture dealer sell a chair that cost $18 so that he can make 40% of the selling price? 6. Find V2 to four significant figures. 7 . Express 237,741 to three significant figures. 8. Factor: {a) 3 a: + 6 y (c) 27 \u2014 18 r^w (&) 14 w + 7 {d) 1.5 a; + 4.5 9 . Stock selling for $45 a share pays $3 a share dividend. What per cent interest on the investment does it pay? 10. Find to the nearest hundredth a leg of a right triangle: {a) If the hypotenuse is 16 and the other leg 13. (&) If the hypotenuse is 20 and the other leg 14. (c) If the hypotenuse is 70 and the other leg 47. 262 SQUARE ROOT AND RADICALS 11. Solve for x: (\u0430) 2 = 14 (c) Vx + 4 = 0 (\u0431) V2 – 7 – 5 = 0 (d) + 2 – 3 = 0 12. Find to three significant figures the square root of: (a) 387 (c) 89540 (e) .005 (d) 21.6 (d) 7.194 (\/) I 13. Express to two significant figures: (a) 3.764 (c) 1.085 (e) 8472 (b) .0356 (d) 2.99 (\/) 698.54 14. If c is the hypotenuse of a right triangle whose base and altitude are b and a, then c = + b\u2018^. In these exercises find the unknown side: {a) a = 3, c = 13 (&) b =20, c = 25 (c) a = 8, b = 15 (d) a = 40, c = 41 (e) c = 83, a = 57 (\/) \u00ab = 61, 6 = 54 (g) a = 8.5,c = 11.6 (h) b = .037, c = .042 16. A ladder 20 ft. long has its base 11 ft. from the wall of a building. How far up the building will it reach? 16. A team of small boys wish to lay out a baseball diamond. As their third baseman cannot throw the ball across the regula- tion diamond, what length should they take for a side of their diamond so that the distance from third base to first base will be 80 ft.? Chapter 12 RATIO AND PROPORTION I N his will a man left his wife $12,000 and his son $8000, but when he died, it was found that his estate amounted to only $15,000. How much should each receive? The probate court decided that the estate should be divided in the ratio of 12,000 to 8000. What did it mean? What is the ratio of 12,000 to 8000? Can you divide $15,000 in this ratio? This is but one of the many problems that occur in life, in law, in business, in art, in chemistry, in home economics, in engineering, in music, in physics and mechanics, and in almost every line of human endeavor, in which a knowledge of ratio is helpful. The ratio of two numbers is the quotient obtained by dividing the first by the second. For example, the ratio of 12 to 20 is ^ or f . A ratio then is just a fraction. Its numerator and denominator are called the terms of the ratio. The ratio y is also written in the form a:b. In either 0 form it is read \u201cc is to Since the ratio is a fraction, it may also be expressed as a decimal. For example, 3:8 = f = .375. Exercises Find the ratio of: 1 . 33 to 44 2. 840 to 930 3 . 12 days to 30 days 4 . 8 ft. to 18 ft. 6. $600 to $900 6. i to f 7. t to 1 8. Y to Y 263 264 RATIO AND PROPORTION Express these ratios as decimals correct to three significant figures: 12. i:| 13. 48:16 14. 13:13 9. 5:6 10. 8:15 11. 48:90 Find these ratios to hundredths, and arrange them with the larger ratio first: 16. 5 to 8 or 2 to 3 16. 13:20 or 16:25 17. 256 to 288 or 288 to 320 18. 341i:426f or 256:320 19. Baseball records are expressed as the ratio of the number of games won to the number played. This ratio expressed to thousandths is called the per cent of the team, although it is not the ordinary use of the word \u201c per cent.\u201d Find the ratio or \u201cper cent\u201d for each team in the American League, if the games won and lost in a certain season were as follows: Team Won Lost Team Won Lost Detroit 93 58 Chicago 74 78 New York 89 60 Washington 67 86 Cleveland 82 71 St. Louis 65 87 Boston 78 75 Philadelphia 58 91 20. A basketball team won 7 games out of 12 played. If it should win its next two games, by how much would its percentage be increased? 21. A flag-pole 60 ft. high casts a shadow 45 ft. long. What is the ratio of the height of the pole to the length of its shadow? 22. In 1920 there were 95,000,000 white persons and 10,000,- 000 negroes in the United States. What was the ratio of the whites to the negroes? 23. In a single year the number of births in the United States was 1,600,000 and the number of deaths was 850,000. Find the ratio of the number of deaths to the number of births. 24. A baseball player\u2019s batting average is the ratio of the number of hits to the number of times at bat. It is expressed RATIO 265 Photo hy Charles Phelps Cushing, R.I. Nesmith, N.Y. A WORLD SERIES GAME as a decimal to three significant figures. Find the batting aver- ages of these National League players for a certain year, and arrange them in order, placing the highest of the averages first. Name Team At Bat Hits Harnett Chicago 413 142 Herman Chicago 657 222 Lombardi Cincinnati 333 114 Med wick St. Louis 626 223 Terry New York 596 203 25. A certain angle contains 75\u00b0. What is its ratio to its supplement? 26. A tax rate is the ratio of the amount of money to be raised to the assessed valuation of the town. Find to 3 sig- nificant figures the tax rate if the amount needed is $87,400 and the assessed valuation of the town is $2,460,000. 266 RATIO AND PROPORTION Ratio Problems If two numbers are in the ratio we can represent them by ax and hx, for ^ ^ . Two numbers in the ratio 3:7 are represented by 3 a: and 7 x. Illustration: Find two numbers in the ratio 4:5 whose sum is 162. Let 4 X = the first number 5 X = the second number 4 X + 5 X = 162 Check: X = 18 72:90 = 4:5 4 X = 72 72 + 90 = 162 5 X = 90 1 . Find two numbers in the ratio 5:8 whose sum is 65. 2. Find two numbers in the ratio 7:4 whose difference is 51. 3 . Separate 108 into two parts in the ratio 5:7. 4 . Three numbers are to each other as 3:5:7, and their sum is 45. Find them. 5 . Three numbers are to each other as 3:5:6, and the sum of the first two is 8 more than the third. 6. Find two supplementary angles in the ratio of 3: 1. 7 . Find the angles of a triangle if they are to each other as 3:7:8. 8 . The length of a rectangle is to the width as 7:5, and the perimeter is 132. Find the dimensions. 9 . Two numbers are in the ratio 6:5. If I add 4 to the smaller, they will be equal. Find the numbers. 10 . Find the smallest angle of a triangle if the three angles are to each other as 1:2:2. 11 . In a class of 35 pupils, the ratio of the number of boys to the number of girls is 4:3. How many of each are there? 12 . Find two numbers whose ratio is 4 and whose sum is 40. 13 . Two numbers are in the ratio 3:8. If 7 is added to the smaller, they are then in the ratio 2:3. Find the numbers. 14 . In surveying, the Egyptians divided a rope into 3 sec- tions that were to each other as 3 : 4 : 5. Into what lengths would they divide a rope 90 ft. long? RATIO 267 The Ratio Applied 1 . An average family spends for food and clothing in the ratio 5 to 2. If the Jones family can afford to spend $840 a year for both items, how much should they spend on each? 2. Solder is an alloy of lead and tin in the ratio 2:1. How much of each metal is needed to make 10 lbs. of solder? 3 . How much antimony is needed to make 100 lbs. of type metal, if lead and antimony are used in the ratio 4 to 1? \u25a0 4 . Mr. Wills left $3000 to be divided among his 3 children in the ratio 3:4:5. Find the share of each. 5 . Common salt is composed of sodium and chlorine in the ratio 23: 35. How many grams of salt must be used to obtain 105 g. of chlorine? 6. A common iron ore contains iron and oxygen in the ratio 7 : 3. How many tons of ore must be reduced to produce 35 T. of iron? 7 . An important ore of zinc contains zinc and sulphur in the ratio 2:1. How much zinc can be taken from 600 lbs. of this ore? 8. Two men went into partnership investing $4000 and $6000. During their first year they made a profit of $1200. If they divide this in the ratio of their investments, how much should each receive? 9. I pay $200 a yr. taxes on a piece of property worth $6000. What should another piece of property be worth if it is taxed $240 a yr.? 10 . John Edwards bought a lot for $6000 and built a house on it costing $4000. During the depression the whole property depreciated to $7000. If he wishes to insure the house for its present value, how much insurance should he carry on it? 11 . A house worth $3000 is insured for $2400 with the under- standing that the insurance company will pay such part of the loss as the insurance is of the value of the property. If a fire damages the house to the extent of $800, how much should the company pay? 12. A building is insured for $3000 in one company and for $4000 in another. How much should each company pay if fire injures it to the amount of $2100? 268 RATIO AND PROPORTION 13 . A bank having deposits of $1,000,000 and assets of $600,- 000 failed. Mr. Adams has $800 in his account. How much should he receive on his claim? 14 . Concrete is made of cement, sand, and broken stone in the ratio 1:2:4. How much of each is needed to make 3500 lbs. of concrete? 16 . In a poorer concrete the cement, sand, and stone are used in the ratio 1:3:6. Find the amount of each needed to make 1200 lbs. of this concrete. 16 . In his will Robert Palmer left $20,000 to his wife, $9000 to his son, and $10,000 to his daughter. At his death his estate amounted to $65,000. How much should each receive? 17 . Three partners invested $8000, $10,000, and $15,000 in a business. If a profit of $44,000 is to be divided in the ratio of their investments, how much should each receive? PROPORTION A proportion is a statement that two ratios are equal. For example, is a proportion. We read \u201c8 is to 10 as 12 is to 15\u201d which means that 8 is the same part of 10 that 12 is of 15. We can write 8:10 = 12 : 15. In a proportion the first and last terms are called ex- tremes and the second and third terms are called means. In the above equation 8 and 15 are the extremes, and 10 and 12 are the means. In the proportion ^ ^ ex- tremes are a and d, the means b and c. Historical Note on Ratio and Proportion: Various forms of proportion were used by early writers, Euclid in particular using it in his geometry. Brahmagupta (628) uses a form of it called the rule of three. Most writers gave an arbitrary rule for solving problems, and it was not until about 1500 that a pro- portion was recognized as the equality of two ratios. Inverse proportion and compound proportion were also studied. Prob- lems that we now solve by simple equations were solved by guessing an answer and then correcting it by proportion. Most of the problems were fanciful and absurd. PROPORTION 269 Since a proportion is simply a fractional equation, it can be solved just as any other fractional equation can. Examine the proportion 8:10 = 12:15. What is the product of the extremes? Of the means? Try several other proportions. Is the same relation true? In a proportion, the product of the extremes equals the product of the means. In the proportion | the common denominator is bd. Multiplying by bd, we get ~ which reduces to ad = be. This property is very useful either in testing a proportion or in solving for an unknown term. Illustration 1. Is f = f a true proportion? The product of the extremes is 4 X 9 = 36. The product of the means is 7 X 5 = 35. Since the products are not equal, the proportion is not true. X 5 Illustration 2. Find x i -^ = -g. By the products of extremes and means, we have ^x =20. Therefore, x = 2|-. Class Exercises 1. Determine which of these proportions are true: 2. Find the value of x in these proportions: 33 l-L 2.4 12 12.8 x X 3 270 RATIO AND PROPORTION {m) 3.5 X (\u00ab) I? = r .06 3. Find the ratio x:y if: (\u00ab) 3 :r = 2 3; (b) x = y (c) 5x – 4y = 0 Optional Exercises 4 . Solve for x: (\u00ab) X 2 5 a (i) X \u2014 a _ c 6 3 b X \u2014 b d ip) x – 1 7 (\/) ^ = m U) ax \u2014 1 _ 3 4 2 n bx -}- 2 \u25a0 4 ic) X 3 5 (k) X + a _ r X 3 4 n X + b s id) X 3 c (\/) mx + n _ g 4 \u2014 X 7 ” d px +q h Honor Work 6 . Solve for x: (a) (b) x–2 X 3 X + a X \u2014 b X – 1 X – 2 X \u2014 c X d , . ax + 2 _ ax + 9 bx + 5 ~ bx -3 ax At P _ ax m ^ \u2019 bx + q bx n The Ratio in Business 1 . Brown and Smith formed a partnership, Brown investing $7200 and Smith $9600. The first year their profits were $2800. If they divided in the ratio of their investments, how much did each receive? 2. A town must raise $22,000 by taxes. If the total valuation of the town is $1,000,000 and the value of Henry Dunn\u2019s prop- erty is $25,000, what tax should he pay? 3. A tax rate is the ratio of the amount of money to be raised to the total valuation of the town. If the valuation is $1,600,000 and the amount to be raised is $27,000, find the tax rate to five decimal places. 4. A house worth $12,000 is insured for $8000. If the in- THE RATIO IN BUSINESS 271 surance company pays in the ratio of the amount of insurance to 80% of the value of the house, how much should it pay for a fire loss of $1800? 6. Mr. Dean invested $9000 in stock and receives a dividend of $270. How much must Mr. Love have invested in the same stock if his dividend is $540? 6. As building lots on the same street generally have the same depth back from the street, their values are in the ratio of the number of front feet. If a lot with a frontage of 28 ft. sells for $400, what price should a lot with a frontage of 49 ft. bring? 7 . A cylindrical tank, standing on end, is 8 ft. high and, when full, holds 680 gals, of gasoline. If the number of gallons of gasoline has the same ratio as the depth, how many gallons are in it when it is filled to a depth of 3 ft.? 8. A certain kind of concrete is made of cement, sand, and gravel in the ratio 2:7:11. How much of each will be required to make 400 lbs. of the mixture? 9 . Mrs. Wells made up this family budget when Mr. Wells was getting a salary of $3000 a year: food, $800; clothing, $450; rent, $600; fuel and light, $200; other expenses, $450; and sav- ings, $500. Now Mr. Wells salary has been raised to $3600. If she makes a new budget in the same ratio, how much will she allot to each item? 10. Burns, Sax, and Long form a partnership, investing re- spectively $9000, $7000, and $8000. (a) How should they divide a profit of $8400? (b) If first they take out a salary of $2000 each, how should they divide the remainder? 11. Three men go into partnership, investing 25%, 35%, and 40% of the capital needed. Each man is to receive a salary of $2000, and the remainder of the profit is to be divided in pro- portion to their investments. How should they divide a profit of $17,000? 12. Mr. Dixon had his house insured for $3000 in one fire- insurance company and for $2500 in another. What part should each pay if he has a fire loss of $2750? 13. In a certain life insurance company, a 20-payment life policy at age 22 costs $30.31 annually or $15.42 semi-annually. Is the rate proportional to the time? Which rate is cheaper? 272 RATIO AND PROPORTION Ratio on the Farm 1. A certain mixed feed is made from cracked corn and oats in the ratio of 3:2. How much of each should be used to make 800 lbs. of the feed? Courtesy of International Harvester. AN ELECTRIC CREAM SEPARATOR The farmer pours the milk into the bowl at the top of the separator. It passes down over rapidly whirling discs which cause the cream to move to the center. Notice the cream coming from the spout on the left, and the remainder of the milk from that on the right. 2 . A mash for chickens consists of 3 lbs. of corn meal to 8 lbs. of bran. How much must a farmer use to make 550 lbs. of the mash? 3 . A cream separator breaks up whole milk into 3 qts. of cream to every 13 qts. of skimmed milk. How many quarts of whole milk are needed to produce 75 qts. of cream? 4 . If a separator is set to furnish cream and milk in the ratio 2:9, how many quarts of cream will be obtained from 418 qts. of whole milk? THE RATIO IN COOKING 273 The Ratio in Sewing and Cooking 1 . Directions for making ice cream say, \u201c Use 4 parts ot ice to 1 part of rock salt.\u201d How much of each should Gladys use to make 30 lbs. of the mixture? 2. A recipe for boiled rice, enough for 6 persons, is: 3 qts. of boiling water, 1 c. rice, and 2 t. of salt. Change the recipe to make enough for 10 persons. 3. Blanc mange is made from 2 c. of milk, i c. corn starch, i c. sugar, with a little vanilla, nutmeg, and salt. Disregarding these last ingredients, how many ounces of corn starch are needed for making 2 lbs. of blanc mange? 4 . A food expert analyzed 2 lbs. of oatmeal and found that it contained 2 ^^ oz. water, 5 oz. of protein, 21 oz. of carbohy- drates, and 3i oz. of other materials. Find the per cent of each ingredient. 6. To make fruit ice cream, use 2 c. of fruit juice, 2 c. of sugar, and 4 c. of cream. How many pounds of each will be needed to make 3 lbs. of ice cream? 6. Margaret is making Parker House rolls. The recipe calls for 4 c. flour, 1 t. salt, 6 t. baking powder, 2 T. shortening, and li c. milk. By mistake she put in 5 c. of flour. What amounts of the other ingredients should she take to make the recipe right? 7 . Hazel\u2019s mother finds that the pattern she has been using will make a dress too small for Hazel now. One section of the pattern is 12 in. long and 4 in. wide. If she must make this section ^ in. wider, how much should she lengthen it so as to keep the dress the same shape as that called for in the pattern? 8. This recipe for cream of tomato soup is enough for 7 people. 2 c. stewed tomatoes i t. baking powder 1 qt. milk i t. pepper i c. flour 1 c. butter 2 t. salt Determine the approximate amount of each ingredient to make enough for 5 people. 9 . A recipe for penoche says: Take 2 c. of brown sugar, f c. milk, 2 c. chopped nuts, and a little butter and vanilla. Estimating the last two together as i c., how many pounds of brown sugar will be needed to make 5 lbs. of penoche? 274 RATIO AND PROPORTION 10 . Here is a recipe for plain cookies: 1 c. sugar 1 T. shortening 1 egg i c. milk 1 t. baking powder 3 c. flour If this recipe furnished enough cookies for 4 children, make a recipe that will supply 10 children. 11. Mrs. Hamilton crochets doilies 1 ft. in diameter which she sells for $1 each. A customer wants to know what price she would charge to make her a doily 20 in. in diameter. If the cost is proportional to the square of the diameter, what price should she ask? The Ratio in Science and Engineering 1. Mr. Barnum finds that 3 T. of coal last 40 days. How much coal will he need for 90 days more? 2. Emily saw the lightning strike a tree 2300 ft. away from her, and she heard the crash 2 sec. later. She then timed another flash and found that the thunder came 7 sec. later. How far away was this lightning? 3 . The pressure exerted by the water on a dam is proportional to the depth of the water above it. At a depth of 7 ft. the pressure on the dam is found to be 434 lbs. What is the pressure at a depth of 16 ft.? 4 . A mining engineer finds that the amount of iron in an ore is to the amount of waste material as 4 : 7. How much of the ore will be needed to produce 20 T. of iron? 5 . On a hydraulic jack it is necessary to use a force of 80 lbs. to raise a weight of 1440 lbs. What force will be needed to raise a weight of 3600 lbs.? 6. A water tank in the basement of a building can safely stand a pressure of 50 lbs. a sq. in. When water rises to a height of 34 ft. it causes a pressure of 15 lbs. a sq. in. on the tank. To how high a building can the tank safely supply water? 7 . If five 50- watt electric lights use i kw. of electricity an hr., how much will twelve 50-watt lights use? 8. A steel cable with a diameter of 3 cm. has a tensile strength of 90,000 lbs. If the strength is proportional to the square of the diameter, what size cable is needed to support 160,000 lbs.? THE RATIO IN CHEMISTRY 275 The Ratio in Chemistry and Medicine (Optional) Nearly all of the substances you see around you \u2014 the wood of your desk, the clothing you wear, the food you eat, the medicines you take, and the air you breathe \u2014 are made up of a few simple elements such as carbon, hydrogen, nitrogen, etc. The chemist studies these elements and from them makes new substances that were never known to man before. Here are a few of these elements with the letters that the chemist uses to stand for them, and with the weight of an atom of each. In writing his formulas, the chemist leaves out plus signs, but we shall not do this because it would be too confusing. He also writes the multiplier after the letter in^ stead of before it as we ordinarily do. For example, his formula for soap is K C 16 H 31 O. This means that soap contains 1 part of potassium (potash), 16 of carbon, 31 of hydrogen, and 1 of oxygen. Atomic weight is simply a ratio for comparing the weights of elements. The atomic weight of hydrogen, the lightest element, is called 1. Then the atomic weight of oxygen, 16, means simply that an atom of oxygen is 16 times as heavy as an atom of hydrogen. How to use chemical formulas. Illustration: The farmer needs nitrogen in the soil to make plants grow, so he buys 400 lbs. of nitrate of potash and spreads it on his land. How much nitrogen does this give him? Nitrate of potash is K + N + O 3. Nitrate of potash = K + N + 03 = 39 + 14 + 16 x 3 = 101 Nitrogen = 14. Element Symbol Atomic Weight Carbon C 12 Hydrogen H 1 Iodine I 126 Nitrogen N 14 Oxygen 0 16 Phosphorus P 31 Potassium K 39 Sulphur S 32 276 RATIO AND PROPORTION CHEMISTRY AT HOME This Boy Scout is making tests with his home chemistry outfit. With an outfit like this, any boy or girl can perform many interesting experiments. Atomic Weight No. of Lbs. Nitrate of potash 101 400 Nitrogen 14 X 101 400 14 ~ X X = 55^ lbs. 1. Water is composed of oxygen and hydrogen. How many pounds of water would be needed to supply 6 lbs. of oxygen? Formula for water: H 2 + O 2. Ammonia contains nitrogen, hydrogen, and oxygen. How many pounds of nitrogen are there in 700 lbs. of ammonia? Formula for ammonia: N + H 5 + O 3. Gas from burning sulphur is often used for fumigating. To fumigate a certain room requires 5 lbs. of the gas. How many pounds of sulphur should be burned? Formula for sulphur gas: S + O 2 THE RATIO IN ART 277 4. A chemist wants to prepare 7 lbs. of iodine from iodide of potash. How many pounds of it should he order? Formula for iodide of potash: K -f I 5. The Gold Match Company ordered 6200 lbs. of phosphorus for making matches. How many pounds of phosphate of potash will the chemical company use in making it? Formula for phosphate of potash: K 3 + P + O 4 6. An important acid used in making glass and soap is called sulphuric acid because it is made from sulphur. How many pounds of sulphur are used in making 980 lbs. of this acid? Formula for sulphuric acid: H 2 + S + O 4 7. Hydrogen, a gas used for filling airships because of its lightness, can be obtained from an acid. How many tons of sulphuric acid would be needed to furnish 2 T of hydrogen gas? The Ratio in Art (Optional) Artists find that the most beautiful proportions for a picture are those of the root 2, root 3, and root 5 rectangles. The ratio of the length to the width of these rectangles is as follows: Ratio of Length to Width Root 2 rectangle V2: 1 or 141: 100 Root 3 rectangle V3: 1 or 173: 100 Root 5 rectangle V5:l or 224; 100 1 . How wide should a root 2 rectangle be if it is 18 in. long? 2. What length should a root 3 rectangle be if its width is 20 in.? 3. To fit the page of a book a picture must be reduced in size so that its length becomes 3^ in. To what will the width be reduced if the picture is a root 5 rectangle? 4. The canvas that Rosemary is painting is 30 in. wide. What length should she use if she wants the picture to be a root 2 rectangle? 278 RATIO AND PROPORTION 6. An artist has a canvas 20 in. wide and of proper length for a root 5 rectangle. If he decides to change to a root 3 rectangle, how many inches should he cut from the length? 6. Show that the two definitions of a root 5 rectangle given in the table are consistent; that is, that if the ratio of the length to the width is V5, then for a rectangle whose width is 100, the length should be 224. 7 . A photograph is 7^ in. long and 5 in. wide. If a copy is made from it 3f in. long, what will be the width of this copy? The Ratio in Music (Optional) When you strike a key on the piano, a little hammer hits a string. The string then makes the tone by vibrating back and forth very rapidly. The pitch of the tone depends on the number of times a second the string vibrates. Below is given the scale in the key of C with the numbers of vibra- tions a second that produce those tones. CDEFGA BCDE 256 288 320 34li 384 426| 480 512 576 640 Three or more notes played together produce a chord. This chord will be pleasant or unpleasant depending on the ratio of the vibration rates of the notes played. If the ratio is reduced to its lowest terms, the smaller the numbers the terms of the ratio reduce to, the more pleasant the chord will sound. This applies to the natural scale and not to the modified scale of which you will learn later. In an interval containing only two notes, you may consider the sound a pleasant one if the terms of the ratio are both smaller than 8. Illustrations. The octave C:C = 256:512 = 1:2 is the most harmonious interval THE RATIO IN MUSIC 279 C 256 2 . , ^ C 256 8 G \u201c 384 = 3 D = 288 = 9 Determine which of these are harmonious: 1. D and G 4. C and F 7. F and A 2. E and F 6. B and C 8 . A and C 3. G and B 6. A and B 9. G and A Which is the more harmonious: 10. C and G or C and F? 12. C and E or F and A? 11. C and E or E and A? 13. C and G or G and D? Are these chords harmonious: 14. C, E, G, and C? 16. D, G, and B? 16. F, A, and C? The modified scale. The numbers of vibrations given above are those of the natural scale. The scale actually in use on the piano differs a little from this and is called the modified scale. We shall now discover why this scale was needed. C D EFGABCD o C do re mi fa so! la ti do. >\u00bb D do re mi fa so! fa ti’ do Investigating the modified scale. 1. Find the ratios in the key of C for do: re, for do: mi, for do: fa. 2. Find the ratios in the key of D for do: re, for do: mi, for do: fa. 3. In the natural scale, is the do: re ratio the same in the key of D as in the key of C? Is the do: mi ratio the same? The do: fa? Do they differ much? Do you think that this difference would matter? 280 RATIO AND PROPORTION Now in music, when we transpose from the key of C to that of D, we do not wish to change the tune. We wish to change the pitch only. But in exercises 1 and 2 above we noticed that the ratios in the key of D were not the same as those in the key of C. This would give an entirely different effect and would change both the chords and the tune. To avoid this, the pitch of D is slightly changed on the piano. The chords then are not quite perfect for either C or D, but they are only slightly off and have the ad- vantage of being the same for both keys. Similarly the other notes of the scale are shifted a little in pitch so that they will give the same effect for one key as for any other. Why we have black keys on the piano. If we transpose to the key of E, E becomes do and F ought to become re. Determine what the do: re ratio is in this key. Is this ratio nearly the same as the do: re ratio in the key of C? Do you think that a slight shift of F would make the two ratios equal? What would be the vibration rate of F if the E:F ratio were made equal to the C: D ratio? In answering these questions, you should discover that the E:F ratio differs greatly from the C:D ratio. The difference is so great that if we attempted to shift F enough to make them alike, we would destroy the interval. To overcome this difficulty, F was left where it was, but an extra note was added to the piano scale between F and G and called Fif (F sharp). The ratio E:Fif was made equal to the ratio C:D. For other keys it was necessary to add other extra notes until five in all were added. Because of this modified scale, certain combinations of notes, particularly in minor chords, have been made har- monious, although in the natural scale the terms of their ratios do not reduce to small numbers. For example, D : F, which equalled 27 : 32, has been transformed into the same interval as E: G, which was 5: 6. Note that the ratio 27 : 32 and 5: 6 differ only slightly. SIMILAR FIGURES 281 Exercises In any natural key in music, the tones should be such that the rates of vibration of its do : mi : sol : do are as 4 : 5: 6 : 8. 1 . If an instrument is tuned to such a pitch that middle C vibrates 240 times a sec., find the vibration rates of the E, G, and C above this C. 2. In the key of G, G is do, and D is sol. Using the value of G just found, find the vibration rate of the D on the third line of the treble staff, 3. Two Boy Scouts have bugles tuned to the same pitch. If the only notes that can be gotten out of the bugles are sol, do, mi, sol, can the boys produce a discord? What notes should they play to get the most nearly perfect harmony? 4 . What note other than the octave will sound the most harmonious with C? With F? With G? If you have a musi- cal instrument convenient, check your results by trial. 5. In writing music, it is customary to make the last chord very harmonious so as to leave a pleasant impression, and to make the chord just before it rather inharmonious so as to bring out the contrast. Determine the ratios for the last two chords of \u201cAnnie Laurie\u201d as given here, and compare the harmony produced by them. 6. An instrument maker wishes to make a low-pitched instrument whose key-note, do, vibrates 48 times a sec. At what rates must he make the other notes of the scale \u2014 re, mi, fa, sol, la, ti, do \u2014 vibrate so the ratios will be the same as those for the natural key of C? THE RATIO IN GEOMETRY: SIMILAR FIGURES Any two figures that have the same shape are similar. Examine the picture and its enlargement. Do you think that they are similar? Does each length in one have the same ratio to the corresponding length in the other that any other two corresponding lengths have? How do the 282 RATIO AND PROPORTION Courtesy of Radio Corporation of America. SIMILAR FIGURES Does enlarging the picture change the size of angles? The length of lines? Do you think that the lengths in one picture are proportional to those in the other picture? The picture shows a radio transmission tower. angles made by corresponding lines in the two pictures compare? Two figures are similar if corresponding lines are pro- portional and corresponding angles are equal. Similar figures have many uses. A map of a state is similar to the state; a plan of a house is similar to the floor section of the house; a model or statue is similar to the object modeled, etc. Indirect measurement, such as finding a distance too great to be measured directly or to an .inaccessible point, depends on the fact that corresponding lines in similar figures are proportional. SIMILAR FIGURES 283 Exercises 1 . Here are two similar polygons. If A’B’ is twice as long as AB, how does B’C compare with 5C? If BC is 8 in. long, how long is B’C’l 2 . ABCDE and A’B’C’D’E’ are similar. AB = 4, BC = 3, CD = 6, DE = 2, and EA = 5. Find the lengths of B’C CD’, D’E’, and E’A’ if: (a) A’B’ = 8 (c) A’B’ = 4 (b) A’B’ = 12 (d) B’C = 90 3. A picture is 3 in. long by 2 in. wide. If it is enlarged so that the length is 7^ in., what is the width? A Ex. 4 Ex. 5 4 . The base of a triangle is 5, and its other sides are 3 and 4. Find the sides of a similar triangle whose base is 15. 6. Triangle ABC is similar to triangle A’ B’C’. Find the lengths of the sides not given. AB AC BC A’B’ A’C B’C (a) 8 14 10 12 (b) 3 21 18 24 (c) 9 lOi 5 15 id) 5.4 6.6 5.1 8.5 284 RATIO AND PROPORTION 6. When a line is parallel to one side of a triangle, it cuts the other two sides into proportional segments. If PQ is parallel to LM, then: KP _ KQ PL QM If KP:PL =4:3, what is the ratio KQiQM? 7. (a) If KP = S, PL = 4, and KQ = 10, what is the length of QM? (b) If KP = 12, PL = 8, and KM = 15, what is the length oiKQ? 8. If the base of one rectangle is 3 times that of another rectangle while their altitudes are equal, how do their areas compare? 9. Calling the areas A and A’, write a proportion. Rectangles having equal altitudes are to each other as their bases. 10. The area of a rectangle is 40, and its base is 7. Another c rectangle, having the same altitude, \/ has a base of 21. What is its area? 11. You learned that the circum- ference of a circle is c = 2 Trr. The circumference of a second circle is c’ = 2 Trr’. Obtain a proportion by dividing one equation by the other. Circumferences are to each other as their radii. SIMILAR FIGURES 285 12. The circumference of one circle is 5 times that of another circle. How do their, radii compare? 13. Find c’ if c = 25, 7 = 8, and r’ = 4. 14. Two rectangles A and A’ have the same shape, but each side of A’ is 3 times as long as the corresponding side of A. How many times as large as A is ^’? 16. If A and A’ are similar, how many times as large as A is A’ if (a) s’ = 5 s? (b) s’ = 6 s? Areas of similar rectangles are to each other as the squares of corresponding sides. A:A’ = A’ Exs. 14, 15, 16 16. If A and A’ are similar and s and s’ are corresponding sides, find A’: (a) ^ = 36, s = 2, and s’ = 5 (b) A = 75, s = 5, and s’ = 3 (c) A = 380, s = 3, and s’ = 4.5 17. A and A’ are similar triangles. If s’ is 3 times as long as s, how many times as large as A is A’? 18. In similar triangles A and A’, how many times as large as A is A’, if- (a) s’ =2 si (&) s’ – 5 s? (c) s’ = 4 s? If a triangle grows larger without changing its shape, which increases more rapidly, the sides or the area? 286 RATIO AND PROPORTION Historical Note: Thales of Miletus (640-546 b.c.), a Greek philosopher and scientist, was probably the first to use similar triangles for measuring the height of a pyramid. His prediction of the time of a total eclipse of the sun in 585 b.c. made him famous, but his real claim to fame was the introduction of de- monstrative geometry. Areas of similar triangles are to each other as the squares of corresponding sides. A: A’ = s^:s’^ 19. By what number is the area of a triangle multiplied when each side is multiplied by (a) 3? (b) 6? (c) n? 20. If \u2014 = what is the ratio 4-? s’ 9 A’ 21. If what is the ratio ? A’ 9 s’ 22. If A and A’ are similar triangles and s and s’ are cor- responding sides, find A’ when: (a) A = 12, s = 2, and s’ = 3 (b) A = 9000, s = 30, and s’ = 100 (c) A = 50, s = 5, and s’ = 3 23. A triangle is growing larger. How many times as great has its area become when its sides have each become (a) twice as large? (b) 3 times as large? 24. The area of a circle is given by the formula A = Trr What is the area of a circle whose radius is (a) 1? (b) 2? (c) 3? 26. By what number is the area of a circle multiplied when its radius becomes (a) twice as great? (b) 3 times as great? (c) n times as great? Areas of circles are to each other as the squares of their radii or as the squares of their diameters. AA’ = AA’ = 26. The area of a circle is 100. Find the area of a circle whose radius is 3 times as great. SIMILAR FIGURES 287 27 . If the area of a circle is 28.26 when r = 3, what is the area when r increases to 5? 28 . We learned that the areas of similar rectangles vary as what? The areas of similar triangles vary as what? The areas of circles vary as what? Then if these two polygons K and K’ were similar and if corresponding sides s:s\u2019 = 2:3, what would you expect the ratio K:K’ to be? If we select polygons regular enough so that we can find their areas, or if we cut them into triangles and find their areas by taking the sum of the areas of the triangles, we find that: The areas of similar polygons are to each other as the squares of corresponding sides. 29 . If = 200, s = 10, and s’ = 30, find K’. 30 . If iC = 98, s = 7, and s’ = 5, find K’. 31 . If a polygon grows larger until its sides are 4 times as long as before, by what number is the area multiplied? 32 . Sometimes the area of an irregular figure is determined by comparing its weight with that of a known area having the same thickness. If the weights of K and K are in the ratio 16:25, what is the ratio of their sides? Note: This fact about the areas of similar polygons has many im- portant bearings on physics and astronomy. It is due to the fact that the areas of similar polygons vary as the squares of corresponding sides that light, sound, magnetism, gravitation, and electric forces vary inversely as the squares of distances. 288 RATIO AND PROPORTION The Ratio in Planning the Home The plan shown in the diagram is that of a building 50 ft. long. By making measurements on the plan, answer these questions: 1. What is the width of the building? 2. Find the dimensions of the living-room and of the kitchen. 3. What is the width of the door- way between the living-room and porch? 4. What width is the hall beside the stairway? 6. Is there room enough in the living-room against the wall between it and the kitchen for a piano, the length of a piano being 4^ ft.? 6. If a single width of carpet is to be laid the full length of the hall, how many yards will be required? The Ratio in Life Insurance To compute the amount of premium you should pay for a policy, a life insurance company must know how long you are likely to live. To determine this, they find how many out of a very large num- ber of people will be alive a certain num- ber of years from now. Here is a table showing how many, out of 1000 who reach the age of 10 yrs., will be alive at the ages given. 1. Out of 2000 at age 10, how many will be alive at 30? At 50? At 90? 2. Out of 44,500 at 30, how many will live to 70? To 90? Age Number alive 10 1000 30 890 50 718 70 382 90 16 RATIO FOR THE SCOUT 289 Ratio for the Boy or Girl Scout 1 . Here are a flag-pole and a post, both vertical; that is, if they stand on level ground, they make right angles with the ground. The sun causes them to throw shadows on the ground. Are the sun\u2019s rays parallel? How many equal angles can you find? Are the tri- angles similar? Can you make a propor- tion from w^hich you can find x, the height of the flagpole? 2. To find the height of a tree, some Girl Scouts measured the length of its shadow and then the shadow of the yardstick. If the tree cast a shadow 30 ft. long when the yardstick cast one 2i ft. long, how high is the tree? 3. A vertical 6-ft. pole casts a shadow 5 ft. long at the same time that a build- ing casts one 30 ft. long. How high is the build- ing? 4 . A Boy Scout found the height of a tree as follows: He placed a 2) mirror horizontally on the ground and walked away from the tree until he could see the top of the tree in the mirror. If the mirror was 20 ft. from the tree and 4 ft. from the boy whose eye was 5 ft. from 4he ground, what was the height of the tree? 5. To measure the height of a building, make a right isosceles triangle ABC, and suspend a weight from a corner, A. Hold the 290 RATIO AND PROPORTION triangle at the level of the eye so that the edge AC will be ver- tical (it will be vertical when the plumb-line hangs parallel to the edge) and walk toward or from the building until the highest point B is in line with B and A. Let e be the height of the eye from the ground, d the distance of the observer from the building, and h the height of the building. Then h = d A- e. What is the height of a building if a girl whose eye is ft. from the ground finds that she is 18 ft. from the building? Ex. 6 6. In a camera the size of a picture is to its distance from the lens of the camera as the size of the object is to its distance from the lens. If the camera is 6 in. deep and is held 10 ft. from a bush 4 ft. tall, what height will the picture have? If it is desired that the picture be 4 in. high, how far from the bush should the camera be held? 7. Robert wanted to find the distance across a river from C to B, so he measured the angles A and C with his protractor and drew a triangle on paper having angles A’ and C equal to those angles. He found AC to be 100 ft. If A’C’ is 2 in. and C’B’ is 5 in., what is the distance CR? 8. The panto- graph is an instru- ment for enlarging drawings. If a drawing is traced by point E, a pencil at F will draw the enlarged figure. The ratio of the lines of the enlarged figure at F to those of the figure at E will equal the ratio ol AC to AB. The Girl Scouts have a pantograph in which AC is 18 in. long. How long should they make A 5 so that the copy will be 3 times the original? RATIO IN GEOGRAPHY 291 The Ratio in Geography Maps and plans are similar to the objects of which they are maps or plans. Consequently all corresponding lines are proportional. 1. A map of the United States is drawn to the scale 1 : 12,000,- 000. Explain what this means, and tell what distance 1 in. represents. 2. How far is it from New York to Cleveland if the distance on this map is 2 in. 3. On this map the distance from Seattle to San Francisco is in. What is the distance in miles? 4. On a map of Missouri, in. represent 50 mi. Express the scale of this map as 1 to some number. 5. On this map of Missouri the distance from St. Louis to Jefferson City is 3 in. What is the distance in miles? 6. On a map of the United States, 1 ft. represents 1000 mi. If the area of the map is 3 sq. ft., what is the area of the United States? 7. A geography publisher wishes to illustrate by areas of similar figures the comparative amounts of wheat produced in different countries. If the ratio for two countries is 4:1, what ratio should he use for the sides of his figures? 8. On page 292 is an airways map of the eastern part of the United States. The air-line distance from New York to Chicago is about 700 mi. By measuring distances on the map, find the air-line distance between: (a) Atlanta and Miami {h) Memphis and New Orleans (c) Buffalo and Chicago {d) Boston and Detroit (g) Cleveland and Washington (\/) Philadelphia and Pittsburgh How things change together. Most things with which we have to deal in life change \u2014 your weight, your age, the number of pounds you can lift, the rate you can run, the AIRWAY MAP OF EASTERN UNITED STATES 292 HOW THINGS CHANGE TOGETHER 293 weather, the temperature, the value of your automobile, all change. And usually a change in one thing produces a change in another. A change in your age is usually accom- panied by a change in your weight and your strength. A change in the direction of the wind is accompanied by a change in the weather. And it is true in mathematics and science generally that a change in one quantity produces a change in others. If a car is traveling at a constant speed, the distance it will go depends on the time it travels. The volume of a cube depends on the length of an edge. The length of a circular track depends on its radius. A quantity that changes is called a variable, one that does not change is called a constant. Exercises Name as many as you can of the principal variables on which these depend: 1. The amount of interest that I receive on an investment 2. The number of dozen oranges I can buy for $1 3. The cost of sending a parcel post package 4. The cost of painting my house 5. The distance a train can travel 6. The time a stone, that I throw, will remain in the air 7. The area of a rectangle 8. The number of pounds of grain a bin will hold 9. The number of feet of fence needed to enclose a rectangular lot 10. The length of the shadow cast by a post 11. The weather 12. The amount of money a salesman will earn 13. The number of rolls of wall paper needed for a room 14. The mark you get in algebra 16. The amount of money you will have when you are 30 16. The number of tons of coal needed to heat a building 17. The price of a pair of shoes 18. The amount of money your parents pay for electricity 294 RATIO AND PROPORTION How to make a formula from a table. The change in one quantity produced by a change in another can often be measured. Then the relation of the one variable to the other can be expressed, either as a rule, a formula, a graph, or a table. We have already made formulas from rules, and tables and graphs from formulas. It is not always possible to reverse this and make a formula from a table or graph, but when a formula can be made from a table, as in science, it is very often valuable. Illustration. Make a formula from which this table might have been obtained. The cost c of sending a package weighing p lbs. by parcel post to a certain zone is given in this table: p 1 2 3 4 5 c 10 12 14 16 18 Examining the table, we notice that c increases by 2 for every 1 that p increases. So c = 2 p some number. The number can be found by tracing the table back to the point where p is 0. Since c decreases by 2 as we go toward the left, the term forp =0 would be c = 8. Or we can obtain the num- ber to be added by subtracting twice the value of p from the corresponding value of c, for if c = 2 \/? + a number, the number is c \u2014 2 p. Then our formula is c = 2 p + S. If either of the letters is squared, the solution is much more difficult. Class Exercises In each of these exercises, the values of the second letter depend on those of the first letter. Make a simple formula from which the table could have been obtained. h 0 1 2 3 4 0 T 8 P 1 2 3 4 5 c Is FORMULAS FROM TABLES 295 X 0 1 2 3 4 y 1 3 5 X 9 * X 0 1 2 3 4 y 5 1 T TT X X t 0 1 2 3 4 s 2 X X X 10 w 1 2 3 4 5 T 7 To Ts X X X 1 2 3 4 5 y 1 X X X X t 1 2 3 4 5 d 5 10 15 20 X X 0 2 4 6 8 y 0 4 8 12 16 X 1 3 5 7 9 y X X 30 X 54 P 1 2 3 4 5 c X X 10 11 X n 1 2 3 4 5 P 4 X X 7 X X 0 2 4 6 8 y 3 X X X X X 1 2 3 4 5 y – 1 1 X 5 7 Optional Exercises X 0 1 1 2 3 4 y 11 9 7 5 \u25a0 3 X 0 1 2 3 4 y 5 0 – 5 – 10 – 15 X 2 4 6 8 10 y 3 6 9 12 15 296 RATIO AND PROPORTION X 3 6 9 12 y 3 5 7 9 \u2022 X 2 4 6 8 y – 3 – 6 – 9 – 12 23. X 2 4 6 8 y 4 3 2 1 X 3 6 9 12 y 11 7 \u201d3~ – 1 Honor Work 33. In exercise 31, if you select any 2 pairs of corresponding values on P and V, is it then possible to rearrange them so as to form a proportion? INVERSE VARIATION The Lever You have already used the lever in some of its forms. The teeter board or seesaw, the scissors, the scales for weighing, the crowbar for lifting weights, and the jack for raising your automobile \u2014 all are familiar forms of lever. Here Rosemary and Margaret are riding on a teeter board. The board is resting on a support called the fulcrum. If Rosemary is heavier than Margaret, should INVERSE VARIATION 297 she sit farther from the fulcrum or nearer to it? How should their distances from the fulcrum compare if Rose- mary weighs just twice as much as Margaret? If just 3 times as much? You can see that the heavier girl should sit nearer to the fulcrum. Perhaps you have also dis- covered that the product of the weight and distance of one girl equals the prod- uct of the weight and distance of the other. Formula: Dividing both sides hy w^d^: Notice that in this proportion, the ratio of the d’s has the sub-two on top whereas that of the w\u2019s has the sub-one on top. These quantities are then inversely proportional. Two variables are said to vary inversely when their product is constant. Inverse variation may be expressed by k the equation y = – or xy = k. Thought Questions If V and y vary inversely, how does y change when x grows larger? If y decreases when x increases, do x and y necessarily vary inversely? Philip D. Gendreau, N.Y. THE TEETER BOARD OR SEESAW 298 RATIO AND PROPORTION Exercises 1. Replace the blank with the word directly or inversely: (\u00ab) The amount of interest I shall receive on an investment varies … as the time. {b) The number of cubic feet of air per pupil in a class- room varies … as the number of pupils in the room. (c) The number of pencils Dorothy can buy for $1 varies … as the price of a pencil. (d) The distance Henry can run varies … as the time. (e) The time it will take a car to run 1 mi. varies . . . as its speed. (\/) The income from a piece of property is $500 a yr. The per cent it will pay on the cost varies … as the cost. (g) The cost of a number of pencils varies … as the price of each pencil. 2. Assuming that the quantities not mentioned are constant, state whether the variation is direct or inverse: (a) The distance a car can go varies … as the rate. (b) The time the car takes varies … as the distance. (c) The rate varies . . . as the distance. (d) The rate varies … as the time. 3. Katherine who weighs 60 lbs. is 4 ft. from the fulcrum of a teeter board. If Gladys, sitting at a distance of 5 ft. from the fulcrum, just balances her, what does Gladys weigh? 4. Emily weighs 80 lbs., and Grace weighs 70 lbs. If Grace sits 6 ft. from the fulcrum of a teeter board, how far from it must Emily sit to balance her? 5. It requires 800 lbs. force to raise a rock. William sets his bar so that the fulcrum is 6 in. from the rock and 5 ft. from his hands. With what force must he press to raise the rock? 6. Sarah wishes to find out if the butcher gave her the right weight of meat. She has a 1 lb. weight, but no scales, so she INVERSE VARIATION 299 suspends a light rod and hangs the meat 2 in. from the fulcrum. She finds that the 1 lb. weight, when suspended 5^ in. from the fulcrum, balances the meat. What was n the weight of the meat? . \u2014 f v \u2014 !l .p 7. John wishes to exert a force of 600 Qj lbs. on a rock with a bar 4 ft. long. If vstj he is able to press down with a force of 120 lbs., how far from the rock should he place the fulcrum? 8 . A teeter board is 12 ft. long. Two girls, weighing 70 lbs. and 50 lbs., want to sit on the ends of it. How far from the heavier should the support be placed? 9. Sarah is cutting tough cloth with a pair of scissors. Will it cut more easily when near the end of the scissors or near the fulcrum? Explain. Inverse Variation in Seienee Mr. Boyle compressed the gas in a tank and made this table showing how the volume changed as the pressure increased: P 1 2 3 4 5 6 V 60 30 20 15 12 10 Before looking at Boyle\u2019s formula, see if you can write it yourself. As P increases, how does V change? Are the values of P and V proportional? Inversely proportional? Can V ever become 0? The relation between the pressure and volume of a gas is given by the formula: PV = K This formula shows another way of expressing inverse variation. 1. Solve this formula for P, for V. 2. If V = 10 and P = 8, what is the value of K? 3. If P = 6 and K = 54, what is the value of VI 300 RATIO AND PROPORTION 4 . If F = 16 when P = 9, what is the value of V when P = 24? First find the value of K. 5. Find the value of V when: (a) P = 15, if F = 30 when P = 9 (b) P = 6, if F = 8 when P = 30 (c) P = 11, if F = 9 when P = 8 (d) P = 6.4, if F = 3.6 when P = 4.8 6. In the formula PF = K, by what number is F multiplied when P becomes 5 times as great? The Law of Inverse Squares 1. Since light from a point travels in straight lines, the same amount of light that would strike a figure 1 ft. away will strike at a distance of 2 ft. a similar figure whose corresponding sides are twice as long. How will their areas com- pare? 2. If the same amount of light is spread out over twice as great an area, we say that the intensity is half as great. How would the intensities compare in Exercise 1? 3. When you study your lesson in the evening, if you hold your book only 3 ft. from the light, how many times as great is the intensity of the illumination as it would be if you held the book 12 ft. from the light? 4 . Write a formula for intensity of illumination using \/i and 1 2 for intensities and di and d 2 for distances from the light. 6. As you move your book away from the light, does the intensity of illumination on it increase or decrease? THE STRAIGHT LINE (Optional) The line y = mx Let us make a graph of the equation y = 2 x. First we shall make the table: X 0 1 2 3 4 5 y 0 2 4 6 8 10 THE STRAIGHT LINE 301 Are the corresponding values of x and y proportional? What is the ratio y:x? What is the value of y when x = 3? When x = 5? Are the triangles OAB and OCD similar? Y D 7 \/ – 7 – f 10 7 \/ 1 6 A c X O 1 2 3 4 5 6 y = 2 X What is the ratio The ratio \u2014 ? Are they equal? OA OC Draw any other vertical line from OC to OD, and deter- mine if the ratio for your line is the same as those you have already found. Now make a graph of the equation y = 3 x. Are the values of y and x proportional for this graph also? When you draw vertical lines as above, are the triangles similar? What is the ratio ^ for this graph? Is the ratio the same for this graph y = 3 % as it was for y = 2 x? Plot the graph oi y = 4 x, oi y = x, of y = ^ x. Then determine the ratio ^ for each of these graphs. Can you find any relation between the coefficient of X and the value of the ratio y:x for each of these graphs? Slope of a line. When an equation is in the form y = mx, m, the coefficient of x, is the ratio of the y value to the 302 RATIO AND PROPORTION X value for any point of the line. It is called the slope of the line. Can you see why? The slope of y = mx is m How does the line move as m increases? What is the slope of y = \u20142×1 Draw this line. What do you observe about its location? If 2 is the angle made by the line y = mx with the X axis, how does z change when m grows larger? Can m change without z changing? Can z change without m changing? If m decreases toward 0, how does z change? When m = 0, what is the value of 2 ? When m be- comes negative, what can you say about 2 ? Tangent of an angle. You have noticed that for each value of Z 2 , the ratio ^ or m has a definite value and does X not change if you take large or small triangles, but that if Z 2 grows larger, this ratio grows larger too. This ratio or m is called the tangent of the angle (written tan 2 ). In a right triangle, the ratio of the leg opposite an acute angle to the leg adjacent to the angle is the tangent of that acute angle. B A b C THE STRAIGHT LINE 303 This is an important ratio about which you will learn more in the next chapter. The line y = m jc + b We have just drawn the graph ofy = 2x. Now suppose a number, say 3, were added, and our equation became y = 2 X + 3, what change would it make in the graph? Making a table: X 0 1 2 3 4 5 y 3 5 7 9 11 13 What is the slope of this line? Does adding a number change the slope of a line? Plot the graphs of y = 2 x -j- 4, of y = 2 x + 5, of y = 2x 4- 10. What effect does increasing the number have on the position of the graph? Does the line turn, or does it always make the same angle with the X axis? Does it move up or down? Does it remain parallel to its original position? At what point is the y axis cut by the graph of y = 2 X? Of y = 2 x 4- 3? Of y = 2 X + 5? Of y = 2 x + b? What is the relation between the value of b and the point where the graph cuts the y axis? In the equation y = mx + b> m is the slope of the line or the tangent of the angle that the line makes with the x axis, and b is the point where the line cuts the y axis (the y intercept). Thought Questions BC If AC = 10 and tan A or \u2014 = 2, what is the length of BC’^ o Do you think this a useful method of finding the length of a line? Y \/ T t t 1 10 \/ 13 \/ fz 5 X O 1 2 3 4 6 304 RATIO AND PROPORTION How the line y = mx h moves when b grows larger. If we make graphs on the same axes ot y = x + b ior different values of b, we get these lines. Are they parallel? Can lines having the same slope meet? Let us see: Solving y = mx + 2 and y = mx \u2014 algebraically for their point of intersection we have : y = mx 2 y ^ mx \u2014 1 0 “^ +3 As it is impossible that 0 should equal 3, these equations cannot have a point in common. Consequently, lines having the same slope never meet, that is, they are parallel. In the equation y = mx b,lm does not change, then as b increases, we obtain a set of parallel lines. Inconsistent equations. We have learned that equations are called inconsistent when they have no common solu- tion. Any pair of parallel lines, therefore, have inconsistent equations. Exercises Find the slope and the y intercept for each of these lines: 1. y=2x + b %.2y = Ax + ^ 11.2x-2y = 10 2. y = x + l 1.2y=x + 2 12.2x–y = 7 Z. y = 2x \u2014 A 8. 5y= \u2014 :r-j-7 13. 3y = 12 y = ^x \u20142 9. 2y=\u2014 5 a; \u2014 8 14. ;c-fy = l 5. y=-2A: + 7 10. 3y=-2A;-4 15. 3A;-2y=-5 Write the equation of the line if: 16. m = 2 and b = 0 18. m = \u2014 1 and b = \u2014 5 n. m = \u2014 3 and 6 = 1 19. m = 0 and 6 = \u2014 5 REVIEW EXERCISES 305 Tell which of the following lines are parallel: 20. y ~ 2 X + S 23. y = 3 X \u2014 6 23. y = 3x 21. y = – X -{-2 24. X + 3; = 7 27. 2 – 3^ = 3 22. 3’=\u2014 4 25. 2a:\u2014 3^=9 28. a:+3’=\u2014 3 Draw these lines by plotting the y intercept and using the slope: 29. 3^ === 3 a: \u2014 5 31. 3^ \u2014 a: = 4 33. a: + 3^ = 8 30. j=a:+3 32. 3; + 2a: = -2 34. a: -33^=9 Show how the point {x, y) must move if it always satisfies the equation: 35. .r+ 3^= 4 37. a: + 23; = 10 39. 2 a: + 33^ = 12 36. 3^ = a: 38. a: \u2014 3^ = 5 40. j = 4 Write the equation of a line parallel to the line whose equation is 3 ^ = 2 a: + 1 : 41. And passing through the origin. 42. And having its y intercept equal to 5. Review Exercises 1. Separate 198 into two parts in the ratio 7:11. 2. Separate 153 into three parts in the ratio 3:5:9. 3. Solve for x: 2 = 5 i = 8 7+3 9 ^^^2a:-5_2a:-8 12 ^ a: + 2 ;c – 1 4. Divide an estate of $46,000 among 4 heirs in the ratio 3:4:6:7. 5. ^Show whether these proportions are true: \/ N O-r 14: jf \u201c 21 2.4 42 A 1.4 6. If concrete contains 1 part of cement and 3 parts of sand to 5 parts of gravel, how many pounds of cement will be re- quired to make 450 lbs. of the mixture? 7. Write these equations as proportions: {a) ab = cd (c) PiVi = ^21^2 (&) Widi = W 2 d 2 id) x^ = ab 306 RATIO AND PROPORTION 8. Complete these tables, d being ‘directly proportional to r: r 2 5 12 d 14 r 3 6 10 d 11 9. Complete these tables, r being inversely proportional to \/: (fl) t 6 10 r 15 (ft) t 4 6 9 r 36 10. A cylindrical pail 10 in. high holds 2 gals, when full. How many quarts of oil does it contain when it is filled to a depth of 4 in.? 11. Are the values of y proportional to x in: (a) y = 3 X? (b) y = 2 X 5? (c) x + y = 10? 12. These polygons are similar. Find w, x, y, and z. 13. These triangles are similar. (a) Find side x. (b) If the area of the small triangle is 4, find the area of the large triangle. 14. Three men entered into partnership investing $8000, $10,000, and $12,000. How should they divide a profit of $4740? 15. A recipe calls for 3 c. of flour, 1 c. of sugar, and i c. of milk. How many pounds of sugar are used in making 3 lbs. of the product? 16. Potassium nitrate TK 4- N 4- O 3) is used in explosives REVIEW EXERCISES 307 because of the large amount of oxygen it contains. If K = 39, N ^ 14, and O = 16, how much oxygen is there in 1010 lbs. of potassium nitrate? 17. A picture is in the form of a root 5 rectangle. If the length must be reduced to 4 in. to fit a page, to what will the width be reduced? 18. Which of these musical intervals are harmonious? C = 256, D – 288, E – 320, G – 384, and B = 480. (\u0430) C and D? (c) D and E? (e) D and B? (\u0431) C and E? (d) D and G? (f) B and C? 19. What is the modified scale, and why was it needed on the piano? 20. When you blow into a harmonica, the notes produced are C, E, G, and C. Can you produce a discord by any combination of these notes? 21. On a certain map a distance of 300 mi. is represented by a length of 5 in. What distance does 3 in. represent? How many inches represent a distance of 240 mi? 22. A water pipe has a circumference of 3 in. What is the circumference of another water pipe whose diameter is twice as large as that of the first pipe? 23. FQ is parallel to LM. Find the length of QM: (a) If KP = 4, PL = 5, and KQ = 6 (b) UKP = 7, PL = 3, and KM = 20 (c) If KL = 12, PL – 5, and KM = 9 id) If KP ^ a, PL = b, and KQ = e 24. A side of a triangle is 10. Find the corresponding side of a similar triangle whose area is 9 times as large. 25. Two angles of one triangle are 50\u00b0 and 55\u00b0. Two angles of another triangle are 50\u00b0 and 75\u00b0. Are these triangles similar? How many degrees are there in the other angles? 26. How high is a building that casts a shadow 42 ft. long when a yardstick casts a shadow 42 in. long? 27. If your camera is 5 in. deep, how far from a person 6 ft. tall should you stand to take a picture of him 4 in. high? 28. Tell whether these quantities vary directly or inversely: (fl) The rate a person walks and the distance he goes in a certain time 308 RATIO AND PROPORTION (b) The rate a man walks and the time it takes to go a certain distance (c) The interest rate on an investment and the income from it (d) The interest rate on an investment and the principal that must be invested to produce a certain income 29 . Make formulas showing the relations in these tables: X y 1 5 o| 3 15 4 20 (c) n s 1 5 2 8 3 11 4 14 h 1 2 3 4 (d) r 1 2 3 4 k T T IT T d T T TT ~16 30 . Robert wants to raise a stone with an iron bar. He places the support 4 in, from the stone and pushes down on the bar with a force of 90 lbs. at a distance of 5 ft. from the support. What force is he exerting on the stone? 31 . A straight line passes through the point (0, 0). (a) Urn =3, what is the equation of the line? (b) If it also passes through (3, 3), what is its slope? (c) If it passes through (2, 5), what value has y when X = 4? 32 . Find the slope and the y intercept of these lines: (a) y = 5 X + ^ (b) y = 2x \u2014 7 (c) y = x 33. A moving picture film has pictures f in. by 1 in. If this is thrown on a screen 9 ft. by 12 ft., what size will a man, i in. high in the film, appear on the screen? 34 . Mr. Grant earns $42 a week and saves $8 of if. Find the ratio of his savings to his earnings. If Mr. Cushman, who earns $28 a week, wants to save in the same ratio, how much a week must he save? c 36. Using the formula m = \u2014 \u2014 find the price (1 – p){l – d) at which a dealer must mark a gas stove that cost $31.50 so he can give a discount of 40% and still make a profit of 30% of the selling price. Check by subtracting the discount and profit from the marked price. TESTS 309 Test on. Ratio 1. Divide 48 in the ratio 9:7. 2 . Divide 100 into three parts in the ratio 5:3:2. 3. Write km = w\/) as a proportion. Solve, using the methods of proportion: 4. _5 _ 4 Z 3 6 . ^ X + 3 1 2 7. If 100 ft. of wire weighs 8 lbs., what is the weight of 350 ft.? Are the values of y proportional to those of x in: 8. j – 2 + 5? 9. 3^ – 3 X? IQ. y xl By Ewing Galloway, N.Y. THE TRANSIT Chapter 13 INDIRECT MEASUREMENT SURVEYING INSTRUMENTS T he instrument shown in the picture is a transit. It is used by surveyors and engineers for laying out streets, railroads, boundaries of fields and plots for build- ings, and for other work in which the accurate measure- ment of angles is needed. To measure angles in a vertical plane, the telescope is tipped up or down, and the number of degrees is read on the large circle that you see on this side of the instrument. The long tube just under the tele- scope is a level. It is used to determine the horizontal line 310 SURVEYING INSTRUMENTS 311 from which the angle is to be measured. The telescope can also be turned to the right or left to measure angles in a horizontal plane. The number of degrees is read on a large circle some distance below the telescope. In the center of this circle is a compass by means of which the surveyor can determine the number of degrees from a north and south line. The transit is an expensive instrument, for the surveyor must make his measurements very accurately. In the measurements that you will want to make, however, such as the height of a pole or the distance across a river, an error of a few inches or even feet will not matter. And for this purpose you can make a cheap instrument yourself. How to make a transit. First, make a frame like that shown here. If you make it of a single piece of metal with the ends bent up to form the supports for the telescope, it will be stronger and more usable than if you make it of JS THE TRANSIT FRAME THE TRANSIT TABLE wood. It should have a circular hole in the center for the scale on the frame to come through. Next, fasten a pro- tractor to the frame, as shown at C, and two medicine vials nearly filled with water, as at D and E. Next, take a cheap telescope or just a tube, and put cross wires on one end, as at B. Attach this strongly to an axis that should be pivoted to the frame with a pointer on the end to move over scale C. This pointer must be parallel to the tube, so it will show the angle made by the tube with the horizontal. 312 INDIRECT MEASUREMENT The second part of the instrument, on which the part already made rests, consists of a horizontal circle supported by three legs hinged to its under side. In the center, put two protractors together to form a circle, as at G, and in the center of this circle a small compass. Another design, that is easy to make, places the levels on the second part of the instrument. The telescope is supported in a fork whose shaft passes down through the center of the second part. The pointer is then attached to the shaft of the fork. If you are a little ingenious, perhaps you can make a better transit of your own design. How to make an astrolabe. A much simpler but very useful instrument for measuring angles in a vertical plane is the astrolabe. This instrument was much used by early explorers, and one lost by Samuel Champlain 300 yrs. ago was recently found. If you can get the face of an old clock, you will have ready-made a scale laid off in arcs of 6\u00b0. Attach a ring to one side of it, and if it is too light to hang down satisfac – torily, attach a weight to the op- posite side. Now fasten a small tube so that it will turn around the point through which the axis of the hands passed, and make a pointer at one end of the tube to pass over the scale. Mark 90\u00b0 di- rectly under the ring so that the line from 0\u00b0 to 180\u00b0 will be horizontal. To measure an angle, hold the astrolabe by the ring, and sight along the tube. The number of degrees is then read where the pointer crosses the scale. The as- trolabe can also be used for measuring horizontal angles by being placed on a horizontal table such as the lower part of the transit. SCALE DRAWING 313 How to make a clinometer. Take a sheet of graph paper, and with one corner as a center and a radius of 10 squares, draw a quarter circle. Lay off the arc in degrees as shown in the figure. Paste this sheet on a board or heavy cardboard, and suspend a weight from the center of the circle. When you have sighted along the clinometer, place a finger on the cord before lowering the in- strument. This retains the angle for you so you can read it after lowering the clinometer. The sextant. This instrument is used in navigation both on the sea and in the air. The observer looks at the horizon through his telescope and at the same time sees, an image of the sun or of a star reflected by mirrors. One mirror is attached to a movable part. As this part, with its mirror, is turned, a pointer moves over a scale. When the sextant is adjusted so that the image seems to touch the horizon, the angle of the sun or star above the horizon is shown on the scale. The observer uses this to help locate his position on the earth. SCALE DRAWING Surveyors and engineers often find a distance by draw- ing a figure similar to that to be measured and then finding the distance by proportion. You remem- ber that in similar figures, corresponding lines are proportional. How to make triangles similar. We can make a triangle similar to another triangle if we make two of its angles equal to those of the other triangle. Triangles ABC A 314 INDIRECT MEASUREMENT and A’B’C are similar because \/.A and \/.A’ are each 70\u00b0 and Z-Band Z B’ are each 65\u00b0. Right triangles are similar if we make one acute angle of one of them equal to an acute angle of the other, for since the right angles are equal, they have then two angles of one equal to two angles of the other. In finding heights the right triangle is very useful. Illustration. To find the distance from Ato B across a pond, the Boy Scouts set up a transit at A and sighted across to B. Then they turned the telescope and sighted C. They found from the scale that Z A was 45\u00b0. In the same way they found that ZC equaled 100\u00b0. Next they measured the length AC. This they found to be 280 ft. Next they drew on paper a triangle in which Z.A’ = 45\u00b0, A’C = 2 in., and ZC’ = 100\u00b0. They found that A’B\u2019 was 3|- in. long. Finally, they wrote a proportion and solved it for x. Here is their computation: a: _ 280 ^ 2 Multiplying by = 490 or about 500 ft. Historical Note on Trigonometry: Practical trigonometry goes back to Hipparchus (140 b.c.), a Greek astronomer, who worked out trigonometric tables for use in astronomy. In 1464 Regiomontanus established trigonometry as an independent science. During the past 300 years, the development of algebraic symbols has aided the analytic treatment of trigonometric functions. SCALE DRAWING 315 Scale Drawing for the Boy Scout To solve these problems, draw on paper with protractor and ruler a triangle similar to the one given in the exetcise. Then measure the sides of your triangle, and make a propor- tion using your measurements and the distance given. 1. To find the distance across a river from C to B, two Boy Scouts made a right angle at C, measured AC 240 ft., and found \/.A was then 50\u00b0. Find the distance across the river. As a convenient scale, use 1 in. = 80 ft. ‘H km 2 . To find the height of her church, Emily measured Z A and the distance from A to the church. If the distance was 300 ft. and ZA was 19\u00b0, what was the height of the church? As a scale, use 1 in. = 100 ft. 3. Two Scouts wanted to know the distance across a river, so they measured 2 angles and a side. They found Z A = 60\u00b0. Z C = 90\u00b0, and AC = 300 ft. From a scale drawing find the distance across the river. (The figure is the same as that of Ex. 1.) 4. To measure the distance from a point L on the shore to an island K, Richard found that ZL = 40\u00b0, Z.N = 35\u00b0, and LN = 700 ft. Make a scale drawing using 1 in. = 200 ft., and from it find the distance KL. 6. Chester who is at C observes that an airplane is directly 316 INDIRECT MEASUREMENT overhead at the same time that Tom 800 yds. away at A finds the angle of elevation (ZA) to be 62\u00b0. Choose a suitable scale, and find the height of the airplane from a scale drawing. 6. To find the distance from A to B across a pond, Paul measured AC 300 yds. directly south. Then he found that B was directly west 180 yds. What was the distance from A to B? 7. Find the height of a building AC if BC = 20 ft. and ZB = 70\u00b0. 8. Find the height of a pole AC li BC is 12 ft. and ZB is, 58\u00b0. 9. Find the height of a tree AC if its shadow BC is 30 ft. and ZB is 48\u00b0. Find the distance across the river from A to B if: 10. ZA = 60\u00b0, ZC = 70\u00b0, and AC = 200 ft. 11. ZA = 80\u00b0, ZC = 75\u00b0, and AC = 450 ft. 12. ZA = 70\u00b0, ZC = 60\u00b0, and AC = B 300 ft. 13. A baseball diamond is a square 90 ft. on a side. Make a scale drawing, and find the distance from third base to first. 14. Make a scale drawing of the lot on which your home stands, putting in the outline of the house. 16. Make a floor plan of your house or apartment. 16. Robert told Richard that he could find the distance from island C to island D without leaving the shore. To prove it, he measured a line AB on the shore 500 ft. long at right angles to the lines AC and BD. Then he found that the Z CBA was 40\u00b0 and Z DAB was 50\u00b0. What distance did he find CD to be? TRIGONOMETRY 317 TRIGONOMETRY: USE OF THE TRIANGLE IN MEASUREMENT Mt. Everest in Asia is 29,002 ft. high. The moon is 238,000 mi. away. The area of the United States is 3,026,789 sq. mi. These are all well-known facts, but how were they determined? No one has yet climbed to the top of Mt. Everest. Certainly no one has been to the moon. And the 3,000,000 sq. mi. in the United States were not determined by stretching a tape measure around the country side. Evidently some other way of doing this was found. We shall now learn how such measurements are made. Have you ever watched a surveyor at work? He uses a transit * for measuring angles of triangles, and then from these triangles he determines the distances. He could, and sometimes does, make a scale drawing, but for much of his work this method is not accurate enough. So he uses the method you are to learn now. Because they have no stationary place for setting up a transit, the ship captain on the sea and the aviator in the air determine their location by using another instrument, the sextant. To understand how they work, we must understand triangles. Some facts we already know about angles and triangles. 1. An angle is formed when two lines meet. We generally read it by a capital letter placed near its vertex. 2. We may think of Z A as having been formed by turning the line AB around point A from the position AC to the posi- ^ tion AB. The size of the angle depends on the amount of this turning and not on the length of the sides. * An inexpensive transit and sextant, accurate enough for your measurements, is made by the Educator Toy Co., Bronxville, N.Y. 318 INDIRECT MEASUREMENT 3. The number of degrees in an angle can be read by- placing the vertex at the center of a circle that is divided into degrees and reading the numbers where the sides cross the circle. 4. The sum of the angles of a triangle is a straight angle or 180\u00b0. 5. A right angle equals 90\u00b0. 6. The sum of the two acute angles of a right triangle is’ 90\u00b0. 7. Vertical angles are equal. 8. Alternate interior or corresponding angles of parallel lines are equal. 9. Two triangles are similar if two angles of one of them equal two angles of the other. 10. Two right triangles are similar if an acute angle of one equals an acute angle of the other. 11. The corresponding sides of similar triangles are proportional. Reviewing the Triangle 1. If the three angles of a triangle are all equal to each other, how many degrees are there in each of them? In triangle ABC, Z. C is the right angle. 2. How many degrees are there in ZB ii ZA equals: 30\u00b0 ? 45\u00b0? 25\u00b0? 38\u00b0? 57\u00b0? 89\u00b0? 3 . If ZA grows larger, how does Z B change? 4 . Find the number of degrees in Z A if: ZA = ZB Z A is 20\u00b0 larger than Z B Z A is 4 times as large as Z B Z A is 10\u00b0 more than twice Z B Z A is to Z 5 as 7 to 11 id the number of degrees in each angle of a triangle if they are in the ratio 2:3:4. In the ratio 11:16:9. 6. Find the number of degrees in each angle of the triangle ABC if ZB IS, 30\u00b0 more than Z A, and Z C is 10\u00b0 more than twice ZB. TRIGONOMETRY 319 A new way to use ratios to find distances. Paul wished to find the distance across the Otter Creek from C to B, so he measured a distance of 100 ft. in the direction C^l at right angles to CB, and then he found with a protractor that Z A was 78\u00b0. He next drew carefully on paper a right AA’B’C, having Z.A’ equal to 78\u00b0, and measured C’B’ and A\u2019C. He found that, if he made A’C 2 in. long, CB’ was about 9.4 in. long. Paul concludes that, since CB’ is 4,7 times as long as A’C, CB must be 4.7 times as long as HC or 470 ft., because AABC and A’B’C are similar. Now Paul measured only one line on the ground and two short lines on paper. And he measured the lines CB’ and A’C on paper only to determine the ratio of CB to AC. Evidently, if he had known this ratio beforehand, he would have needed to measure only one line. Since all right triangles, having an Z A equal to 78\u00b0, are similar, this ratio could be computed once for all for a 78\u00b0 angle. Therefore it would not be necessary to draw a triangle on paper every time we wished to find the ratio. Then, however large or small \/ABC is, this ratio will remain unchanged so long as the size of Z A remains unchanged. But if Z A grows larger or smaller, the ratio will change with it. This ratio is, therefore, called 3. function of Z A. In A ABC, in which Z C is a right angle, the ratio of the leg opposite an acute angle to the leg adjacent to that angle is the tangent of the angle, written tan A or tan B. Tan A =- = opposite leg b adjacent leg Tan B = – = oPPOsite leg a adjacent leg The tangent of B is called the cotangent of A, written cot A. a! C B 320 INDIRECT MEASUREMENT Exercises 1. Using a protractor, draw a right triangle having Z A equal to 10\u00b0. Measure a and h carefully and compute the value of tan 10\u00b0 to the nearest hundredth. 2. From the same triangle compute tan 80\u00b0. 3. By drawing a triangle having LA equal to 20\u00b0, compute tan 20\u00b0 and tan 70\u00b0. 4 . Using the same method, fill the blanks in the following table: Angle Tan Angle Tan 10 \u00b0 50 \u00b0 20 \u00b0 60 \u00b0 30 \u00b0 70 \u00b0 40 \u00b0 80 \u00b0 6. As the angle increases from 10\u00b0 to 80\u00b0, does its tangent increase or decrease? 6. Find tan 45\u00b0 without measuring lines. 7 . Henry finds that a wire from the top of a telegraph pole touches the ground 20 ft, from the foot of the pole and makes an angle of 50\u00b0 with the ground. Using your table, find the height of the pole. If we know any side of a right tri- angle and an acute angle, we can find the other two sides by using the tan- gent ratio, but sometimes this method would be very inconvenient, as in the following example: Robert is flying his kite. After let- ting out 100 ft. of string, he notices that the string makes an angle of 40\u00b0 with the ground. Assuming that the string is straight, how high is the kite? Robert says that he could determine the height very easily if he knew the ratio of a to c. TRIGONOMETRY 321 Evidently to solve problems such as this, we need an- other ratio, that of the opposite side to the hypotenuse. . In a right triangle, the ratio of the side opposite an acute angle to the hypotenuse is the sine of the angle, written sin A or sin B. Sin A – ^ c hypotenuse Sin B \u2014 ^ c hypotenuse The sine of B is called the cosine of A, written cos A. Notice that in both the sine and the tangent, the side opposite the angle is always the numerator. If the denom- inator is the hypotenuse, the ratio is the sine; if it is not the hypotenuse, the ratio is the tangent. Exercises 1. Draw a right triangle having Z A equal to 10\u00b0 and meas- ure a and c carefully. From their lengths, find sin 10\u00b0 to the nearest hundredth. 2. From the same triangle, find sin 80\u00b0. 3. By drawing right triangles and measuring their sides as in the last group of exercises, fill the blanks in the following table: Angle Sin Angle Sin 10\u00b0 50\u00b0 20\u00b0 60\u00b0 30\u00b0 70\u00b0 40\u00b0 80\u00b0 4 . Using the value from your table, find the height of Robert\u2019s kite. 5. Make a table for cosines like that for sines in Exercise 3. How can you find the cosine of an angle from a table giving only sines of angles? 322 INDIRECT MEASUREMENT 6. John has a ladder 12 ft. long on which he wishes to climb to the eaves of his house. If for safety, the ladder must make an angle of 70\u00b0 with the ground, how high will it reach? Use your table. How to use the tables. You have already made a two-place table of the sines and tangents of angles at intervals of 10\u00b0. In the same way you could have filled in these functions for angles differing by one degree. On page 324 such a table is given. In the first column is the number of degrees in the angle. In the second column, opposite the number of degrees, is the sine of the angle, and in the fourth column the tangent of the angle is given. Illustration 1. Find tan 18\u00b0. Look down the column headed degrees until you come to 18. The number .3249 found in the tangent column opposite 18 is the answer. Illustration 2. Find sin 37\u00b0. The required answer is the num- ber .6018 found in the sine column opposite 37. Exercises Use the table on page 324 for these exercises: 1 . Verify the following: {a) sin 13\u00b0 = .2250 {d) sin 86\u00b0 = .9976 ib) tan 77\u00b0 = 4.3315 {e) cos 65\u00b0 = .4226 (c) tan 24\u00b0 = .4452 (\/) cos 20\u00b0 = .9397 2 . Find the value of the following functions: (\u00ab) tan 25\u00b0 (\/) tan 5\u00b0 (k) sin 10\u00b0 iP) tan 3\u00b0 {b) tan 43\u00b0 (g) sin 31\u00b0 (1) sin 68\u00b0 iq) cos 12\u00b0 ic) sin 16\u00b0 (h) sin 45\u00b0 (m) tan 70\u00b0 (r) tan 41\u00b0 id) cos 37\u00b0 (i) tan 1\u00b0 (n) tan 89\u00b0 (s) sin 72\u00b0 ie) sin 75\u00b0 U) cos 51\u00b0 (0) cos 83\u00b0 (0 sin 56\u00b0 TRIGONOMETRY 323 3 . Verify these angles: (a) sin ;c = .7660 = 50\u00b0 (c) tan x = 28.6363 ;c = 88\u00b0 (b) sin X = .9962 x = 85\u00b0 (d) tan x = .5543 x = 29\u00b0 4 . Find the value of x for these functions: {a) sin X = .4540 {d) sin :r = i (&) tan X = .7002 {e) sin x = .8090 (c) tan X = 2.1445 (\/) tan x = 1 6. For what values of x is sin x nearly equal to tan xl 6. For what value of x do sin x and tan x differ most? 7. For what value of x is tan x about twice sin xl 8. Is the sine of a 60\u00b0 angle twice the sine of a 30\u00b0 angle? Is tan 80\u00b0 twice tan 40\u00b0 ? 9. In a right triangle, the side opposite an angle of 64\u00b0 is . . . times the side adjacent to the angle. 10 . In a right triangle, the side opposite an angle of 22\u00b0 is . . . times the hypotenuse. 11 . In the right triangle ABC, \u00ab is 3 and b is 4. (a) tan A is . . ip) sin A is . . (c) side c is . . . . a-3 {d) tan B is . (e) sin B is . (\/) cos A is . 12 . In the right triangle ABC, ZA = 45\u00b0. Without the table find tan A, tan B, sin A, sin B. Angle of ( elevation. Henry, who stands on the ground, is looking up at James, who is at an upstairs window. The angle x that the line of sight from Henry to James makes with the horizontal line is called the angle of elevation. Angle of depression. The angle y that the line of sight from James to Henry makes with the horizontal is called the angle of depression. Trigonometric Table Angle Sin Cos Tan Angle Sin Cos Tan 1 \u00b0 .0175 .9998 .0175 46 \u00b0 .7193 .6947 1.0355 2 \u00b0 .0349 .9994 .0349 47 \u00b0 .7314 .6820 1.0724 3 \u00b0 .0523 .9986 .0524 48 \u00b0 .7431 .6691 1.1106 4 \u00b0 .0698 .9976 .0699 49 \u00b0 .7547 .6561 1.1504 6 \u00b0 .0872 .9962 .0875 60 \u00b0 .7660 .6428 1.1918 6 \u00b0 .1045 .9945 .1051 61 \u00b0 .7771 .6293 1.2349 T .1219 .9925 .1228 62 \u00b0 .7880 .6157 1.2799 8 \u00b0 .1392 .9903 .1405 63 \u00b0 .7986 .6018 1.3270 9 \u00b0 .1564 .9877 .1584 64 \u00b0 .8090 .5878 1.3764 10 \u00b0 .1736 .9848 .1763 66 \u00b0 .8192 .5736 1.4281 11 \u00b0 .1908 .9816 .1944 66 \u00b0 .8290 .5592 1.4826 12 \u00b0 .2079 .9781 .2126 67 \u00b0 .8387 .5446 1.5399 13 \u00b0 .2250 .9744 .2309 68 \u00b0 .8480 .5299 1.6003 14 \u00b0 .2419 .9703 .2493 69 \u00b0 .8572 .5150 1.6643 15 \u00b0 .2588 .9659 .2679 60 \u00b0 .8660 .5000 1.7321 16 \u00b0 .2756 .9613 .2867 61 \u00b0 .8746 .4848 1.8040 17 \u00b0 .2924 .9563 .3057 62 \u00b0 .8829 .4695 1.8807 18 \u00b0 .3090 .9511 .3249 63 \u00b0 .8910 .4540 1.9626 19 \u00b0 .3256 .9455 .3443 64 \u00b0 .8988 .4384 2.0503 20 \u00b0 .3420 .9397 .3640 66 \u00b0 .9063 .4226 2.1445 21 \u00b0 .3584 .9336 .3839 66 \u00b0 .9135 .4067 2.2460 22 \u00b0 .3746 .9272 .4040 67 \u00b0 .9205 .3907 2.3559 23 \u00b0 .3907 .9205 .4245 68 \u00b0 .9272 .3746 2.4751 24 \u00b0 .4067 .9135 .4452 69 \u00b0 .9336 .3584 2.6051 26 \u00b0 .4226 .9063 .4663 70 \u00b0 .9397 .3420 2.7475 26 \u00b0 .4384 .8988 .4877 71 \u00b0 .9455 .3256 2.9042 27 \u00b0 .4540 .8910 .5095 72 \u00b0 .9511 .3090 3.0777 28 \u00b0 .4695 .8829 .5317 73 \u00b0 .9563 .2924 3.2709 29 \u00b0 .4848 .8746 .5543 74 \u00b0 .9613 .2756 3.4874 30 \u00b0 .5000 .8660 .5774 76 \u00b0 .9659 .2588 3.7321 31 \u00b0 .5150 .8572 .6009 76 \u00b0 .9703 .2419 4.0108 32 \u00b0 .5299 .8480 .6249 77 \u00b0 .9744 .2250 4.3315 33 \u00b0 .5446 .8387 .6494 78 \u00b0 .9781 .2079 4.7046 34 \u00b0 .5592 .8290 .6745 79 \u00b0 .9816 .1908 5.1446 36 \u00b0 .5736 .8192 .7002 80 \u00b0 .9848 .1736 5.6713 36 \u00b0 .5878 .8090 .7265 81 \u00b0 .9877 .1564 6.3138 37 \u00b0 .6018 .7986 .7536 82 \u00b0 .9903 .1392 7.1154 38 \u00b0 .6157 .7880 .7813 83 \u00b0 .9925 .1219 8.1443 39 \u00b0 .6293 .7771 .8098 84 \u00b0 .9945 .1045 9.5144 40 \u00b0 .6428 .7660 .8391 86 \u00b0 .9962 .0872 11.4301 41 \u00b0 .6561 .7547 .8693 86 \u00b0 .9976 .0698 14.3007 42 \u00b0 .6691 .7431 .9004 87 \u00b0 .9986 .0523 19.0811 43 \u00b0 .6820 .7314 .9325 88 \u00b0 .9994 .0349 28.6363 44 \u00b0 .6947 .7193 .9657 89 \u00b0 .9998 .0175 57.2900 46 \u00b0 .7071 .7071 1.0000 324 TRIGONOMETRY 325 Notice that the angles of elevation and depression are the same size, for they are alternate interior angles of the two horizontal lines. The only difference is the position of the observer. If you are below looking up, it is an angle of elevation, but if you are above looking down, it is an angle of depression. How to find distances by using the tangent and sine. If you know the length of one side of a triangle and the ratio of that side to another side, you can find the length of the other side. Therefore, to find a side, you must choose that function of the angle which is the ratio of the known side and the side to be found. Illustration. 1. li A A = 40\u00b0 and AC = 200 ft., find BC. You must select that function oi A A that is the ratio of BC to AC. That function is tan A. So write: B Since Z A is 40\u00b0, tan A is .8391, so we have Now solve your equation for the unknown side, and perform the multiplication and division. – 200 X .8391 = 167.82 = 168 ft. to the nearest foot To check, find Z B. Then find side b by using the values of a and of tan B just found. Compare this result with the given value of b. Illustration 2. Mr. White wishes to brace an antenna pole in his back yard by fastening a wire from its top to a post 23 ft. from the foot of the pole. He finds that the angle at T is 41\u00b0. What length of wire will he need? 326 INDIRECT MEASUREMENT Here h is 23 and Z A is 41\u00b0. To find c. The ratio – is the sine c of Z B. So we must first find Z B. AB = 90\u00b0 – ZA = 90\u00b0 – 41\u00b0 = 49\u00b0 sin J5 ^ c Multiplying by c: c sin B = b Dividing by sin B: c = . ^ ^ sin B Since b is 23 and sin 49\u00b0 is .75 to two significant figures, Exercises In triangle ABC, \/.Cis a right angle. 1 . If c = 30 and ZA = 28\u00b0, find b. 2 . If c = 10 and AB = 53\u00b0, find b. 3 . If & = 2.5 and ZA = 75\u00b0, find a. 4 . lia = 20 and ZA = 35\u00b0, find b. 5 . Itb = .42 and AB = 20\u00b0, find c. 6 . lia = 350 and UB = 84\u00b0, find c. Trigonometry for the Seout 1. Girl Scouts, who planned to conduct a swimming contest from point C on the shore to point B on an island, found the distance as follows. They measured CA 340 ft. along the shore at right angles to CB, and then found that A A was 38\u00b0. What is the distance from C to B1 2. To find the distance CB across a river, the Boy Scouts measured a line AC nt right angles to CB which they found to be 120 ft. Then with a transit they found Z.A to be 82\u00b0. Find for them the length of CB. 3 . A steep mountain road makes an angle of 36\u00b0 with the TRIGONOMETRY 327 horizontal. How many feet have I risen when I have traveled 385 ft. up the road? 4. The angle of elevation of the top of a mountain known to be 6000 ft., high is 19\u00b0. How far is the top of the mountain from the ob- server? 6. A rope 80 ft. long from the top of a flag-pole makes an angle of 75\u00b0 with the ground when pulled tight. Find the height of the pole. 6. Sarah found the height of her home by noting that it cast a shadow 24 ft. long when the angle of elevation of the sun was 57\u00b0. How high was her home? 7 . A tower on the bank of a river is known to extend 163 ft. above the water. If the angle of elevation of the top from a point on the opposite bank is 15\u00b0, how wide is the river? 8 . If the angle of elevation of the top of a tower 800 ft. away is 9\u00b0, what is the height of the tower? 9 . Charles\u2019 father told Charles that it was not safe to climb a ladder if it made an angle of more than 65\u00b0 with the ground. How far from the house can Charles place the foot of a ladder 14 ft. long if it is to lean against the house? 10. The Boy Scouts climbed a mountain 4000 ft. high. From its top they saw a lake at an angle of depression of 18\u00b0. How far from them in a straight line is the lake? 11. Some boys cut 12-ft. poles to build themselves a wigwam. If the poles meet the ground at an angle of 68\u00b0, how high is the center point of the wigwam? 12. From the top of a cliff 300 ft. high, the angle of de- pression of a ship is 36\u00b0. How far is the ship from the foot of the cliff? 13. John found that from his classroom window the angle of depression of the opposite side of the street was 14\u00b0. If the window is 24 ft. above the ground, how far is it in a straight line from the window to the opposite side of the street? 14. At noon on June 21, a certain flag-pole 100 ft. high casts a shadow 30 ft. long. What is the angle of elevation of the sun? A INDIRECT MEASUREMENT Courtesy of Boy Scouts of America. BOY SCOUTS MAKING A MAP These boys are using a plane table and an alidade for plotting points on their map. The plane table is a drawing board set up on three legs. The alidade is a ruler with upright sights. The boy sights a point along the alidade and then draws the line along the edge on the paper. After sighting all the required points in this way and drawing lines, he moves his plane table to another position and sights the same points again. The intersections of the lines to the same point locate that point on his map. 328 TRIGONOMETRY 329 Trigonometry for the Practical Man 1. A forest ranger has an observation post on top of a moun- tain 3816 ft. higher than the surround- ing territory. He sees smoke rising from a fire in the forest at an angle of depression of 11\u00ae. How far away in a straight line is the fire? 2. The \u201cairman\u2019s ceiling\u201d is the un- der side of the clouds as seen from the earth. To find its height at night, the observer causes a powerful light to throw its rays vertically upwards. Then from his post 600 ft. from the light, he reads the angle of elevation of the spot of light thrown on the cloud. If he finds this angle to be 64\u00b0, what is the height of the \u201cairman\u2019s ceiling\u201d? 3. Later when the clouds lower, he finds that the angle of elevation has become only 18\u00b0. What now is the height of the airman\u2019s ceiling? What angle of elevation must the observer have found, if at one time he reported \u201cceiling 3000 ft.\u201d? \u201cCeiling 500 ft.\u201d? \u201cCeiling zero\u201d? 4 . Two artillery officers want to locate the position of the enemy\u2019s battery by the flash of the guns. Lieut. a. Homer observes that Z.BCA is a right angle, @ whereas Capt. Meyer, 468 ft. from him at A, finds i that Z.BAC is 82\u00b0. How far is the battery B from C? 6. An astronomer at D observes that the moon is directly overhead. Six hours later, when the earth has turned so that he is at B, he finds \/.CBA to be 89\u00b0. If the radius of the earth is about 4000 mi., how far is the moon from B? 6. The captain of a ship, S, ob- served that a lighthouse, L was di- rectly east. After sailing 5 mi. north, he noticed that the lighthouse was 33\u00b0 east of south ( Z A ^ 33\u00b0). How far was he from the lighthouse at first? 330 INDIRECT MEASUREMENT 7. A railroad engineer needs to know the radius of a curve so he will be able to raise the outside rail the right amount to keep the train from jumping from the track. He cannot measure it directly as the track passes between a cliff and a river, so he determines that an arc of 20\u00b0 has a chord AB 340 ft, long. Find FB the radius of the curve (ZF = 10\u00b0 and BC = ^ AB). 8. A war observer in a captive balloon at a height of 2000 ft. above the earth finds that the enemy is concentrating at a point whose angle of depression is 22\u00b0. How far away from the war observer is the enemy? 9. Two towns A and B are on opposite sides of a mountain range and are 15,000 ft. apart in a straight line. The altitude of B above sea level is 800 ft. greater than that of A. If a tunnel is to be built through the moun- tain, at what angle of elevation should it start from A so that it will come out at the level of F? 10. The Empire State Building on 34th St,, New York, is 1248 ft. high. One day a man with a large telescope on 42nd St. let people look at the top of the building for I noticed that his telescope was set at an angle of elevation of 31\u00b0. How far is 34th St. from 42nd St.? 11. A pilot at an altitude of 450 ft. wants to land at an aviation field 3000 ft. distant from his present position. If the safe angle of landing should not exceed 10\u00b0, may he descend to the field in a straight line or must he circle it first? Trigonometry in Athletics (Optional) When a heavy body leaves the ground, the distance it will go in a horizorital direction is given by the formula sin A sin F d = 16 where v is its starting velocity, A is its angle of elevation and F = 90 \u2014 A. TRIGONOMETRY IN ATHLETICS 331 1 . In the broad jump, Paul can start with a velocity of 20 ft. a sec. How far will he jump if he leaves the ground at an angle of elevation of (\u00ab) 30\u00b0? (b) 45\u00b0? (c) 60\u00b0? (d) At what angle of elevation should he jump to go the farthest? 2. In the Olympic record broad jump, the winner jumped with a velocity of about 28 ft. a sec. How far would he jump if his angle of elevation was (a) 30\u00b0? (b) 45\u00b0? (c) 65\u00b0? (d) At which angle do you think he made the record jump, and what was his distance? 3. The world\u2019s record holder can throw a 16-lb. shot at a velocity of 41 ft. a sec. How far can he throw it if it leaves his hand at an angle of elevation of (a) 20\u00b0? (b) 45\u00b0? (c) 70\u00b0? (d) At which angle must he have thrown the shot when he made his record and how far did he throw it? 4. In the Olympic contests, the winner in the 16-ib. hammer- throwing contest could throw his hammer with a velocity of 76 ft. a sec. If he threw it at the most advantageous angle, what is his record? 5. Find, by trying several angles, at what angle of elevation a baseball player should try to hit the ball if he is attempting to make a home run. Do you think that the angle of elevation at which a body will travel farthest is always the same? The line y = mx + b (Optional) To find the angle made by the line y = mx + b with the x axis, we first solve the equation for m. y = mx + b mx = y \u2014 b X {^,y) From the figure, you can see that ^ is X jdC m = tan z The tangent of the angle made by the line with the x axis is m. 332 INDIRECT MEASUREMENT How to find the angle z. Illustration: Find z for the line 2 x \u2014 ?> y = b. Solving for \u20142y= \u2014 2 x + 5 y = % X \u2014 ^ m = tan z f = .6667 From the table: z = 34\u00b0 Exercises Find the angle z made by these lines with the x axis: 1. y=x+A 3.y=ix + 4 5. 2x-5y=-4: 2. y = 2 X \u2014 5 4:. 3 X \u2014 y = 2 6. y = 3 INTERPOLATION (Optional) How to find the functions of angles that are not given in the table. If sin 34\u00b0 = .5592 and sin 35\u00b0 = .5736, what is the value of sin 34^\u00b0 or sin 34\u00b030′? How would it compare with the values for 34\u00b0 and 35\u00b0? Can you now find sin 34\u00b015′? How much bigger is sin 35\u00b0 than sin 34\u00b0? What part of this difference should correspond to 15′? Find sin 34\u00b010′. What part of a degree is 10′? What part of the difference between sin 34\u00b0 and sin 35\u00b0 should correspond to this? The difference in the angles is 60′, and the difference in the sines is 106. (Omit the decimal point.) d should be the same part of 106 as 16′ is of 60′. To the nearest unit, d = 28. Since 16′ corresponds to a dif- ference of 28, sin 52\u00b016′ will be 28 more than 7880 or will be 7908. Verify these functions: (a) sin 8\u00b040′ = .1507 (c) sin 37\u00b025′ = .6076 (b) tan 27\u00b012′ = .5139 (d) sin 83\u00b048′ = .9941 Illustration. Find sin 52\u00b016′. 1 sin 53\u00b0 = .7986 , 60′, sin 52\u00b016′ = 106 I 16′ d ‘-^=sin 52\u00b0 = .7880=^ 1 _ d 60 106 60 J = 1696 d = 28.3 sin 52\u00b016′ = .7908 INTERPOLATION 333 Exercises Find the value of: 1. sin 10\u00b015′ 2. sin 68\u00b020′ 3 . sin 44\u00b045′ 4 . sin 67\u00b040′ 6. sin 4\u00b050′ 6. tan8\u00b030′ 7 . tan22\u00b045′ 8. taneriO’ 9 . tan33\u00b012′ 10. tan84\u00b06′ 11. sin 12\u00b057′ 12. tan69\u00b014- 13. tan42\u00b028′ 14. sin 20\u00b038- 15. tan46\u00b046′ How to find angles when their functions are not in the table. 1.2349 = tan 51\u00b0 and 1.2799 = tan 52\u00b0. Of what angle is 1.2574 the tangent? How much do the tangents of 51\u00b0 and 52\u00b0 differ? How many minutes does this dif- ference correspond to? If tan x were 225 larger than tan 51\u00b0 (neglecting the decimal point), how many min- utes larger than 51\u00b0 must x be? Of what angle is .3906 the tangent? What two values of the tangent in the table are nearest to .3906? By how much do they differ? Of what angle is the value just above it the tangent? That just below it? By how much does .3906 exceed the smaller of these tangents? What part of the difference above is this? How many minutes should correspond to it? Illustration. Find y if tan y = .3906. I .4040 = tan 22\u00b0 | In the table, the nearest tan- 201 |\u2014 .3906 = 1 60′ gents to .3906 are .3839 and i=.3839 = tan 21\u00b0: .4040. Their difference is 201. The difference between 3906 and 3839 is 67. d must be the same part of 60′ that 67 is of 201. y is 20′ larger than 21\u00b0. 201 60 d ==20 y = 21\u00b020′ Exercises Find the value of x if: 1. tan a: = .4770 3. sin x = .4962 2. tan% = .9111 4. sin x = .8923 6. sin X = .9629 6. tan a: = 2.1875 334 INDIRECT MEASUREMENT Trigonometry for Boys and Girls 1 . Ruth and Mildred found the height of a flag-pole by meas- uring a distance of 50 ft. from its base and by finding that from that point the angle of elevation of the top was 48\u00b040′. How high was the pole? 2. Allan found that his school cast a shadow 20 ft. long. In an almanac he found that the angle of elevation of the sun at that time was 43\u00b035′. How high was the school? 3 . If the radius of the earth is 4000 mi., and if Z.B h 89\u00b04′, how far is the moon from B1 4 . When airplane B was directly over C, Paul at A, 1200 ft. from C, found its angle of elevation to be 74\u00b055′. How high was the airplane? 6. From the top of a mountain 6000 ft. high, Sarah found that the angle of de- pression of a lake was 28\u00b015′. How far was she from the lake? 6. Robert wants to run an aerial from his window to the top of a pole 32 ft. away. He finds the angle of elevation of the top of the pole to be 42\u00b048′. What length of wire will he need? 7 . John was told that it was unsafe to climb a ladder unless the angle it made with the ground was less than 65\u00b0. He placed a 14-ft. ladder against the house so that the foot of it was 5 ft. 3 in. from the house. Find to the nearest minute the angle it then made with the ground. Thought Questions 1. In triangle ABC, a = IQ, \/LB = 48\u00b0, and Z.A = 37\u00b0. By using the two right triangles CD A and CDB, find b. 2. Write a general solution, that is, a formula for b in terms of a, B, and A. Can you write a formula for finding h in each triangle and then eliminate hi 3 . Using the formula that you have derived, find b whin a = 50, ZA = 66\u00b0, and ZB = 48\u00b0. REVIEW EXERCISES 335 Review Exercises 1. In triangle ABC, ZC = 90\u00b0, Z.A = 53\u00b0, and AC = 100. Find the length of BC to the nearest tenth. 2. In triangle ABC, Z C – 90\u00b0, AB – 50, BC = 30, and AC = 40. Express sin A and tan A as decimals. 3. As angle A increases from 0\u00b0 to 90\u00b0, does sin A increase or decrease? Does tan A increase or decrease? Cos A? 4. Find the value of: (a) sin 12\u00b0 (c) sin 82\u00b0 (e) tan 45\u00b0 (b) tan 67\u00b0 (d) tan 5\u00b0 (\/) sin 53\u00b0 5. Find the angle whose sine is: (a) .4384 (b) .9903 (c) .5 6. Find the angle whose tangent is: (a) .0349 (b) 9.5144 (c) 1 7. Find to the nearest degree, the angle whose: (a) sine is .9801 (b) tangent is .1504 8. A wire attached to the top of a telephone pole reaches the ground 10 ft. from the pole and makes an angle of 74\u00b0 with the ground. How high is the pole? 9. An airplane is 800 ft. above the ground and directly over a school. The angle of elevation of the plane as seen by a boy at a point on the ground some distance from the school is 22\u00b0. How far is the boy from the school, and how far from the plane? 10. The ratio of a leg of a right triangle to the hypotenuse is Find to thousandths the sine of the opposite angle. Find the angle to the nearest degree. 11. As an angle increases from 0\u00b0 to 90\u00b0, which increases faster, the sine or the tangent? 12. From the top of a cliff 700 ft. high, Paul observes that the angle of depression of a boat is 42\u00b0. How far is the boat from the foot of the cliff? 13. Find the angle of elevation of the sun when a building 50 ft. high casts a shadow 46 ft. long. 14. Find the length of the shadow cast by a 12-ft. pole held vertically when the sun is 28\u00b0 above the horizon. 15. How long a wire is needed to reach from the top of a pole 30 ft. high to the ground and make an angle of 64\u00b0 with the ground? 336 INDIRECT MEASUREMENT 16. A 20-ft. ladder is to be placed against a building so that it will make an angle of 75\u00b0 with the ground. How far from the building should the foot of the ladder be placed? 17. Do E, F, and G together form a pleasing chord? Show how you know. 18. Solve algebraically: 5x + 2y = 8, 4x + 3y = 5 19. Separate 300 into two parts in the ratio 11:4. 20. A pole 4 ft. tall casts a shadow 5 ft, long at the same time that a building casts a shadow 65 ft. long. How high is the building? 21. Solve: ^ = ;c + 12 3 5 22. Find x if AB is parallel to CD. 23. Find the price goods must be marked, if the cost is $8.40, a profit of 25% of the cost is to be made, and a discount of 40% of the marked price is to be given. Formula: m = 1 \u2014 a 24. Solve for ;r: 5 :r \u2014 2 = 10 + 2 x 25. A grocer has 100 lbs. of 40^ coffee. How many pounds of 25^ coffee must he use to make a mixture worth 30j^ a lb.? 26. Using F = ^ + 32, find F when C = 20. 27. In the triangle ABC, the length of BC is a and of AC is h. h, d, and e are as shown in the figure. Show that the formula for \u00ab IS \u00ab = e. a 28. If AC = 5 in., BC = Z in., 5F = 5 ft., and EF = 24 ft., find the length of DF. 29. If the triangle has an altitude of 9 in. and a base of 3 in. and the boy, whose eye is 5 ft. from the ground, stands 20 ft. from the building, what is the height of the building? 30. The horsepower of an automobile is given by the formula ^ = .4 d^n. Find h when = 5 and w = 8. D TESTS 337 Test In triangle ABC, Z C is the right angle. Find all of the parts not given in the table, lines to three significant figures and angles to the nearest degree. Chapter 14 LOGARITHMS How we can use exponents to save us work. Here is a table of the powers of 2: 21 = 2 21 = 128 213 = 8192 22 = 4 26 = 256 214 = 16,384 23 = 8 29 = 512 216 = 32,768 21 = 16 210 = 1024 216 = 65,536 25 = 32 211 = 2048 217 = 131,072 26 = 64 212 = 4096 218 = 262,144 By using this table we can do certain computations easily. Illustration 1. Multiply 8192 by 16. From the table we see that 8192 = 2^^ and 16 = So: 8192 X 16 = 213 X 21 = 2^7 Looking again at the table, we see that 2^’^ = 131,072. So: 8192 X 16 = 131,072 Illustration 2. Divide 65,536 by 2048. 65,536 2048 = 2i6 — 2\u201c = 2^ = 32 Illustration 3. Find the square root of 16,384. VT6:384 = V2^ = 2^ = 128 Exercises Fmd the value of: 1. 32 X 512 2. 256 X 128 3. 64 X 4096 4. 8 X 16,384 5. 8192 X 32 6. 131,072 8192 7. 16,384 ^ 512 8. 262,144 — 65,536 9. 4096 256 10. 1282 11. V262,144 12. V4096 13. V65OT 14. ^32,768 15. 16^ 338 THE FRACTIONAL EXPONENT 339 It may occur to you that this table is not of much use, for you may want to multiply numbers that are not in it. Let us see if we can discover a way of using exponents for multiplying or dividing any two numbers. To do this we must study exponents that are not integers. What is a fractional exponent? Does mean anything to you? When you multiply numbers, what do you do with their exponents? What is \u25a0 a^l x^ \u2022 x^l bi \u2022 bi? What is the value of V9? Does this have any relation to the fact that 3×3 = 9? What is V49? Is it related to the fact that 7 X 7 = 49? If the square root of a number is multiplied by itself, what is the result? Find: V25 \u2022 V25, V8l \u2022 V8l, Vi \u2022 Vi, ylabcd \u2022 ^iabcd, V894 \u2022 V894. The square root of a number is one of the two equal factors of the number. V25 = 5 because 5 X 5 = 25. If = X, what is the relation of x^ to x? Does xi = Vi? What is the value of 9i? 16i? 36i? 8D? 144i? What is the value of x^ \u2022 x^ \u2022 xa? If the cube root of X (^yx) is one of the three equal factors of x, is xi the same as V^x? Find the value of 81^, 27i, 64i, What is the value of at \u2022 at? Express at as a root of a. What does the numerator, 3, tell you? What does the denominator, 2, tell you? When a quantity has a fractional exponent, the numer- ator of the fraction indicates the power, and the denomina- tor indicates the root. For example, xt means the cube root of x^, or xt = -v^x^. Illustrations: at \u2014 9t = V9 = 3 16t = Vl65 = 23 = 8 Vib’ (8 a3)t – {IW\u00a5Y = (2 ay = 4 340 LOGARITHMS Exercises Write these exercises with roots instead of fractional ex- ponents: 1. 6. 9. 3 x^ 2. cl 6. rl 10. 4 3. rm 7. ti 11. 7Z)I A 3 4. 8. Ai 12. 5xi Write these exercises with fractional exponents instead of roots: 13. Vc 17. ^ylb 21. 2 ^lt 14. Vm3 18. 22. a 16. 19. 23. 5 16. ^xy 20. 24. yjTdb Illustration. Find the value of 8^: 8^ = = (“V8)2 = 2^ = 4 Find the value of: 26. 4I 29. 4^ 33. 26. 16l 30. 16^ 34. 27. 25I 31. 8l 36. 27^ 28. 64I 32. 25^ 36. 32I Find the value to three significant figures of: 37. 2I 39. 355I 41. 5^ 38. 7I 40. 89.47I 42. 2^ The zero exponent. yT Find the value of ^x^ . x^^ of , of x^\u2019 x^ What did you do to the exponents to get your answer? When you divide one power of a letter by another power of that letter, do you always subtract exponents? Then what x^ exponent has x in the answer to ^ ? But what is the value oO of , that is, of any number divided by itself? THE ZERO EXPONENT 341 Any quantity to the zero power equals 1. Sometimes 0\u00b0 is an exception. X3 By the law of exponents: ^ ^ ^ x3 But by ordinary division: ^ ^ ^ Consequently: = 1 Exercises 1. What is the value of: (2 6)\u00bb? (x + 1)\u00ab? (x + y)\u00b0? x^ + y\u00ab? 3w\u00ab? 2’\u2019? lOV 2. What power of 10 is 1? 3. Express as powers of 10: 100; 10,000; 1; 1,000,000; 10; 1000 4. One followed by 26 zeros is what power of 10? 5. 26,000,000 = 2.6 X 10,000,000 = 2.6 X 10 Express in the same way, that is, with but one digit on the left of the decimal point: 38,000,000; 74,000 ; 6,500,000; 24 6. How many thousand is 9840? How many hundred thou- sand is 864,300? Express the latter using a power of 10 for the hundred thousand. 7. Express these numbers as numbers between 1 and 10 multiplied by a power of 10: (a) 8000 (d) 834,000 (g) 53,400 \u2019 (d) 40,000 (e) 62,840 (h) 3 (c) 7600 (\/) 1,960,000 (i) 81,659 8. What is the value of: 1″? 1″? P””? 14? It? 1\u00ab? Express 2 as a power of 8. Since 2 X 2 X 2 = 8, 2 = 84. We could write this exponent as a decimal by changing i to .3333, so 2 = 8*3333^ You have just seen that numbers which are not exact powers of a number can be expressed as decimal powers of the number. A little over 300 yrs. ago, a Scotchman named John Napier discovered that all numbers could be ex- pressed as powers of a single number. He called these exponents logarithms. With the help of another man, Briggs, he made a table by means of which you can change 342 LOGARITHMS Courtesy Port of New York AvthorUy. ‘ MATHEMATICS AT WORK All construction work involves a great deal of mathematical computation. all multiplication exercises into easy addition exercises, and all division exercises into subtraction exercises. This table expresses all numbers as powers of 10. We shall now learn how to use it. How to use a table of logarithms. The table of loga- rithms is found on pages 344 and 345. These logarithms are decimals, but as all decimal points have been omitted from the table, you will have to put them in yourself. The first two digits of the number are found in the column headed N, and the third digit is found across the top of the page. We shall understand that there is a decimal point after the first digit. The rest of the page gives the exponents or logarithms, and in these we shall understand that there is a decimal point before the whole number. LOGARITHMS IN COMPUTATION 343 Illustration. Find the logarithm of 3.87. . As the table contains no decimal points, we shall look first for 38 on the left. Then we look across that line until we come to the column headed with the third digit, 7. There we find 5877. Now put the decimal point before this, and we have 3.87 = 10-\u00ae\u00ae’^’^. (Do not forget that these logarithms are ex- ponents of 10.) Exercises Here are numbers with their logarithms. Look them up in the table to see if all of them are correct: 1. 2.5 = 10- 3 9 7 9 6. 1.42 = 10′ ,15 2 3 2. 5.4 = 10- 7 324 7. 8.91 = 13’ .949 9 3. 7.2 = 10- 85 73 8. 3.65 = 10 .562 !3 4. 9 = 10-” 5 4 2 9. 7.28 = 10 .862 1 1 5. 3 = 10-^ 7 7 1 10. 4.59 = 10 .6 6 1 8 Write the logarithms of: 11. 2.8 16. 5.0 21. 2.41 26. 8.66 12. 7.6 17. 2.0 22. 3.82 27. 3.33 13. 4.3 18. 4 23. 7.84 28. 2.01 14. 8.5 19. 9 24. 9.68 29. 1.11 15. 1.2 20. 6 25. 6.52 30. 4.99 Write the number that corresponds to: 31. j ^ q .5185 36. 10- 8457 41. 10- 9996 32. 10.6812 37. 10- 6590 42. 10- 0043 33. 10.2788 38. 10- 8797 43. 10- 8500 34. 10. 3032 39. 10- 8488 44. 10- 7642 35. 10.4393 40. 10- 9805 45. 10- 0000 How to use logarithms in computation. When you multiply numbers, what do you do with the exponents? When you divide numbers, what do you do with the ex- ponents? What should you do with their logarithms, when you multiply numbers? What should you do with their logarithms, when you divide numbers? \u2018344 LOGARITHMS Common Logarithms of Numbers N 0 1 2 3 4 5 6 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 19 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 N 0 1 2 3 4 5 6 7 8 9 LOGARITHMS IN COMPUTATION 345 Common Logarithms of Numbers {Continued) N 0 1 2 3 4 5 6 7 8 9 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 69 8388 8395 840 ^ -8407 8414 8420 8426 8432 8439 8445 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 8751 8756 8762 8768 8774 8\/79 8785 8791 8797 8802 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 N 0 1 2 3 4 5 6 7 8 9 346 LOGARITHMS Illustration 1. Multiply 3.12 by 2.47. 3.12 = 2.47 = 10-“‘\u2019”” 3.12 X 2.47 = + .3927 _ IQ. 8869 The nearest logarithm to this in the table is 8871, and this corresponds to 7.71, so: 3.12 X 2.47 = 7.71 Illustration 2. Find the value of 6.84 4.73. 6.84 = 4.73 = 6.84 4- 4.73 = 10-8351 – .6749 _ 10.1602 The nearest logarithm to 1602 is 1614, and this corresponds to 1.45. So: 6.84 ^ 4.73 = L45 Illustration 3. Find the square root of 3.29. V3.29 = VIO-5 172 _ | 0’2586 = 1.81 Exercises Find the value of: 1. 2.84 X 3.17 10. 8.79 ^ 3.64 19. V7.37 2. 8.34 X 1.18 11. 4.53 3.68 20. V5.44 3. 6.45 X 1.32 12. 5.58 ^ 4.97 21. V9.98 4. 2.98 X 3.16 13. 8.96 ^ 8.14 22. 2.073 6. 7.79 X 1.15 14. 6.71 4- 4.38 23. 1.862 6. 2.32 X 3.41 15. 7.82 – 3.69 24. ^6:92 7. 3.64 X 1.72 16. 5.91 4.15 25. ^3.35 8. 4.68 X 2.12 17. 3.38 2.74 26. V7.69 9. 1.94 X 3.85 18. 1.76 1.51 27. V2.63 In these exercises the decimal point came after the first digit. But suppose you wanted to multiply numbers in which it occurred somewhere else. What would you do then? Let us examine such a number. LOGARITHMS IN COMPUTATION 347 Illustration 1. Find the logarithm of 47.8. 47.8 is 10 times as large as 4.78. 4.78 = 10 = 10 ^ 47.8 = + 1 _ j^Ql.6 7 94 Illustration 2. Find the logarithm of .00736. .00736 is 7.36 ^ 1000 or by 10^ 7.36 = 10-\u00ae\u00ae\u00ae\u00ae 1000 = 10 ^ .00736 = Notice that now our logarithm is made up of two parts, a whole number and a decimal. The whole number is called the characteristic, and the decimal is called the mantissa. How to find the characteristic. You have already learned to find the mantissa in the table. Now let us examine the illustration above to see how we could find the characteristic. When the decimal point was after the first digit, as in 4.78, the characteristic was 0. When we moved the deci- mal point one place to the right, as in 47.8, we multiplied the number by 10, and this made the characteristic 1. If we moved the decimal point another place to the right, it would again multiply the number by 10 and so add another 1 to the characteristic. So you see then that for every place the decimal point is moved to the right, the charac- teristic is increased by 1. To find the characteristic, it is only necessary to count the number of places the decimal point is moved to the right. In the second illustration, the decimal point in .00736 is three places farther to the left than in 7.36. This divides the number by 10^ and so subtracts 3 from the logarithm. Therefore, for every place the decimal point is moved to the left, we subtract 1 from the logarithm. 348 LOGARITHMS To find the characteristic, first think of the decimal point as being after the first significant figure of the num- ber. Then count the number of places from this imaginary decimal point to the real decimal point. If you count to the right, add that number to the logarithm. If you count to the left, subtract that number from the logarithm. Illustration 1. Find the characteristic of the logarithm of 384. If the decimal point were after the 3, the characteristic would be 0. Since it is two places to the right, the characteristic is 2. Illustration 2. Find the characteristic of the logarithm of .061. If the decimal point were after the 6, the characteristic would be 0. Since it is two places to the left, the characteristic is \u2014 2. Illustration 3. Find the characteristic of the logarithm of 7.35. If the decimal point were after the 7, the characteristic would be 0. Since it is there, the characteristic is 0. When you know the characteristic of a logarithm, you can use this same rule reversed for finding the decimal point in the number. Illustration 4. The characteristic of the logarithm for the set of digits 942 is 5. Put in the decimal point. If the characteristic had been 0, the decimal point would have been after the 9. Since it is 5, we count five places to the right, adding O\u2019s when needed, and the number is 942000. Illustration 5. The characteristic of the logarithm for the set of digits 346 is \u2014 4. Put in the decimal point. If the character- istic had been 0, the decimal point would have been after the 3. Since it is \u2014 4, we count four places to the left, adding O\u2019s when needed, and the number is .000346. Exercises Give the characteristic of the logarithm of: 1 . 6 2. 5.8 3. 7.94 4 . 84 6. 63.1 6. 125 7 . .7 8. .34 9. .828 10 . .0045 11 . .0354 12. .0001 LOGARITHMS IN COMPUTATION 349 Using the set of digits 6452, put in a decimal point for each of these characteristics: 13 . 0 15.-3 17 . – 5 19 . 3 14 . 2 16 . 1 18 . 6 20.-1 You have now worked with logarithms long enough so that you know that they are exponents of 10. From now on, we shall omit the 10 and write the work in a more con- densed form. We shall use \u201clog\u201d to stand for \u201cthe logarithm of.\u201d For example, log 2 = .3010 means \u201cthe logarithm of 2 = .3010.\u201d Illustration. Find N li N = 7420 X .0653. Log N = log 7420 + log .0653 Before looking up the logarithms in the table, make a complete form for your work: log 7420 = + log .0653 = log N = N = Then use a table to fill in the blanks, and complete the work: log 7420 = 3.8704 -1- log .0653 = .8149 – 2* log N = 4.6853 – 2 = 2.6853 N = 485 Find the value of: 21 . 34 X 5.3 22 . 87 X 45.5 23 . 42.3 ^ 1.58 24 . 5.93 X .0042 25 . 85.7 3.82 26 . 3600 X 6.52 27 . .047 ^ .293 28 . .0569 4- 842 29. 546* 30. V76 31. V45,600 32. ^72.4 * When the characteristic is negative, it is customary to change it to a number \u2014 10. \u2014 2 becomes 8 \u2014 10, and the logarithm .8149 \u2014 2 is written 8.8149 \u2014 10. 350 LOGARITHMS Logarithms in Business You have already found compound interest by means of a compound-interest table. This was very easy, but you could solve exercises for the rates and times given in the table only. By means of logarithms you can find interest for any rate and any time. The compound-interest formula is A = P{1 + r)\u201d. Illustration. Find the amount of $347 at 4% for 8 yrs. A =P(1 Ary log 1.04 = .0170 – 347(1 + .04)\u00ab log 347 = 2.5403 = 347(1.04)8 + 8 log 1.04 = .1360 log A – log 347 -f 8 log 1.04 log A = 2.6763 A = 475 If interest is compounded annually, find the amount of: 1 . $400 at 6% for 10 yrs. 2 . $760 at 4% for 5 yrs. 3. $645 at 8% for 8 yrs. 4 . $192 at 2% for 6 yrs. 5. $873 at 6% for 9 yrs. 6. $84.50 at 4% for 5 yrs. 7. $1340 at 6% for 3 yrs. 8. $11,600 at 2% for 4 yrs. 9 . $94.60 at 4% for 25 yrs. 10. $2630 at 6% for 100 yrs. 11. In 1623 the Indians sold Manhattan Island for $24. If they had placed that money at 6% interest, compounded annu- ally, how much would it have amounted to in 1923? 12. If every person who landed at Plymouth from the May- flower in 1620 had 3 descendants and each of these averaged 3 descendants in 25 yrs., etc., down to the present, the number of persons who might have had an ancestor on the Mayflower would be more than 3^2 How many people does this represent? How does it compare with the population of the United States in 1920 (105,000,000)? Logarithms in Music (Optional) In planning the tempered scale for the piano, Bach found it necessary to make all ratios the same. Let x be the ratio of any two consecutive half tones in the chromatic scale. Then if C = 256, C# = 256 x, D = 256 x^, D# = 256 and so on to the next C which would be 256 x^”^. THE SLIDE RULE 351 C D D# E F F# G G# A B C 256 256x\u2019 256x= 256x^ 256x’\u00bb 256x^ 256×6 256x\u2019 256×6 356 x 6 256x^\u201c 256x” 256x\u2019* But the next C is double the C below or 512. So 256 = 512, and = 2. 1. Using logarithms, find x. 2. Find D on this modified scale, and determine by how many vibrations it differs from the natural D, 288. 3. Find G on the modified scale, and determine how much it differs from the natural G, 384. THE SLIDE RULE (Optional) Engineers and scientists multiply, divide, extract square root, find sines, tangents, and logarithms, solve ratio and proportion; and perform other computations on a ruler called a slide rule. By simply pulling a slide out the proper distance, the answer is found without the trouble of figuring. How the slide rule is made. Let us draw a line 10 in. long and lay off logarithms on it. From our table, we find ] I I I J I ^ ^ l\u2014J 1 2 3 4567891 that log 1.0 is .0000, so we shall put 1 at the end of the rule. Log 2.0 is .3010, so we shall take .3010 X 10 in. or about 3 in. as the length to represent 2. Log 3 is .4771, so we take .48 X 10 in. or 4.8 in. to represent 3. What length shall we take to represent 4? 5? 6? 7? 8? 9? We can also put on the second figure of the number. For example, log 1.1 = .0404; therefore, 1.1 would be placed about .04 X 10 in. or .4 in. from the end. 352 LOGARITHMS Where would you put 1.2? 1.3? 1.4? 1.5? Here is the way the section from 1 to 2 looks when we put on the ^ 2 second digit. In the same way we J \u2014 : \u2014 I I I I , I I I I could divide these parts to show the 1 2 3 4 5 6 7 8 9 ^ digit is on the slide rule as the marks would be too close to each other at the right end of the rule. The slide rule has two scales, C and D, like those we have just made. They are exactly alike, but D is on the body of the rule, whereas C is on the slide and can be made to fit on D. How to divide on the slide rule. Let us divide 3 by 2. You remember that to divide by logarithms, you subtract 2 , ZTT 4 5 6 7 8,9 lliul 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 hlllilllllllllllllllllllllllil 1 1 11 1 1 1 1 1 IlirilMMinilliiiilllilliiillllilllii \u25a0^1 2 3;r 4 5 C ,,|J, ,,,,,,, If,,,,, ,1^,1, ^ i E liiiiliiiil 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 h 1 i1ilililililtlili1ililili|ililililililiM 1 1 1 1 hi 2 37r 4 J C ^ 1 2 |3 4 i5 [6 7 8 9 j 1 < I t 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 1 1 1 111 1 tl il MIllinilllllllMlIlMlIhllltMlIltljIllltllHlIlltllllllllllllll\/ xy = 4 – 2 xy + 5 y 2 = 16-4 + 5= 17 5×2- xy + 2 y2 = 20 – 2 + 2 = 20 < \u2014 ^ 20 Class Exercises Add: 1 . 3fl\u2014 46\u2014 5c \u2014 3b 2&-3m; + 3\/? \u2014 5 & + w \u2014 h + 2& \u2014 3m; + 4\/z 10. -2x\u2018^y xy^ – x^ +2 x^y + 3 xy^ -2x^ +4 x^y+ xy^ + 3x^ – x^y -5 xy^ 11 . 2 ab \u2014 4 ac \u2014 3 be ab 3 ac \u2014 be 2 ae \u2014 be \u2014 4 ab 3 ae 12 . 3 m \u2014 2 p w 3 p \u2014 2 w 4 m 4- P + \u2014 3 m \u2014 2 p Optional Exercises Put like terms in columns. Usually it is best to arrange with the highest exponent first and then run down the powers, or to arrange letters in alphabetical order. Then add: 13. 3fl-2&-c, 2\u00ab+3&-2c, 26-4a+3c 14. -2 ab 4- b a^ – 2b^ 4- ab, 4- b^ + ab – 3 a^ 16. 4;c” -5x + 7, + 3 -2^:^ 4- x,2 x – 3 x^ -3 16. x^ 4- y^ 4- 2^, x^ – – 2 y^, y^ – x^ 4- 4 z^ 17. kl \u2014 Im 4- rnn, Im \u2014 mn + kl, mn \u2014 kl 4-lm, la. 3r \u2014 1 q 4- 3 p, \u2014 3p4-4q\u2014 3r, \u2014 2 r 4- q \u2014 p Honor Work 19. 3.2 ;ic \u2014 2.3y \u2014 4.3z, 1.7 a; + 1.4 3 ; \u2014 3z, \u2014x 4-y 4- 5.2z 20. .02 X \u2014 .001 3 ^ \u2014 .5 2 , .3 a; + .01 3 ^ \u2014 2 , \u2014 x \u2014 .008 3 ^ + 2 2 21. .3 \u00ab 4- .4 & – 1.6 c, – 2.3 \u00ab – & + c, – .4 + .3 & + 1.2 c 22. ax 4- by 4- cz, bx \u2014 by 4- 2 , \u2014 ax 4- y \u2014 bz 23. (a 4- 3 b)x 4- (a \u2014 b)y, (a \u2014 2 b)x 4- (\u2014 2 a 4- 3 b)y 24. 2^ a: \u2014 l-g- 3 ^ 4- 2 , \u2014 f x +2^ 3 ^ \u2014 -|\u2022 2 , a: \u2014 i-y 25. 3 x^^ \u2014 2 x^ \u2014 3 x4 \u2014 4 a:\u201c + 3 at*\u2019 + a:^ 3 a:^ \u2014 3 a:* + 4 a:*” SUBTRACTION OF POLYNOMIALS Subtraction is like addition except that in every term subtracted, we think of the sign as changed. We can check as in addition by substituting numbers for the letters, or we can add the quantity subtracted to the answer. SUBTRACTION OF POLYNOMIALS 361 Illustration: Subtract, and check, letting a = 3: 5a^ -7a -h 4 =45-21+ 4 = 28 3 a2 + 2 a – 12 = 27 + 6 – 12 = 21 2 a2 – 9 a + 16 = 18 – 27 + 16 = 7^7 Class Exercises Subtract\u2019 1. yr. mo. da. 1938 8 21 1933 5 12 2 . 3 8 5 2 6 3 3 . 3 + 8 \/ + 5 2\/2 + 6^ + 3 4 . 5a\u2014 3& + 6 3a – 4& + 7 5. 2K-3L-M SK – 2L + M 6 . 3x^ -ix + 7 – x^ + 2x + 3 7 . 5 cwt. 18 lbs. 12 oz. 3 cwt. 9 lbs. 7 oz. 8. 5 c + 18 w + 12 2 3c + 9^ + 7 2 9. 3x^ -4 xy -4 y^ 2x^ + 3 xy – y^ 10. 3 8 6 7 4 0 5 11. 3\/2 + 8\/2 4- 6 \/ + 7 4\/2 +5 12. x^-2x^y +7y 2 \u2014 3 x^y + xy^ \u2014 y^ 13. a + b 14. a \u2014 b + c 2a + 3b\u2014 4c 15. 0 3 a:2 – 2 y – z Optional Exercises 16. From 7 a + & + 5 c take 2a \u2014 3&\u2014 4 c. 17. From \u2014 3 a2 + 9 at + 7 &2 take \u2014 a6 + 2 &2 _ 6 a2. 18. From 4 x^ + 5 xy \u2014 6 y^ take \u2014 6 a :2 + 10 \u2014 8 y^. 19. From 6a;\u2014 43^ + 22 take 2a: \u2014 3y \u2014 5 z. 20. Subtract 4 A \u2014 5 B + 6 C from A \u2014 2 B \u2014 4 C. 21. Subtract 5 a:i \u2014 2 a :2 \u2014 3 a ;3 from a:i + 4 a :2 + a: 3. 22. Subtract \u2014 2 a \u2014 & \u2014 4 from 5 a + 4 & \u2014 1. 23. From 0 take 4 a :2 \u2014 5 a:^ \u2014 3 3^2, 24. From 2 a + d take 5 a – 3b \u2014 c. 362 POLYNOMIALS Subtract: 25. 3.8 X + 7.34 3; – 2.2 X + 3.21 3; 26. 2^Jx + 3^lx \u2014 y -r2y) 23. {3x – 3+2x\u2018^){3x – 1) 24. {x^ +x^y^ +y^)(x^ -y^) 25. (3x^ -4)(2 +3x) 26. (x + 2)(x – 3)(x – 1) 27. (x^ – X ~ 4)(x^ -2x) 28. (2\u00ab +36 + c)(2a – 6) 29. (2:\u00bb: + 3 3^ – 4)(xy – 1) 30. (x^y^ -2x^y^ +3y^)(x^y^ – 2 xy^) 31. ‘(2x-y)^ 32. (3i?-2)” Optional Exercises 33. (3 \u00ab” + 2 a – 2)(a” – 3 \u00ab + 1) 34. (2 X \u2014 y \u2014 3 z)(x + 2 y \u2014 z) 35. (6″ +6c + c”)(6″ -6c + c”) 36. (x^ +2x^ + 4 x^ +8x + 16)(x – 2) 37. (a” -2fl6+6″)” 38. (x^ \u2014 2 xy + y^ \u2014 z^)(x + y \u2014 z) 39. (5 \u00ab – 2 fl” + 1 – a^)(a – a” – 3) 40. (x^y^ – 2 x^y^ + 3 x^y^)(x^ + xy – 2 y^) 41. (w\u00ae \u2014 \u2014 w + l)(w^ \u2014 \u2014 2) 42. (2x^ – x” – 3)” Honor Work 43. (x^ + 3)(x^ – 2) 49. (x^ + ^ – 2 x^ – x^ – ^)(x^ – 3) 44. (3^\u201c+” -23;)(3^\u201c + 3) 50. (a^ – 6^)” 45. (2^\u201c -4)(z\u00ae\u00ab+4) 51. (A;”^+3^^”)(;t;”+3^”\u201d)(;c”-3’^”) 46. (xi – 1)(X2 + 1) 52. (6t _ + l)(6i + 1) 47. (\u00abi – 3)(ai + 2) 53. (ki – 48. (m^ \u2014 pi) (m^ + 3 pi) 54. (3^ + 3^^ + 3^^ + 1)(3’^ \u2014 1) 364 POLYNOMIALS DIVISION OF POLYNOMIALS Division of polynomials in algebra is practically the same as long division in arithmetic. Compare these examples: 2x + 1 2 X -l\u201c3|4jr^ 4~3 4 x^ -j- 6x 2 X -f- 3 2 % 3 1. Arrange the terms of both dividend and divisor in descending (or ascending) powers of some letter. 2. Divide the first term of the divisor into the first term of the dividend, and write this answer as the first term of the quotient. 3. Multiply the whole divisor by this quotient, and sub- tract from the dividend. 4. Repeat this process using the remainder as the dividend. Illustration. Divide 6 x”^ \u2014 19 a: + 15 by 2 x \u2014 3. 3x \u2014 5 2x \u2014 3|6 x^ \u2014 19 X 15 Divide 2 x into 6 x^, getting 3 x. ~ 9^ Multiply 3 a: by 2 X \u2014 3, and subtract. \u2014 10% + 15 Divide 2 x into \u2014 10 x, getting \u2014 5. \u2014 10 % + 15 Multiply \u2014 5 by 2 % \u2014 3, and subtract. 21 ^|4 83 23 23 Class Exercises Divide and check by multiplication: 1. -f 9 \u00ab + 20 by \u00ab 4 8. + lOy – 24 by y + 12 2. – 7 b + I0hyb – 2 9. z” – 8 2 + 15 by 2 – 3 3. – 5 c + 6 by c – 3 10. + ife – 12 by ^ – 3 4. \u2014 3 m \u2014 lOby m \u2014 5 4 %2 -p 12 % 9 6. _ 5 ^ _ 14 by + 2 2% + 3 6 . + 9 % \u2014 22 by % + 11 y2 +22 _ 2yz 7. \u2014 w \u2014 56 by w \u2014 8 y – 2 DIVISION OF POLYNOMIALS 365 13. 14. 16. 16. 17. 18. 19. 20 . 21 . 4 ;c 2 -5x – 6 22. 25fl2 – 9 62 X -2 5 \u00ab + 3 6 12 c2 \u20146 \u2014 a 23. 16 m2 \u2014 49 \/)2 2 -{- 3 a Am \u2014 7 p 5 -3x -2x^ 24. a^b^ \u2014 5a^b \u2014 24 2 X 5 \u00ab26 \u2014 8 \u2014 27 62 + 6a2^ 25. 16×33’^ + 14 x33;2 \u2014 15 \u00ab2 \u2014 3 6 8×33^2 -5 8 m2 + 24 \u2014 38 m 26. 9 +4962 \u2014 42fl26 4 m \u2014 3 3fl2 -76 20×2 + 20 >^2 +41×3^ 27. 6r4s6 \u2014 6 \/2 \u2014 Sr^s^t 53^ + 4 X 2r2s3 – 3\/ 10 – 4 + 6 X2 28. 8×5 – 10 a;3’^ +11×33;2 3ii: – 1 8 x3 \u2014 5 X3^2 16 – 27×2 – 6x 29. 9×2 _ 163;2 2 ~ 3x 43 ; \u2014 3 X 4×2-1 30. 4fl3 \u2014 25 ab^ 2x – 1 2 + 5 62 Optional Exercises 3 \u2014 5 + 1 fl \u2014 1 \u201e\u201e 15×3 +x^y -3xy^ +2y3 3x+2y 22 6w^\u2014 5w3\u2014 4w2+23w \u2014 20 2 \u2014 3 w + 4 g . 2a^ -a3b -Ua^b^ + 13ab3 +3b^ 2 fl2 -^ab – 36 8 + 14 m3 \u2014 15 + 13 m \u2014 2 2 m2 + 5 m \u2014 1 36. X – y 8fl3 -2763 2a – 3b 38. X + j 22 r2 \u2014 rs + s2 oi\u00ab + 3265c15 a2 + 2 6c3 Honor Work Divide: 41. a \u2014 jy2 6 6 42. (2 a- 2 \u2014 1 + 1 x\u00ae \u2014 3^’ 366 43. POLYNOMIALS 22 n + 3 + 12 22 n + 2 27 ^2 n + 1 44. 45 2 + 3 \u00ab3 + 3 fl2^ + 3 ab^ -f &3 _|. c3 X + 3^ a + & + c 46. x3 _|_ ^3 _|_ 23 \u2014 3 xyz X + y z In these exercises find the first four terms of the quotient in ascending powers of x. 47. 48. 1 +2x 49. 1 + a:2 1 -hx 50. 2 + x + x^^ DIFFERENCE OF TWO SQUARES Illustration. Multiply a + b by a \u2014 b. If we actually perform the multiplication, we get: (a + b) (a \u2014 b) = \u2014 \u00a5 Rule. The product of the sum and difference of two numbers is the difference of their squares. Exercises Find the answers by inspection: 1. (c + d){c – d) 2. {m + n){m \u2014 n) 3. {x – y){x +y) 4. {x + 5)(:r \u2014 5) 6. (2 a – b){2a + b) 6. {2 X + A y){?> X – ^ y) 7. (3^ – 1)(3\/ + 1) 8. {2x\u2018^ +3)(2x” -3) 9. (7 -3y)(7+3y) 10. (a^ + b^){a^ – b^) 11. {xy^ \u2014 4)(xy^ + 4) 12. (x + .l)(x – .1) 13. (3 a – .05) (3 a + .05) 14. {.lx + .01)(.7;r – .01) 15. {t^ -2v\u2018^){F +2v^) 16. (s +i)(s -i) Illustration. Multiply: 47 X 53 47 X 53 = (50 – 3) (50 + 3) = 2500 – 9 = 2491 If the number half way between the two numbers to be multiplied ends in 0, this method is often much easier than DIFFERENCE OF TWO SQUARES 367 multiplying out the numbers, and it has the advantage that it can more easily be done mentally. Exercises Find the product mentally: 1. 39 X 41 2. 56 X 64 3. 32 X 28 4. 37 X 43 6. 97 X 103 6. 78 X 82 7. 81 X 79 8. 104 X 96 9. 123 X 117 10. 76 X 84 11. 124 X 116 12. 85 X 75 13. 71 X 49 14. 115 X 105 15. 1203 X 1197 16. Charles Drake and Company sold this bill of goods: 48 doz. eggs at 52(2^ a doz. 73 bu. potatoes at 67^ a bu. 102 yds. cloth at 98^ a yd. 31 lbs. coffee at 29(zf a lb. 64 lbs. tea at 56^ a lb. Find the total amount of the bill. FACTORING THE DIFFERENCE OF TWO SQUARES What is the product oi a y- b and a \u2014 P. What then are the factors of \u2014 ^2? Rule. When an expression is made up of two squares separated by a minus sign, take the square root of each square. One factor is the sum of these square roots, and the other factor is their difference. a2 – 62 _ (a + _ ij) Factor: 1. Class Exercises 8. 4 – 1 15. 36 – 25 y^ 2. nF’ \u2014 yF 9. 25 – 9 16. 9 a^b^ – 1 3. -9 10. -4:F 17. 49 – 16 s^F 4. \u2014 4 11. \u25a0 18. 36 – 49 5. -1 12. a\u00ae – 19. -9b^ 6. 16 – 13. 9 – 16 20. 4x^ -y^^ 7. 9 – x^ 14. 4x^ – 25 21. 9 – 64 b^c^ 368 POLYNOMIALS 22. x^ – .01 29. X2 1 34. 4 9 23. y” – \u25a0 T 25 4 X2 \u25a0y2 24. 2 ^ – .09 30. – .0121 fl2 62 26. .01 X ” – 1 31. 100 c ^ – .01 d^ oD. X2 \u201c \u25a0y2 26. 4 m” – .25 32. 1.44 -x\u00ab 36. .01 49 27. 28. i – 1 % 4 x” \u25a0 m” 33. d^ – 25 49 22 64 Optional Exercises Find the prime factors: 37. 41. – I 38. 42. 16 – 81 39. x^ \u2014 16 43. ax’^ \u2014 a 40. – 1 44. – & 45. 2 – 18 46. 27 – 12 xy^ 47. x^ – X 48. ttR^ \u2014 Trr^ 49. {a -VhY – I 60. {a \u2014 x)^ \u2014 61. {x -y)^ – 4 52. 1 – (x +yY 63. 4 – (a – bY 64. 9 \u2014 4(x + Honor Work 56. (X + -{y + bY 66. (x -3yY – (4 +2)^ 67. (2fl -3&)^ – (3x -4.dY 68 . A{a–bY -9{c^-dY 69. x^^ – y\u2018^^ 60. x^ \u201d + ^ \u2014 x^ Applied Problems Illustration 1. Find the value of 243 ^ \u2014 2422. 2432 – 2422 = (243 + 242) (243 – 242) = 485 X 1 = 485 Illustration 2. Find the value of 5622 _ 4332^ 5622 _ 4332 = (562 + 438) (562 – 438) = 1000 X 124 = 124,000 If the difference of the two numbers is small, or if either the sum or the difference ends in 0, this method is very easy and can often be done mentally. In most cases it reduces the labor of multiplying. DIFFERENCE OF TWO SQUARES 369 Do these exercises without multiplying out the squares: 61. 362 _ 262 62. 1482 – 522 63. 4352 – 4342 64. 13442 _ ; l 3422 66. 6782 – 3222 66 . 86532 _ 36492 67. 34722 – 32722 68. 3.14 X 5512 _ 3 14 5492 69. 3.602 _ 2.312 70. 3.14 X 1562 – 3.14 X 1442 71. The figure shows a metal square, having a side a, from which a square, having a side b, has been cut. {a) Write a formula for the area A in terms of a and b. (b) Find the area of the metal if a = 3.84 and b = .84. (c) Find the area if \u00ab = 7.83 and b = 2.17. ZI \u2014 a > Ex. 72 72. A tin pan is made from a square of tin, side a, by cutting from each corner a square, side b, and folding along the dqtted lines. (\u00ab) Write a formula for the area of the outside of the pan in terms of a and b. (b) Find the number of square inches of tin used to make a pan if a = 9.38 in. and b = 2.19 in. (c) Find the area if a = 73.6 and b = 3.2, if a = .83 and b = .065. 73. We have already used the fact that in a right triangle the square of a leg b equals the square of the hypotenuse h minus the square of the other leg a. {a) Write a rormula for finding a leg of a right triangle when the hypotenuse and the other leg are known. {b) Find b for each of these triangles: h = 4:1 and a = 40 h = ?>1 and a = 35 h = 626 and a = 624 h = 743 and a = 257 h a Ex. 71 370 POLYNOMIALS 74. The area of a washer is found by subtracting the area of the small circle from that of the large circle. {a) Using the letter tt for 3.1416, write a formula for the area of a washer whose radii are R and r. (6) Find the area of a washer if \/? = 1.75, 7 = 1.25, and tt = 3.14. How to find prime factors. 1. Look for a monomial factor. If there is one, remove it. 2. See if the remaining expression is the difference of two squares. 3. See if any of these factors can still be factored again. 4. Your answer will consist of all factors that cannot be factored again. Illustration. Factor: 3 \u2014 48 x. 3 – 48 X = 3 x(x^ – 16) = 3x(x” +4)(x” -4) = 3x(x^ +4)(x + 2)(x-2) Exercises Factor: 11 . 12. 16 x^ – 1 13. – 81 14. x^ – 1 16. – 16 16. – k 17. – xy^ 18. 2^ – 19. 3 a^b^ – 48 20 . tcR^ – irr^ 31. Using fractional exponents, factor as the difference of two squares: (a) X \u2014 1 (b) a \u2014 4 (c) r \u2014 s 32. Find the prime factors of x” \u2014 y” if: (_a) n = 2 (b) n = 4 (c) n = S or \u2014 a 9 3. 3 -3 b^ 4. 5a^b – 20 b^ 5. 2 -8y^ 6 . ax^ \u2014 ay^ 7. 18 2^ – 8 8. x^y – xy^ 9. 45 – 5 10. 9 x^ – X 21. x^\u00ae – 1 22 . 23. i x”* – 4 X 24. .5 – 2 25. 2 rR^h – 2 irr^ 26. i x^ – 3 27. 7y” – ly 28. – .01 29. – .0016 30. x^\u201d + ” – X\u00ae REDUCING FRACTIONS 371 REDUCING FRACTIONS What effect does it have on the value of a fraction if I multiply its numerator and denominator by the same number? Does the expression f equal Does the expression f equal If? What effect does it have on the value of a fraction if I divide its numerator and denominator by the same number? Does the expression equal f ? Does the expression equal f? What effect does it have on the value of a fraction if I add the same number to its numerator and denominator? Does the expression f equal f ? Does the expression f equal f ? What effect does it have on the value of a fraction if I subtract the same number from its numerator and de- nominator? Does the expression f equal f ? Does the expression jf- equal Rule. We can reduce a fraction to lower terms by dividing its numerator and denominator by any of theii common factors. Cancelling means dividing by a common factor. We can reduce if to f because 3 is a factor of both 9 and 15. Can we reduce to f by cancelling the 2\u2019s? Why not? 2^ + 1 Notice that 21 is 2 tens + 1. Can we reduce by ^ t “j- o dividing the 2 t from the numerator and the denominator? Why not? Is2t a factor of 2 ^ + 1? Of 2 \/ + 3? Illustration 1. Reduce to lowest terms. 3 – We can divide 3 into 9 and 15 because 3 is a 5 factor of both 9 and 15. 5 372 POLYNOMIALS 3 _ 3 _^2 Illustration 2. Reduce > \u201e , ^ \u2014 to lowest terms. ^ _ 3 – 3 – X) 6o\u2019 + 6-5i = lia(^\u2014 2 Cancel like factors: = ^ ~ \u2014 2 a Exercises 1. Which of these fractions can be reduced and which is in its lowest terms already: ff? fi? ff? |4? 2. Tell what was done to the left member to obtain the right member: A = ? ^ = 2^ ^ = y 12 3 9w 3 w txz t Reduce these fractions to their lowest terms: 3. 18 11. ax + ay 19. m \u2014 n 24 bx by m2 \u2014 W2 4. ax 12. 3x^ – 3;c2 20. fl2 _ 62 ay 6x^y \u2014 6 xy fl 2 -f ab 6. x\u2018^ 13. 4 w2 \u2014 6 \u00ab 21. 3y + 6 x^ 2 \u00ab2 3,2-4 6. -2x^ 14. 12 %2j;3 22. 3 a: + 3 y 3x\u2018^ 3xy – 6 xy^ 6 ac 2 \u2014 6y2 7. \u2014 5 – 15 16. 3x + 12 2a: + 8 23. 1 1 8. 8 ax^ 16. a \u2014 b 24. CO 1 – 6x^y fl2 – &2 4 – 2a: 9. \u2014 a^b^c 17. 5 c2 + 10 c 25. c2 – i a^bc^ 15 ca: + 30 a: c ^ 10. 2;c 18. 6 y + 2 3 + 9y 26. 7r\/?2 \u2014 7rr2 2 tvR \u2014 2 7rr How to find the product of two fractions. We multiply the numerators of two fractions together for the numerator of the product, and multiply the denominators together MULTIPLICATION OF FRACTIONS 373 for the denominator of the product. We can think of an integer as a numerator with 1 as its denominator. Illustrations. ^ = ^ _ ay 7\/ 7 ^b\/ b -3^ 5 5 is b b by bx Z 6x -12 x^ -4 _Z(x – 2)(;t\u2014 _ x – 2 2x +4 \u20199;c-18 Z(x-h^)Z(x–^) 3 3 To multiply fractions: 1. Find the prime factors of all numerators and de- nominators. 2. Write as a single fraction with all the numerators above the line and all the denominators below the line. 3. Divide numerator and denominator by common factors. Never cancel anything except a factor of the whole ex- pression. A term or a factor of a term must never be cancelled. Exercises Multiply, and reduce to lowest terms: 1 3. ^ 15 6. – y 9 -3 \u2018 x^ys – 5 5.9 6 10 g 3 \u00ab 2 b ‘ 4b’9a jg 5%2^8y22 4 yz 15 3. 3(-i) \u201e ax . aby by^ X axy . byz bxz ay^ 8 5x -2y y X 12. y z x 13. * be . ab 15. 2 \u2022 \u2014 X 5 ac \u00ab2 ~3~’x 14. 3 xz 4 y2 . 2 yz \u2019 3 T 2 . 2 xy .y^ . 2 22 ;c3 374 POLYNOMIALS 17. 3 4,. 1 24. 2 c 6 a63 18. 2×2 -1 26. X + 1 6 X 19. 3 w? \u20222\u00ab2 m^n 26. 20. ax \u2014 ay bx + by ab x2 \u2014 y2 27. 21. 28. 22. 2x – 2y 9×2 + 9xy 3x + 3y 4×2\u2014 4yx 29. 23. {a + &)2 {a -2 &)2 fl2 _ 9 62 fl2 _ 62 30. 2 x ^ – a:_4;c + 2 4 – 1 \u2019 ex 3x^y-12y3^ 2 x^y^ exy^ –ey^ 6x^\u201412x^y m + n ^ m{m \u2014 n)^ m \u2014 n 2 m3 \u2014 2 mn^ mx + my ^nx \u2014 ny px \u2014 py r% + ry c- ^ o I, 18 a^h + 6 a&2 * g2 \u2014 16 &2 ^ 3 a^h \u2014 ea^b^ (\u00ab \u2014 2 &)2 — 4: ba 22 _ 4 ^ 22-9 9 + 32*42 + 2 z2 DIVISION OF FRACTIONS In algebra the division of fractions is performed in the same way as in arithmetic, that is, we invert the divisor fraction and proceed as in multiplication. Illustrations. 3 . 9 _ 3 4 _ 1 8 \u25a0 4 8*9 6 a _^2 _ a ^ b _ 1 \u00a5 ‘ \u00a5’T\u00a5 ~ T\u00a5b 3 X – 6 . 4 X – 8 _ 3(x – 2) . x(2 X + 3) _ 3 X 4 X + 6 \u25a0 2 x2 + 3 X 2(2 X + 3) * 4(x – 2) 8 Exercises Divide: – a ^ c^d^ be ‘ \u2019 10 f \u2019 15 ^ 6 . 23 ‘ \u2014 Z 8.f -^(-3.) 15xy223 10x2y^3 14 m^n \u2018 21 mn^ ADDITION OF FRACTIONS 375 .. -I- ab . _ ab a -b \u2018 a -b .. \u2014 1 . 3 c \u2014 3 a – 3 ‘ 2a – 6 -1 . X 2 +1 X – 1 ‘ x+1 13 ^3^ ^ -xy A a — A b . 8fl+86 3a -6b 9 \u00ab – 18 6 w2 \u2014 4 mw \u2014 2 w * m2 + 2 w ‘ 3 mn jg n \u2014 3 ^3 \u2014 n w -f- 3 3 w 17. f ~ — (2s2 -8 S3) 5 _j_ 4 52 ^ ADDITION OF FRACTIONS Md these fractions: Notice that if the denominators of fractions are the same, we add the fractions by adding their numerators and writing this result over this common denominator. Find the missing numerators: 10. i = . 1 1 CO 16. ? _ 3 6 X 6x b 2b 11. 1 = \u2014 14. ? = \u2014 17. 5 5 15 y ay d dx 12. ^ = K 15. – = -r 18. – _ _ 2 8 a ab y Notice that we can multiply the denominator of a frac- tion by any number we please without changing the value of the fraction provided we multiply the numerator by that same number. Perform the additions: 19. 20 . xy xy xy 376 POLYNOMIALS 21 .- + – = _ + _= _ a: .V xy xy xy 22 . – + – d 3 w 2p X 3y Notice that to add fractions whose denominators are not alike: 1. We find the smallest number into which each denomi- nator will go a whole number of times. This is called the lowest common denominator or L.C.D. 2. We multiply each denominator by that factor that will give the L.C.D. 3. We multiply each numerator by the same number by which its denominator was multiplied. 4. We add the numerators and write the result over the L.C.D. Exercises Add: 4. – – – y V 2 . 2 2 a 2 a 11 . b – 2 ^ + 1 y y 3w \u2014 2 , 2w \u2014 1 + n \u2014 m 3 c – 2 3 – 2c 5 5 19. \u00ab _ 2 – L 5 20 . 21 . REVIEW OF FRACTIONAL EQUATIONS 377 1 + 22 2 \u00ab 3 _^3\u00ab 2 _ ^ 6 4 :r+3.2A: \u2014 1 A \u2014 3 x 2^3 6 xy yz zx Illustration. Add: 1 ?L + ? \u2014 . \u20149 + 3 X X – 2 3 x2 \u2014 9^x\u2018^–3x X ~2 3 {x + 3)(;c – 3) x{x + 3) x{x – 2) 3(;c – 3) x{x + 3){x – 3) x{x + 3)(x – 3) ^ x^-2x + 3x-9 x{x + 3)(a: \u2014 3) ^ + X – 9 x{x + 3)(a; \u2014 3) x^ -1 ,x + 3 28. 3 4 x^ ~2 X X 4r2 -9 6r 4- 9 4 3 29. CO 1 LO 2 +3z X + 1 ^ X \u2014 1 6 z + 9 4z + 6 5 2 30. 5 4- ^ y^ \u20149 y\u2018^ \u25a0-3y 3 w + 12 ^ 6n – 3 X – 2 x^ – 2 X 31. r2 +2s2 r \u2014 s X – 3 x^ ~ 9 r2 \u2014 s2 r + s REVIEW OF FRACTIONAL EQUATIONS Class Exercises Solve and check: 3. = 2 6. ? = 1 9. 2 = – – 5 h X 378 POLYNOMIALS 11 . 3 2 5 2 k k 3 + 2\u201c^ U. 1 + 4 = 6 CO 1 16. ^ – 3 = 2 5 >\u2022\u25a01 = .-l 20.7- Optional Exercises ‘r=’ 2 \/ + 7 5 CO rH 1 32. \u201d + 5 – 5s – 7 1 oo 2 s – 3 6 2 66. ^ 16. – 5 = 0 17. J 3 = – 5 8 u o 1 R 5y _2y _1 18 9 3 24. ? -2.1 X X 25. 4 – = 0 26. 2 – = 1 5 4 – X \u00ab; + 1 ^ ^ 2 = 1 \u2014 V ^- 1=2 1 r _ 9 2 2 \/ – 1 _ 5 6 2 ^ _ 3 + 2 2 ‘ s + 1 _ 3 \u201c6 4 34. l(y + 4)-l(y-2)=l 36. ?(>\u25a0 + 7) – i(j- – 2) = 6 m+2 2m-l \u201e 36. ^ g\u2014 = 2 37. i^-2*+4 = l^ o ^ 38. + 8 – = 0 39 ^ \u2019 X \u2014 1 X \u2014 3 40. -A- = 2 \u2014 5 y \u2014 4 _ 4x – 5 + 2 6x – 2 2 _ c +4 3 c + 9 Honor Work orK 3t _ 3^+21 2t -1 2t 44 -j- 45 . 46 . FRACTIONAL EQUATIONS 379 3 m + 4 1 _ 3 47. 2 5 5 m m 2 m + 10 y-2 y + 2 w 1 0 1 00 1 48. 1 4 X \u2014 1 1 – a; 2 + 1 22-1 FRACTIONAL EQUATIONS IN TWO UNKNOWNS Solve: ( 1 ) ( 2 ) If we clear the first equation of fractions, we shall have the equation 3 y + 2 x = xy,m which all three terms have unknowns. To avoid this, it is better to get rid of a letter before clearing of fractions. To get rid of y: Multiply (1) by 3: Multiply (2) by 2: Add: Now clear of fractions: Substituting = 6 in (1): ^ _6 ^7 X y 6 25 _ 25 6 25 X = 150 X = 6 3y + 12 = 6y y = i Check by substituting in both original equations: 8 _ 3 6 4 16-9 12 7_ 12 7 _ 12 7_ 12 7_ 12 380 POLYNOMIALS Exercises Solve and check: Work Problems 1. Three houses are built alike. Mr. Walsh painted his house in 8 days. It took Mr. Peach 10 days to paint his. If both should work together, how long would it take them to paint the third house? Time Part per Day Part in X Days Mr. Walsh 8 1 8 X 8 Mr. Peach 10 1 X 10 lb Notice that the whole piece of work is represented by 1. 2. One tractor plowed a certain number of acres in 6 hrs. Another tractor plowed an equal-sized field in 9 hrs. How long Courtesy of J. I. Case. PLOWING WITH A TRACTOR In modern farming the tractor is used because with it the farmer can plow several times as fast as he could with a team of horses. 381 382 POLYNOMIALS should it take both tractors together to plow a third field of the same size as each of the first 2? 3. Two machines do a piece of work in 5 hrs. Before the new machine was installed, it used to take the old machine 15 hrs. How long would it take the new machine alone? 4. The stream of water flowing into the new reservoir filled it in 50 days. Then the aqueduct to the city was opened. If this aqueduct can carry off a reservoir full of water in 45 days, how long will it be before the reservoir is empty? 6. Sarah can type a report in 6 hrs. Elizabeth takes 8 hrs. for the same work. After Sarah has worked for 2 hrs., Elizabeth is sent to help her. In how many more hours should the two girls complete the work? 6. Mr. Richmond worked 4 days on a piece of work. Then Mr. Hunt took it over and finished it in 3 days. Later he took over a similar piece of work on which Mr. Richmond had worked 6 days and finished it in days. How long would each alone have taken to do one of those pieces of work? No. of Part Done by Each Days 1st Job 2nd Job Mr. Richmond …. X 4 6 X X 3 3 Mr. Hunt y y 2y 7. Working together, Chester and Tom hoed an acre of potatoes in 3 days. On a second acre, Tom worked alone for 4 days, and then Chester, working alone, finished the work in 2i days. How long would each boy alone take to hoe an acre of potatoes? 8. One week John and Robert delivered their magazines in 4 hrs. The next week Robert worked alone for 3 hrs., and then John joined him, and they finished the work in 3 hrs. How long would it take each boy alone to deliver all the magazines? Chapter 16 THE QUADRATIC EQUATION A short cut in multiplication. Multiply 3a: \u2014 2by2x + 5. 3 a: \u2014 2 From what did you get the 6 ac^? The \u2014 10? 2 a: + 5 What two quantities were added to give 11 a:? 6 a:^ \u2014 4 X Where did the \u2014 4 x come from? The + 15 a:? 15 X – 10 6a:2 + 11 a: – 10 -6×2-1 (3x 6 x2 -}- 11 X – 10 I + 15 X- 1. The first term of the answer is the product of the first terms. 2. The second term of the answer is the sum of the outer and inner products. 3. The last term of the answer is the product of the last terms. class Exercises Multiply: (x + 2)(x -f 3) (X + 4)(x + 6) (X -f l)(x + 8) (X + 5)(x + 5) (3 -f- x)(4 + X) 6 . 7. 8 . 9. 10 . (X – 3)(x – 1) (X – 4)(x – 4) (X – 9)(x – 2) (1 – x)(2 ~ X) (5 \u2014 x) (3 \u2014 x) 11. (x+4)(x-3) 12. (x + 3)(x – 5) 13. (x – 7)(x + 2) 14. (x + 2)(x – 2) 15. (6 – x)(3 + x) Thought Question Can you discover an easier rule for multiplying binomials when the coefficient of x in both of them is 1? 383 384 THE QUADRATIC EQUATION Optional Exercises 16. (2x + 3)(x + 5) 17. (3 A +2)(5yl +3) 18. (2 b + 3)(2 & + 3) 19. (3n – l)(n – 5) 20. (5 a – 3)(4a +7) 21. (5 m + 4)(5tw – 2) 22. (7 – k)(4: -i-3k) 23. (w^ + 5) (2 m;, – 3) 24. (4 \u2014 n)(3 \u2014 4n) 25. (5h – 4)(4h + 5) 26. (3ab – c)(2ab + 3 c) 27. (4x^ -2y)(3x^ +4y) 28. (M2 -pq)(3M^ -^pq) 29. (abc – 1)(3 abc + 4) 30. (2x, +X2)(3 a;, \u20142x^) 31. (a + 1.2) (.5 a + .4) Honor Work 32. {2x+^)(^x -2) 33. (I -5m)(i + 3m) 34. (5\/ -4)(.2; – .05) 35. (.5 m; + 4)(2m; – .3) 36. (ax + l)(a^x \u2014 a) 37. (5x^ – 4)(3x^ -f 5) 38. (3 A* -2B^)(2A^ -3B^) 39. (ax + b)(cx + d) 40. (ax + b)(cx \u2014 d) 41. ((\/m;, + sm;2)(sm;, + dw^) FACTORING TRINOMIALS Can you discover what two binomials I multiplied to get + 5 a: + 4? What must I have multiplied to get What could I have multiplied to get + 4? Wpuld 2 and 2 give the middle term? Would 4 and 1 do it? If the product is + 4, are the signs in the factors alike or unlike? If the product had been negative, what could you say of the signs in the factors? If they are alike, must they be + or \u2014 to give 4- 5 %? Class Exercises Factor: 1. x’^ + 3 X + 2 2. x^ + 5 X + Q Z, x^ A- A- 3 4. x2 \u2014 5 % + 6 5. x^ – 4x A- 4 6. ^2 -5×4-4 7. x2 – 7 X 4- 12 8. x2 – 9 X 4- 20 9. x2 4- 7 X -F 6 10. x2 -9×4-8 11. x2 – 3x – 4 12. x2 4- X – 2 13. m2 \u2014 5 m \u2014 6 14. P A-3k – 28 15. m;2 – 4 m; 4- 3 16 . y 2 _ 53 , _ 14 THE QUADRATIC EQUATION 385 n. +d,x + 7 19. – 2 w – 15 18. a\u2018^ + a -12 20. – 2 R- 10 Optional Exercises 21 . 2 a;”” + 5 a: + 2 22. 2 + 5 fl + 3 23 . 2 6^ + 11 6 + 15 24. 6 + 13 M + 6 26. 2 + 11 + 5 26. 5 + 17 r + 6 27 . 4 ~ 12 :\u00bb: + 9 28 . 5 – 11 ;c + 2 29 . 12 – 19 A: + 6 30 . 8 Z” – 22 ^ + 15 31. 8 – 14 fl – 15 32. 6 \u2014 11 w \u2014 10 33. 12 3\/2 + 3 , _ 6 34. 16x^ + 8 x -15 35. 6 r2 – 13 r – 15 36. 15;c2 -{.I9xy – 10 37. 82 ^ -6 zy -9 y^ 38. 6 b^ +by – 35^2 39. 16 – 34 M – 15 40. 6 – a; – 12 a:2 Honor Work Find the prime factors of: 41. 6 ;c2 – 15 % + 6 42. 4 i?2 + 22 i? + 10 43. 3 + 9 a;2 – 120 % 44. 18 – 3 ^2 – 45 46. x^ 2 x^ – 3 46. 4y^ – 253\/2 +36 47. 32 + 14 m2 – 1 48. 8x^ -89 x^ 108 a; 49. a:2 – a: – .24 50. .03 a:2 – .4 a: – f THE QUADRATIC EQUATION An equation in which the highest power of the unknown letter is its square is called a quadratic equation. 3 + 2x + 5 = 0isa quadratic equation. If it is written equal to any other quantity than 0, we can always transpose so as to have the right member 0. A root of an equation is the quantity that when sub- stituted for the letter will make the equation an identity. When the equation is in the form 3A[:2-|-2A[:-f5 = 0, the root or answer should make the left member 0. If any quantity is multiplied by 0, the product is 0. Consequently, if a quadratic equation can be factored, any number that makes either factor 0, will make the left member 0, and will satisfy the equation. 386 THE QUADRATIC EQUATION Illustration. Solve: 2 \u2014 3 x \u2014 2 = 0 Factoring: (2 x + l){x – 2) = 0 Since a number that makes either factor 0 will make the product 0, we can get the roots by solving the two equations: 2 a; + 1 =0 and x \u2014 2 = 0 2x=-l X = 2 X = – i Therefore, we have two answers: x = \u2014 i or 2 Exercises Solve by factoring: 1. x\u2018^ – 3x + 2 = 0 2. a:^-5a:+6=0 3. A:^-4;r+3=0 4. – a; – 2 – 0 6. x ^ 3 X 2 = 0 6. 5a;\u2014 6=0 7 . a:^ – 12 = a: 8. 7 ;r + a:^ = – 10 9. 3 = 2 a: + a:^ 10 . 5 a: \u2014 + 6 = 0 11. 2 x”” – 3 X + 1 = 0 12. 3 x” – 7 X + 2 = 0 13. 6 x^ – 5 X + 1 = 0 14. 3 x^ + 8 X = 3 16. 2 – 9x – 5x^ = 0 16. 15x” + 17x + 4 = 0 17. 6x^ = 7x + 3 18. 4 x^ + 7 X + 3 = 0 19. 19x – 6 = 15 x” 20. 7 X = 6 x^ The plus or minus sign. How is it that the same equa- tion can have two different answers? Consider = 9. What is the value of x? Evidently x = 3, for 3^ = 9. But X could also equal \u2014 3 for ( \u2014 3)^ also equals 9. So x = + 3 or \u2014 3. Instead of writing the two answers separately, we often write x = dz 3, which we understand to mean that X can equal either + 3 or \u2014 3. We read it \u201cx equals plus or minus three.\u201d THE PERFECT TRINOMIAL SQUARE By the method you have just learned or by actual multi- plication, find the square of \u00ab + (c -{- ^)^ = a^ 2 ab -f- b^ THE PERFECT TRINOMIAL SQUARE 387 In the product: The first term is the square of a. The last term is the square of b. The middle term is double the product of a and b. Exercises Square these binomials: 1. X + y 3. a \u2014 b 2. 2 m n 4 . 5 x \u2014 2y 5 . 3a – 5b 7 . 6 m^ -3 y^ 6. 4x + 3y 8. 5 a^b^ + 1 Find the square root of: 9. 4 + 4 ;c + 1 10. + 6 m + 9 11 . 16 \/^ – 8 ^ + 1 12.^v^~l2v + 4 13. 4y^ + 20 yz+ 25 2 ^ 14. 1 \u2014 8 m + 16 16. 9x^ -30xy^ +25y^ 16. 36 + 60 + 25 In a perfect square, the middle term is double the product of the square roots of the end terms. Find the middle term that will complete the square: 17. x^ + +3\/2 18. +4 ^2 19. 9 m2 + +4 n 20. 4 3\/2 – + 25 21. 9x^ – +49 22. 4 ;?2 _l_ 9 ^2 23 . 163 ^ 2 ^ ^25 2 ‘ 24. 64 – + 9 r2 Illustration. Complete the square: 9 x”^\u2014 12 x The first term of the square root is 3 x. Since \u2014 12 x is double the product of 3 a: and another quantity, we can find the othei quantity by dividing \u2014 12 a: by double 3 ai: or by 6 a:. The sec- ond term, then is \u2014 2, and its square is + 4. So the square is 9 a: 2 – 12 ;c + 4. Complete the square: 26. a:2 + 6 a: + 26. m2 \u2014 10 m + 27. jfe2 _ 14 _{_ 28. + 8 y + 29. 4 a:2 + 12 a: + 30. 9 a:2 – 30 a: + 31. 4 a:2 – 4 a: + 32. 4 a:2 + 2 a: + 388 THE QUADRATIC EQUATION Solving the quadratic equation by completing the square. Not all quadratic equations can be solved by factoring. Consequently we must learn a method that can be used in all cases. This method is called completing the square. Illustration 1. Solve: \u2014 6 x \u2014 7 = 0 Transpose the \u2014 7: x^ \u2014 6x =7 Add 9 to complete the square: \u2014 6 :r + 9 = 16 Since it is now a perfect square, take the square root of both members: x \u2014 3 = \u00b14 Transpose the \u2014 3: A:=db4+3 Using first + 4, then \u2014 4: x = 7 or \u2014 1 Illustration 2. Solve: \u2014 3x \u2014 4 = 0 Transpose the \u2014 4: 5 x^ \u2014 3 x = 4 Divide by 5: x^ \u2014 ^ x = ^ id) = ^ \u2022 Square and add: x^-^x–Th=i + T%o Combine terms: = Take the square root: a: \u2014 = =b ^V89 a; = \u00b1 iV \u2019’\/S9 _ 3 =h V89 10 Exercises Solve by completing the square: 1. ;c”-6x + 5= 0 2. – 8 ;r + 15 = 0 3. a:” – 10 a: – 24 = 0 4. a:^+4a:-5=0 6. a:^ – 9 a: + 20 = 0 Q. x^ \u2014 X \u2014 12 = 0 7. 2×2-5a:-3=0 8. 3a:2-7a: + 2= 0 9. 3 + 8 a: = 3 10. 2 + 5 = 11 a: 11. – 4 a; – 6 = 0 12. + 3 a: – 5 = 0 13. – 4 X = 2 14. x^ + 7 = 9 X 15. 2x^ – 6x + 3 = 0 16. 3 x^ – 5 X – 7 = 0 17. 2 x^ + 7 X = 11 18. 5 x^ = 3 X + 1 19. 3×2 – 11 = 4x 20. 2 X – 3 x2 = – 4 21. 5 X – 2 = 2 x2 22. 4 x2 + 1 = 4 X 23. x2 – 6 X = 0 24. x2 – ^ X – 1 = 0 THE QUADRATIC FORMULA 389 THE QUADRATIC FORMULA Solve the equation: ax^ bx -{\u25a0 c = 0, for ;c b c Dividing by a: x^ – x + – =0 a a Transposing + – ;c \u00a3 a a a square: Completing the &2 ^ \u00a3 4^ a b^ \u2014 A ac 4 \u00ab2 Taking the square root: x ^ 2a \u00b1 ^lb^ – 4 ac 2a Transposing x & dr 4 ac 2 \u00ab 2 a This is a general solution, for a, b, and c can stand for any numbers. For example, lia = 2,b = \u2014 5, and c = 3, the equation ax^ bx + c = 0 becomes 2 ;i :2 – 5a;+3 =0, and the answer + 5 \u00b1 V25 – 24 \u2014 & =b V 62 \u2014 4 ac 2 ~^ becomes x = x = 4 + 5=bl 4 \u2022 Check by substituting the answers in the original equation. Historical Note on Solving the Quadratic Equation: Although the quadratic equation was not entirely unknown in early times, and a few special cases were solved geometrically by Euclid (300 B.C.), the Hindu writers deserve most of the credit for the general solution. Brahmagupta (628) gave a rule equivalent to a formula, and Sridhara (1025) explained the method of completing the square. Long afterwards Harriot (1631) solved quadratic equations by factoring. Vieta (about 1580) made the advance of solving these equations analytically instead of geometrically. 390 THE QUADRATIC EQUATION The formula. We can then solve the equation: ax^ hx c = 0 where a, b, and c are any numbers by substituting in the formula: \u2014 b \u00b1 \u2014 4ac Illustration. Solve : 3 \u2014 ix \u2014 4\u2014 0 Here a = 3, b = – 4, and c = – 4. Then – 4 ac = + 48. \u201e _ \u2014 & zb ylb^ \u2014 4 ac _ + 4 zb V16 + 48 6 _ + 4 zb 6 + 4 zb 8 6 2 or \u2014 Class Exercises Solve by means of the formula, and check: 1. x^+3x–2=0 2 . – 6 = 0 3. x^\u2014 4x + 4= 0 4. 2 x^-5x + 2= 0 6. 6%2-2a:-4=0 6. – 4 =4y 7. 4 – 12 Jfe + 9 = 0 8. 3 + 2 m; = 8 d. 2 =3+9 h 10 . 17 x = 12%2 +6 * Note: If you always make the coefficient of x^ positive, you can check your signs at this point by noting that the signs of \u2014 b and of \u2014 4 ac are always the opposites of those of the second and third co- efficients of the equation. 3x^- 4x -4 = 0 T t i i ^ + 4 \u00b1 V16 + 48 “” 6 THE QUADRATIC FORMULA 391 Optional Exercises 11. – 2 a: – 1 = 0 12 . + X – 5 = 0 13. – 6 ;\u00bb: + 2 = 0 14. 3 _ 5 X – 4 = 0 16. 2 + 7 a; + 2 = 0 16. x{x – 4) = 7 17 ^ =-A_ ^^3 x + 1 18. X \u2014 2 X – – = 3 19. % + 7 + – = 0 X 20 . X 2 X – 3 Honor Work Illustration. Find the roots nearest hundredth. of 2×2-4x + l= 0 ^ _ + 4 =b V16 – 8 2. 8 2 8 4 V8.000000 _ + 4 d= V8 4 _ + 4 \u00b1 2.828 4 48 1 4 00 3 84 562 1 1600 4 1124 6.828 1.172 5648 1 47600 4 4 45184 = 1.707 or .293 = 1.71 or .29 Find the roots of these equations to the nearest hundredth: 21. _ 4 _ 7 =3 0 2^2. 5 x2 _ 3 – 4 = 0 23. 4 – :r2 = ^ 24. _ .5 + .01 = 0 25. .08 %2 ~2x – .03 = 0 26. 3×2 – .42 X = 1.46 27. – 3 X + 1 = 0 28. x^ =1 – X 29. _ 5 = 2 30. X + 5 = 31. X = 8 X – 5 32. X – 2 – X ^ 2 33. 34. X – 2 1 1 X \u2014 35. X – 2 \u2018 X 7 X + 5 36. ^ ^ ^ = 2 = 5 391A THE QUADRATIC EQUATION How to solve quadratic equations graphically. In Chap- ter 9, we learned how to estimate the roots of first degree equations from their graphs. In much the same way, we can find the roots of quadratic equations or of higher equations from their graphs. Illustration: Solve x\u2018^\u2014 ix\u2014 5 = 0 graphically. We shall first draw a graph of y = x^ \u2014 i x \u2014 5 by substi- tuting values of x and finding the corre- sponding values of y. When% = l, y = 12 -4-1-5= -8 Whenx = 2, y = 22-4-2-5=-9 Making a table of these values, we have: r-2-1 0 1 2 3 456 y 7 0-5-8-9-8-507 Now we plot these points and draw a smooth curve through them, as shown in the figure. This curve is called a parabola. To solve the equation x^~Ax \u2014 5 = 0 from the graph of \u2014 4 X \u2014 5 = y, we notice first that y must equal 0. Now all points whose y coordinates are 0 lie on the ;r-axis, so evidently the roots will be the x values of the points in which our curve x”^ \u2014 Ax \u2014 5 = y cuts the x-axis. These values, \u2014 1 and + 5, are then the roots of our quadratic equation. From this graph we can see more clearly why a quadratic equation has two roots. Thought questions. As x increases from \u2014 2 to +6, how does y change? For what values of x is y negative? How are these values related to the roots of the equation? What value of X gives the lowest point on the curve? How is this value related to the roots? Has this curve an axis of symmetry? What is the relation of this axis of symmetry to the line segment joining the points for which x = \u2014 1 and x = + 5? GRAPHIC SOLUTION OF QUADRATICS 391B Exercises Solve graphically: 1. x^-^x–2> = 0 2. a:2-6a; + 8= 0 3. a :2-6 a ; + 5 = 0 x-^-2x-?>=0 6 . – 6 = 0 6 . x^ + 6x–S = 0 7. x^ + 5x-{-4 = 0 8. x^ -3x = 0 9. x^ – 2x = S 10 . x^ + 2x = 3 11 . 2x^ + x- 6 = 0 12. 3x^-2x-3 = 0 13. 2 ;c2 + 3 a: – 5 = 0 14. 6x^~5x\u2014 6 = 0 15. 5x^-x-6 = 0 10. x^ – 9 = 0 The parabola in war. What will the bomb hit? If an aviator drops a bomb when he is directly over his target, will he hit the target, or should he have dropped it some time before arriving overhead? As the plane is moving forward, the bomb is moving horizontally with the speed of the plane, say 300 ft. a sec. But it is also pulled down by gravity a distance d, given by the formula 6? = 16 where d is the distance in feet and t the time in seconds. Let us make a table of values for t and d. t 123456789 10 Horizontally 300 600 900 1200 1500 1800 2100 2400 2700 3000 Vertically 16 64 144 256 400 576 784 1024 1296 1600 392 THE QUADRATIC EQUATION The Quadratic Equation in Geometry 1 . Find the dimensions of a rectangle whose area is 104 if its base is 5 more than its altitude. 2. Find the side of a square whose area is doubled when the side is increased by 3. 3. A square rug is 2 ft. from each wall of a room and covers half the floor. Find the length of the rug to the nearest tenth of a foot. 4. The base of a triangle exceeds the altitude by 4, and the area is 48. Find the altitude. 5. A square piece of tin is to have small squares cut from its corners and the sides bent up to make a pan 2 in. deep. What dimensions should the piece of tin have at first if the pan is to hold 98 cu. in.? 6. If AB is a tangent to the circle, it is found that the square of AB equals the product of BC and BD. If AB is 6, and CD exceeds BC by 1, find BC. 7. When two chords, AB and CD intersect in a circle, the product of the segments, AE X EB and CE X ED, are always equal. If AE and EB are each 4, and if the total length of CD is IG, find the segments of CD. 8. The area of a field is 96 sq. rds. What are its dimensions if it requires 40 rds. of fence to enclose it? 9 . The sides of an equilateral triangle are increased respec- tively 1, 3, and 5. If it then becomes a right triangle, what was the side of the equilateral triangle? Remember that in a right triangle A – where a, b, and c are the sides. 10 . The diagonal of a square is 1 in. longer than a side. Find the side of the square to the nearest tenth. 11. The length of a picture is to be 20 in. Artists believe that EQUATIONS LEADING TO QUADRATICS 393 the picture would be the most pleasing if the width w were such that \u2014 = \u2014 . To the nearest tenth of an inch, what should w 20 – w be the width of the picture? 12. Using the formula in exercise 11, find the width of a picture whose length is 4 in. The Quadratic Equation in Science 1. If a ball is shot upward at a speed of 128 ft. a sec., the height it will reach in i sec. is given by the formula 128 t \u2014 16 1\u201c^. After how many seconds will it have risen 256 ft.? 2. When an aviator is traveling over sea or level ground, the greatest distance on the earth that he can see is given by the formula ^ 8000 h + where h is his height in miles. How high must he ascend to see 100 mi.? 3. The height of a circular arch is given by the formula 4: \u2014 8 hr = 0, where h is the height, r the radius, and s the span. How high must an arch be built to have a span of 32 ft. and a radius of 20 ft.? EQUATIONS LEADING TO QUADRATICS Sets of two equations. You have found that it is often easier to solve a problem by using two unknowns than by using only one. Sometimes one of these equations is of the second degree, that is, contains a letter squared or the product of two letters, so that it is necessary to solve a quadratic equation. How to solve sets of equations in which one is of the second degree. Illustration: Solve: = 10 2x + y = 7 1. Solve the first-degree equation for one letter. It is better in this example to solve for y, since it has no co- efficient and we can avoid fractions: y = 7 – 2x 394 THE QUADRATIC EQUATION 2. Substitute this value of y in the second-degree equa- tion: jc” + {7 -2 xy = 10 3. Now solve this equation: + ^9 – 2Sx + 4 =10 5 a: 2 – 28 X -f 39 = 0 5 a:- 13=0 :r-3=0 4. Substitute these values in the first-degree equation so as to find y, and group your answers: When X = -1^ y = 7- When X = 3 y = 7 – 6 = 1 5. Check in both equations: Answers X 3 1 3 5\u201c y 1 9 5 (\u00a5)^ + (1)^ i 32 + 12 1 169 1 81 25~ 1 25 1 10 9 + 1 1 10 10 = 10 10 = 10 \u00a5- + f 1 7 6 -fl i 7 7 = 7 7 = 7 Exercises Solve and check: 1. X -f y = 5 x” -hy” = 13 2. xy = 15 3 . X -f y = 8 3 M -h W = 8 _ 3;2 ^ 24 wy A- wy = 2x -1- y = 11 Wy \u2014 = xy + y^ =8 9 . x^ -h xy -f- y” X \u2014 2 y = 10 X -y = 2 -h2 6″ = 11 10 . d = 2 \u00ab -h & = 5 -f \/ = 66 6. c – <\/ = 1 -3^ = 17 7. + MW = – 10 = 4 RADICALS 395 Equations solved by factoring. Illustration: Solve: 2 \u2014 20 — 1% x = 0 Factoring: 2 \u2014 10 + 9 ) = 0 2x{x'^ – 9 )( a ;2 – 1 ) = 0 2 x{x + 3)(x – 3)U + l)(x – 1) == 0 Setting each factor equal to 0: 2 X = 0 X = 0 a:+3=0 X = – 3 X \u20143 = 0 X = + 3 a: + 1= 0 X = ~1 X -1 =0 X = + l Exercises Solve by factoring: 1. x^ -3x =0 2. x^ 5 X 3. x^-4x = 0 4. x^ ==9 X " 6 . 7x^ – 28 = 0 e. 3 x^ = 48 7. x^ -x^ =0 8. + 2 = 35 ;c 9. 3;c^ + 12:r = 15 x^ 10. 4x^ ==x^ 11. – 10 a;'* + 9=0 12. 4a;^ -37 a;'* = – 9 13. 3a;^ – 15a;^ + 12 = 0 14. a;^ – 2 + 1 = 0 15. 17a;2 = + 16 16. 4 a;^ + 1 = 5 17. 5 a;^ – 25 a; 3 + 20 ;r = 0 18. 3 a;\u00ae + 3 X = 6 19. A;f – 5 + 6 = 0 20. – 3 A:i + 2 = 0 RADICALS One day Dorothy did these square-root exercises for home work: V2 = 1.4142 V3 = 1.732 V8 = 2.8284 Vl2 = 3.464 VI8 = 4.2426 V27 = 5.196 Katherine pointed out that the square root of 8 was exactly twice as large as the square root of 2, and that the square root of 18 was just 3 times as large. Dorothy then noticed that the same relation held for the V3, Vl2, and 396 THE QUADRATIC EQUATION V27. She remarked, \u201cWhat a lot of work I could have saved myself if I had only known this in advance.\u201d Now there are many square roots that can be obtained from others by a simple multiplication. Can you discover the secret for yourself so you can avoid all the labor of extracting these square roots? Let us examine these radicals. Since 8 = 4×2 and 4 is a perfect square, V8 = V4 X 2 = 2V2, for we can take the 4 out of the radical. Similarly VT8 = V9 X 2 = 3V2; so you see why V8 and VTS were 2 and 3 times the V2. This process of taking a factor that is a square out of the radical is called simplifying radicals. Of course if we had a cube root instead of a square root, we should have to take out a factor that was a cube. Illustration. Simplify V75. The largest factor of 75 that is V75 = V25 X 3 a square is 25. Then V25 = 5 = 5 V3 Class Exercises Simplify: 1. V12 6. V44 11. V96 16. ^54 2. V20 7. V45 12. V54 17. ^32 3. V32 8. V98 13. ^16 18. ^80 4. V28 9. V24 14. ^24 19. 2VI2 5. V40 10. V48 15. ^40 20. 5V27 Optional Exercises 21. ' 23. 25. 27. 22. Vl35 24. V1.8 26. V.08 28. V.24 If yl2 = 1.4142, V3 = 1.732, and ^J5 = 2.236, find cor- rect to three significant figures: 29. V50 30. V20 31. 32. V75 33. V^8 34. V98 35. V80 36. V48 RADICALS 397 Honor Work Simplify: 37. 39. 41. 5ay9a^ 43. 2 m^20 m'^n^ 38. V18g3 40. 42. ^24 44. ^32 How to get a radical out of a denominator or a denom- inator out of a radical. _ They are really the same thing under two names, for is equal to \/I. V3 Vs Illustration 1. 4 Find to four decimal places the value of \u2014 V2 Since V2 = 1.41421, we obtain 4 -\u25a0 1.41421. By long division this gives 2.8284. This division is hard work. Can we find an easier way? If we could first get the radical out of the denom- inator, we would not have to divide by this long decimal. And here is the way to do it. Multiply both numerator and denominator by V2: A- = 4V2 V2 V2 V2 = 2V2 Illustration 2. Simplify: Multiply both numerator and denominator by 3: = iV15 Exercises Simplify: 398 THE QUADRATIC EQUATION <o 11 001 11.^ 13.^ 16.^ V3 V18 V5 V32 10. 12. ^ 1 , 3^ 16. V2 V20 V8 V50 PUZZLE PROBLEMS Coin Puzzles I have 16 coins, nickels and dimes, and they are worth $1. How many nickels have I? Kind of Coin Number Value in Cents n 5 n d IQd n -r and Z E. CONGRUENT TRIANGLES Two triangles are congruent (^) if one of them can be placed so that every part of it will fit exactly on the cor- responding part of the other. Corresponding sides and corresponding angles of con- gruent triangles are equal. This is one of the most im- portant ways of proving that a line equals another line or that an angle equals another angle. 406 DEMONSTRATIVE GEOMETRY How to copy a triangle. Here are three methods of con- structing a triangle congruent to a given triangle, ABC. First Method. 1. On the line EH, measure a length EF equal to BC. 2. With E as center and BA as radius, draw an arc. 3. With F as center and CA as radius, draw another arc cutting the last one at D. 4. Draw DE and DF. \/ Do you think that ADEF is \/ Nv congruent to AABCl How many \/ 1 sides of A.ABC did we measure? H Do you think we would always get ‘ a triangle congruent to A ABC if we make its three sides equal to those of AABCi Second Method. 1. On EH measure a length EF equal to BC. 2. At E make an angle equal to ZB. Now how can we finish the tri- angle? If we measure a length from E on ED equal to BA and complete the triangle, do you think it would be congruent to AABC? How many sides did you meas- ure? How many angles? Is the angle that you measured included by the sides that you measured? Do you think that a triangle would always be congruent to AABC if you made two of its sides and their included angle equal to those of AABCl Third Method. After you had made EF equal to BC and Z E equal to ZB m Method 2, could you have completed the work differently? If you made an angle at F equal to Z C and extended the line until it crossed ED, do you think your tri- angle would be congruent to AA5C? How many sides did you measure? How many angles? Is the side included by the angles? Do you think that a triangle A Courtesy of Tennessee Valley Authority. TRANSMISSION TOWER Notice the similar and congruent triangles which have been utilized by- engineers in the construction of this tower. 407 408 DEMONSTRATIVE GEOMETRY would always be congruent to AABC if you made two angles and their included side equal to those of AABCi If you have constructed these triangles correctly, you have discovered that: 1. Two triangles are congruent, if two sides and the included angle of one equal respectively two sides and the included angle of the other, {s.a.s.) 2. Two triangles are congruent, if two angles and the included side of one equal respectively two angles and the included side of the other, {a.s.a.) 3. Two triangles are congruent if the three sides of one equal the three sides of the other, (s.s.s.) Exercises 1. Draw an angle of about 70\u00b0. Now construct a triangle having two sides respectively 2 in. and 3 in. and this angle as their included angle. 2. Draw an angle of about 100\u00b0. Repeat the above construc- tion using this angle. 3. Construct a right angle. Then construct an isosceles tri- angle having this angle as vertex angle and a leg equal to 3 in. 4. Can you construct a right triangle {a) Which has all sides unequal? (6) Which is isosceles? (c) Which is equilateral? 5. Construct angles of 90\u00b0 and 45\u00b0. Then construct a tri- angle having a base 3 in. long included by these angles. 6. Construct a right triangle whose legs are li in. and 2 in. 7. Construct an isosceles triangle and bisect its vertex angle. Is this line perpendicular to the base? Does it bisect the base? 8. Construct a triangle that is not isosceles and bisect its vertex angle. Is this line perpendicular to the base? Does it bisect the base? 9. Can you construct a triangle whose sides are: (\u00ab) 2 in., 3 in., 4 in.? (c) 2 in., 3 in., 6 in.? (&) 2 in., 3 in., 5 in.? {d) 2 in., 3 in.. 2 in.? AXIOMS 409 THE THEOREM In geometry, we prove that a statement is true the conditions given are true. This statement is called a theorem. It consists of two parts \u2014 (1) a part that you are to accept as a starting point called the \u201cgiven,\u201d and (2) the part you are to prove. The given par;t is usually either a clause beginning with \u201cif,\u201d or it is the subject of the sentence. In the latter case, the predicate is the part to be proved. When there is an \u201cif\u201d clause, the independ- ent clause, that not containing the \u201cif,\u201d is the part to be proved. Of course, if the part given is not true, the conclusion \u201cproved\u201d may not be true. This is the way many ad- vertisers fool people. They begin with a false assumption and then make statements that would be true ij the assumption had been true. You must therefore examine carefully the statements assumed as facts before you be- lieve the conclusions drawn from them. Read some advertisements, and think over whether their \u201cfacts\u201d are necessarily true. Then see if the con- clusion drawn would follow from these \u201cfacts.\u201d Most arguments result from a disagreement as to the facts to begin with. One says that a thing is true; the other denies it; and they get nowhere. If you wish to con- vince a person, be sure to start with something that he will admit is true. Then be sure that you have correct reasons for your conclusions. AXIOMS If we are to give a reason for every step in a proof, we must agree on some facts that we can give as reasons. Let us list a few such statements, most of which you have already used. They are called axioms. The axioms. 1. In any process a quantity may be substituted for an equal one (called \u201csubstitution\u201d). 410 DEMONSTRATIVE GEOMETRY 2. If equals are added to equals, the results are equal. 3. If equals are subtracted from equals, the results are equal. 4. If equals are multiplied by equals, the results are equal. (Special case: Doubles of equals are equal.) 5. If equals are divided by equals, the results are equal. (Special case: Halves of equals are equal.) 6. The whole equals the sum of all its parts. Exercises 1. lia = b and b — 2. i a = b, c = d. = c, why does a = ci and b = d, why does a = c? 3. If a = c and b = d, d K M does a + 6 = c + . 8. If AD = BC and DC = AB, prove that AABC ^ AACD. 9. If the lines FG and KH bisect each other at M, prove AFKM ^ AGHM. Exs. 7, 8 Ex. 9 Ex. 10 10. If QS bisects Z PQR, and PR \u00b1 QS prove APQS ^ ARQS. 11. If ZABC is AABD ^ ACBD. 12. If ZABC is bisected by BE, and AB = BC, prove bisected by BE, and Za: = Zy, prove AABD ^ ACBD. 13. If FG _L FL and KL \u00b1 FL, and KG passes through the middle point of FL, prove that AFHG ^ ALHK. 14. The bisector of the vertex angle of an isosceles triangle divides the figure into two congruent triangles. 15. Two right triangles are congruent, if a leg and the adjoining acute angle of one equal respectively a leg and the adjoining acute angle of the other, 16. Two isosceles triangles are congruent, if a leg and the vertex angle of one triangle equal a leg and the vertex angle of the other. 17. Two right triangles are congruent, if the legs of one equal respectively the legs of the other. 18. If the bisector of an angle of a triangle is perpendicular to the opposite side, the triangle is isosceles. 19. If two opposite angles of a four-sided figure (quadrilateral) are bisected by the line joining their vertices, the quadrilateral has two pairs of equal sides. 20. Prove the construction for bisecting an angle. PARALLEL LINES 415 PARALLEL LINES Facts you already know. 1. Parallel (||) lines are straight lines in the same plane that cannot meet however far ex- tended in either direction. 2. A transversal is a line that cuts two or more other lines. 3. If two straight lines are cut by a transversal, the angles are named as follows: A w x’, y, z, are interior A. A w, X, y z’, are exterior A. A pair of angles are alternate, when they are on opposite sides of the transversal, and one at each vertex, as ^ y and w’, z and x’, w and y’, and x and z’. The pairs y and w’, and z and x’ are alternate interior angles. The pairs w and y’, and x and z’ are alternate exterior angles. The pairs w and w x and x’, y and y’, and z and z’, are corresponding angles. When alternate interior angles, or any of the above pairs of angles are mentioned, it is understood that there are two straight lines cut by a transversal. 4. Two lines are parallel if their alternate interior angles are equal. 5. Two lines are parallel if their corresponding angles are equal. 6. Alternate interior angles of parallel lines are equal. 7. Corresponding angles of parallel lines are equal. 416 DEMONSTRATIVE GEOMETRY Parallel line assumption. Through a point not more than one line can be drawn parallel to a given line. A quadrilateral is a figure formed by four straight lines that enclose a part of the plane. Exercises Prove that: 1. Lines perpendicular to the same line are parallel. 2. AB\\ CD ii: {a) Ap = 65\u00b0 and Z.w = 65\u00b0 {b) Z.p = 65\u00b0 and Z;r = 115\u00b0 (c) Z-p = 70\u00b0 and Zy = 70\u00b0 id) Zp = 75\u00b0 and Z 2 : = 105\u00b0 {e) Zr = m\u00b0 and Zx = 180\u00b0 \u2014 m\u00b0 3. Two lines are parallel if their alternate exterior angles are equal. E Ex. 2 Ex. 5 Ex. 6 4. Alternate exterior angles of parallel lines are equal. 5. If Zw = Zx, then Zy = Zz. 6. If PQ II RS and QR |1 ST, then ZQ = ZS. 1. i Zw = Zz and Zx = Zy, then AB jj FD. 8. If AB J_ BF and DF J_ BF, BH bi- -A sects ZFBA, and FC bisects ZBFD, then BH and CF are parallel. 9. If GH and KL bisect each other, prove that GK | LH. 10. A line perpendicular to one of two parallel lines is per- pendicular to the other. 11. Lines parallel to the same line are parallel. (Draw a transversal and use corresponding angles.) 12. If the opposite sides of quadrilateral ABCD are equal, THE SUM OF THE ANGLES OF A TRIANGLE 417 they are parallel. Draw a line joining the opposite vertices of the quadrilateral. 13. If AD and BC are equal and Ax = Ay, then AB is parallel to DC. ^ 14. If the opposite sides of a quadrilateral ^ are parallel, they are equal. \/ \/ 16. If two sides of a quadrilateral are both ^ \/ equal and parallel, the other two sides are ^ equal and parallel. A p 16. Prove that the sum of the angles of triangle ABC equals a straight angle. 17. Prove that the ex- terior angle FHl equals the sum of angles F and G. The sum of the angles of a triangle is a straight angle. A B c Given: A ABC. To prove: AA-{- AB– Proof: Statements Extend BC to D and draw CE | 1. Ax + Ay + Zz = I St. Z. 2. Zy = AA. 3. Zz = AB. Ax = I straight Z . Reasons BA. 1. A st. Z is an Z whose sides lie in a st. line, etc. 2. Alternate interior A of parallel lines are equal. 3. Corr. A of parallel lines are equal. 4. Substitution. 4. Ax+ AA-h AB = 1st. Z. Exercises 1. If two triangles have two angles of one equal respectively to two angles of the other, their third angles are equal. 418 DEMONSTRATIVE GEOMETRY 2. In A ABC, find the number of degrees in Z C if: {a) A = 38\u00b0 and B = 74\u00b0 {b) A = B = C (c) B = 48\u00b0 and A = C (d) C = A+B (e) A = m\u00b0 and B = rf if) C + A = 125\u00b0 and C + ^ = 130\u00b0 3. If CD is perpendicular to the hypotenuse of right AABC, then ZACD = ZB. 4 . If the sum of two angles of a triangle equals the third angle, the triangle is a right triangle. 6. If two angles of a triangle are 30\u00b0 and 60\u00b0, what angle is formed by their bisectors? 6. A triangle cannot have more than one right angle or more than one obtuse angle. 7. In AABC find the number of degrees in each angle, if: (a) Z 5 is twice ZA, and Z C is 20\u00b0 more than ZA (b) ZB is 3 times Z A, and Z C is twice Z B (c) Z B exceeds Z A by 5\u00b0, and Z C exceeds Z B by 5\u00b0 id) The sum of Z A and ZB is 130\u00b0, and 4 times Z A ex- ceeds 5 times Z 5 by 16\u00b0 ie) ZB is twice ZA, and ZC is ^ the sum of ZA and ZB. 8. Two triangles are congruent, if two angles and a side opposite one of them of one triangle equal two angles and the corresponding side of the other. Are the third angles equal? Why? 9. Two right triangles are congruent, if the hypotenuse and an acute angle of one equal the hypotenuse and an acute angle of the other. 10. If two angles of a triangle are equal, the bisector of the third angle bisects the triangle. 11. If in AABC, ZB = ZC, then AB = AC. 12. In congruent triangles, corresponding altitudes are equal. 13. If two sides of a triangle are equal, the bisector of their included angle cuts the figure into two congruent triangles. 14. Perpendiculars to the sides of an angle from a point on its bisector are equal. THE ISOSCELES TRIANGLE 419 THE ISOSCELES TRIANGLE The base angles of an isosceles triangle are equal. Given: AB = AC. To prove: LB = ZC. Proof: Statements Reasons Let A Z) bisect LA. 1. In ^ABDBndACD, AD = AD. 2. AB = AC. 3. Z% = Z>;. 4. AABD ^ AACD. 5. LB = LC. 1. A line equals itself. 2. Given. 3. The bisector makes two equal A. 4. Two triangles are congruent if 2 sides and the included angle of one equal, etc. 5. Corr. ^ of ^ A are =. Exercises 1. An equilateral triangle is equiangular. 2. In an isosceles triangle, the exterior angles made by pro- ducing the base are equal. 3. Two isosceles triangles are congruent if the base and a base angle of one equal the base and a base angle of the other. 4. The lines from the vertex of an isosceles triangle to the trisection points of the base are equal. (To trisect is to ^ divide into three equal parts.) 6. If the base BC of isosceles triangle ABC is extended so that BD = CE, then LD = LE. A Exs. 5, 10 420 DEMONSTRATIVE GEOMETRY If two angles of a triangle are equal, the sides opposite them are equal. Given: AB = AC. To prove: AC = AB. Proof: Statements Let AZ> bisect AA. 1. In AABD and ACD, AD = AD. 2. Ax = Zj. 3- AB = AC. 4. AADB = A ADC. 5. AADB ^ AADC. 6. AB = AC. Reasons 1. A line equals itself. 2. The bisector makes two equal A . 3. Given. 4. If equals are sub- tracted from equals, etc. 5. 2 A are ^ if 2 A and the included side of one, etc. 6. Corr. sides of ^ A are =. Exercises 6 . An equiangular triangle is equilateral. 7. The legs of a right triangle are equal if one of its acute angles equals 45\u00b0. 8. If two angles of a triangle are 70\u00b0 and 40\u00b0, the triangle is isosceles. 9. If any angle of an isosceles triangle is 60\u00b0, the triangle is equilateral. Prove two cases. 10. i AB = AC and A DAB = ABAC, then BD = CE. (See figure for Ex. 5, p. 419.) 11. Construct an angle of 60\u00b0, 30\u00b0, 120\u00b0. ^ 12. Trisect a right angle. eX 13. Find the number of degrees in each X. angle of an isosceles right triangle. 14. If BD bisects A ABC and EB = ED, thenEDll^C. 15 . If bD bisects A ABC and ED H BC, ^ then AEBD is isosceles. 16. If ED 1 1 BC and EB = ED, then BD bisects Z ABC. 17. If PQ is parallel to a leg KL of isosceles triangle KLM, then APQM is isosceles. SIMILAR TRIANGLES 421 SIMILAR TRIANGLES Facts you already know. 1. In two similar triangles, the corresponding angles are equal, and the corresponding sides are proportional. 2. Two triangles are similar if two angles of one equal two angles of the other. 3. Two right triangles are similar if an acute angle of one equals an acute angle of the other. 4. In a proportion, the product of the means equals the product of the extremes. Exercises 1. Are triangles KLM and PQR similar if: (\u00ab) AK = 41\u00b0, ZL = 73\u00b0, ZQ = 73\u00b0, and Zi? = 86\u00b0? ip) \/LK = 67\u00b0, ZL = 54\u00b0, ZQ = 54\u00b0, and Z.R = 59\u00b0? 2. A tower casts a shadow 200 ft. long when a vertical 8-ft. pole casts a shadow 10 ft. long. How high M is the tower? 3. Triangles similar to the same triangle | are similar to each other. K o L 4. In A KLM, NO is perpendicular to KL, and M is a right angle. Prove that AKNO is similar to AKLM. 5. Two isosceles triangles are similar if a base angle of one equals a base angle of the other. 6. Two isosceles triangles are similar if the vertex angle of one equals the vertex angle of the other. 7. A line bisecting one side of a triangle and parallel to a second side bisects the third side and equals i the second side. 8. For each of these proportions, read the triangles that must be proved similar, without attempting to draw a figure: GC GF PQ PS HK LH An altitude of a triangle is a perpendicular from a vertex to the opposite side. 9. In the acute triangle ABC, the altitudes BD and CE are , BD AB drawn. Then ^ 422 DEMONSTRATIVE GEOMETRY 10 . If in the same figure, the altitudes BD and CE cross at F, then 11. In similar triangles bisectors of corresponding angles have the same ratio as a pair of corresponding sides. 12. A line parallel to one side of a triangle divides the other two sides proportionally; that is, a side is to a segment of it as the other side is to its corresponding segment. 13 . AABC is a right triangle. CD is the altitude on the hypotenuse. Prove that; {a) AACD is similar to AABC (b) ACBD is similar to AABC (c) AACD is similar to ACBD 5 = c (\u00ab)\u201c = – ^ ^ a c (J) From the result of (d), what does equal? (g) From the result of (e), what does equal? (h) Using the results of (\/) and (g), find the value of + b In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. Review Exercises 1. Find the hypotenuse of a right triangle whose legs are 3 and 4. 2. Find the diagonal of a rectangle the sides of which are 21 and 28. 3. Find the diagonal of a square whose side is 8. 4 . Find the other leg of a right triangle if the hypotenuse and one leg are 26 and 24. 5. The base of a rectangle is 12, and the diagonal is 15. Find its altitude. 6. Find the altitude of an isosceles triangle whose base is 10 and whose legs are each 13. 7. Find the altitude of an equilateral triangle whose side is 20. THE INDIRECT PROOF 423 8i The bottom of a ladder 17 ft. long is 15 ft. from a wall. How high up the wall is the top of the ladder? 9. The perimeter of a rectangle is 28 ft., and its diagonal is 10 ft. Find the sides of the rectangle. 10. Find the side of a square whose diagonal is 10. 11. To find a distance AB, I measure AC 102 ft. and CB 136 ft. at right angles with each other. What is the length of ABl 12. A boat travels 40 ft. across a river while the current carries it 30 ft. down-stream. Draw the path of the boat, and find the distance it has moved. 13. A man walks at the rate of 5 mi. an hr. across the deck of a boat that is traveling 12 mi. an hr. What is his actual speed? 14. A force of 60 lbs. is pulling directly north, and a force of 80 lbs. directly east. If the resultant force is rep- resented by the diagonal of the rectangle whose sides are pro- portional to the given forces, find its amount. 15. Emily is swimming 1 mi. an hr. across a river that flows 3 mi. an hr. At what speed is she moving? THE INDIRECT PROOF Pete Hill\u2019s father forbade Pete to use the new car except for doing errands for his mother. Last night Mr. Hill accused Pete of disobeying him. Mrs. Hill inter- ceded, \u201cHe went to town on an errand for me. I don\u2019t believe he went anywhere else.\u201d \u201cWell,\u201d said Mr. Hill, \u201cwe had the tank filled last night, didn\u2019t we?\u201d \u201cYes,\u201d admitted Mrs. Hill. \u201cAnd you see it\u2019s about empty now,\u201d continued Mr. Hill. \u201cYes,\u201d answered Mrs. Hill. \u201cIf he had only gone to town, he wouldn\u2019t have used that much gas, would he?\u201d \u201cNo.\u201d \u201cThen he must have disobeyed me,\u201d said Mr. Hill. And Mrs. Hill had to admit that he was right. This type of argument is very common and very effec- tive. It is called the indirect proof. Instead of starting out to prove your statement directly, you can suppose the opposite to be true and then show that this contradicts 424 DEMONSTRATIVE GEOMETRY known facts. If there are more than two possibilities, you must show that all but the one you wish to establish con- tradict known facts. Let us apply it to geometry. Illustration 1. The sides of a triangle are 3, 4, and 6. that it is not a right triangle. Suppose the triangle is a right triangle. Show 1. Then equal 6^. 32 42 would 2. But this is impossible. 3. Therefore, the triangle is not a right triangle. 1. The sum of the squares of the legs of a right triangle equal, etc. 2. 9 -h 16 = 25 and not 36. 3. The supposition that it is a right triangle contradicts known facts. Illustration 2. This triangle is made of three pieces of wood pinned together at the corners. Prove that it is rigid, that is, that it cannot change its shape. Suppose it could change its shape. Then we would have different shaped triangles with the same three sides. But this is impossible, for two triangles are congruent if the three sides of one equal the three sides of the other. Therefore the triangle is rigid, for it is impossible that it could change its shape. Exercises Use the indirect method. Assume that the conclusion is not true, and then show that this leads to an impossible result: 1. If two angles of a triangle are unequal, the sides opposite them are unequal. 2. If any angle of one triangle is not equal to some angle of another triangle, the triangles are not congruent. 3. If a triangle is not isosceles, the bisector of an angle is not perpendicular to the opposite side. 4. Two sides of one triangle equal two sides of another tri- angle. If their included angles are not equal, their third sides are unequal. 6. A triangle cannot have more than one right angle. THE INDIRECT PROOF 425 jX+130, 6. Two lines perpendicular to the same line are parallel. Suppose they meet, and consider the sum of the angles of the triangle formed, a \u25a0\/\u2014\u2014 b 7. The point (2, 3) is not on the line 2x + 3y = 12. 8. The lines 2 x + 3 y = 5 and \u2014 -Y \u2014D 6 a; + 9 y = 10 are parallel. Suppose \/ they are not parallel, and solve for their point of intersection. 9. AB and CD cannot be parallel. Suppose they are parallel, and solve for x and the angles. 10. A triangle cannot have two obtuse angles. 11. The sides of a triangle are 6, 8, and 11. Is it a right tri- angle? Prove your conclusion. What fact do you know about the sides of a right triangle? 12. Is it possible to construct an equilateral right triangle? Prove your conclusion. The Indirect Proof in Life Situations Use the indirect proof to convince the person that he is wrong. Illustration. A customer returned the radio you sold him, saying that the loud speaker rattles. You try a new detector tube and it works quietly. Convince him that it was not the fault of the loud speaker. Proof: \u201cIf it were the fault of the loud speaker, changing the detector tube would not correct it, would it?\u201d \u201cNo.\u201d \u201cBut you see it does correct it, don\u2019t you?\u201d \u201cYes.\u201d \u201cThen it is not the fault of the loud speaker.\u201d 1. When the steam was on last night, water leaked from the radiator on the floor. The janitor says that the radiator is all right. Convince him that he is wrong. 2. A very heavy package was stolen from John\u2019s room. From the time it was seen there until it was missed, it is known that only Peter and the crippled Henry had been in there. You are the prosecuting attorney. Convince the jury that Peter is guilty. 3. The clock you sold yesterday has stopped. The customer returns it. saying that her maid wound it last night, but it will 426 DEMONSTRATIVE GEOMETRY not run. You find that the spring is entirely unwound, so you wind it and it runs. The spring would be unwound if the clock had run down, or if the spring had broken. Convince the lady that her maid forgot to wind the clock. 4. You are the doctor. The child has a fever, rash, sore throat, and his tongue is red. He was vaccinated last year. In measles there is fever, rash, tongue coated white, no sore throat. In scarlet fever there is fever, rash, tongue red, and sore throat. In smallpox there is fever and rash, but the disease is prevented by vaccination. In other diseases there is no rash. By the indirect method, diagnose the case. Chapter 18 GENERAL REVIEW Statistical Graphs 1. Make a bar graph of these statistics: The number of thou- sand miles of railway in the various countries is as follows: Canada, 42; France, 27; Germany, 33; Great Britain, 20; India, 42; Italy, 10; Japan, 13; Russia, 48; and U.S., 248. 2. Make a smooth-curve graph: The size of the American merchant marine in millions of tons was: 1830, 1; 1850, 3.5; 1870, 4; 1890, 4.4; 1910, 7.5; and 1930, 16. 3. Make a broken-line graph: The number of deaths in New York City from alcohol was: 1914, 660; 1915, 562; 1916, 687; 1917, 560; 1918, 252; 1919, 176; 1920, 98; and 1921, 119. 4 . Make a circle graph: The number of million acres of ir- rigated land is: No. America, 27; So. America, 7; Europe, 15; Asia, 141; and Africa, 10. 5. Find the average, median, and mode of these test marks: 5, 6, 6, 6, 6, 7, 7, 8, 8, 9, 10, 10. 6. Make a frequency polygon and a normal frequency curve: In a certain school the height in inches of 9-yr.-old children was: Height in inches 46 47 48 49 50 51 52 53 54 Number of pupils 3 10 42 112 143 114 47 9 2 What height was the mode? Formulas 7. In \/ = \u00ab -f (w \u2014 )d, find \/ when a = 7, n \u2014 20, and d = 2. 8. In K = ^ mv’^, find K when m = 5000 and v = 12. 9. In s = \u2014 \u2022, find s when 1 = 380. m = 19. and a = 12, ma 428 GENERAL REVIEW 10 . In \/ = 2 find t when s = 98 and g = 32. 11 . In A = yls(s \u2014 a)(s – b)(s -c), find A when a = 12, b = 9, and c = 11, if s = b + c). 12. In a certain state, fire-insurance companies are required il to pay partial losses according to the formula p = where p .O V is the amount they must pay, i the amount of insurance on the house, I the amount of the loss, and v the value of the house. If a house valued at $12,000 is insured for $7200, what should the company pay in case of a fire loss of $2800? 13 . A merchant sold a suit for $47.20 making a profit of 18% s of the cost. What was the cost. Formula: c = \u2014 ; \u2014 1 + \/> 14 . Make a graph of the formula s = 5 \/ + 12 from t = 0 to \/ = 5. Geometric Measurement 16. Find an angle that is 30\u00b0 more than four times its supple- ment. 16 . How many degrees are there in the angle made by the hands of a clock at 1:15 p.m.? 17 . When an angle grows larger, what change takes place in its supplement? In its vertical angle? Are supplementary angles ever equal? Explain. 18 . The sum of the angles of a polygon (a figure having any number of straight sides) is s = (\u00ab \u2014 2)180, where n is the num- ber of sides. Find the sum of the angles of a polygon having 10 sides, having 12 sides. 19 . Draw an angle, and construct an angle 3|- times as large. 20. Construct an angle of 22|-\u00b0, one of 135\u00b0. 21. Find the square root of 3847 to hundredths. 22. Could you find the square root of 384.7 by simply point- ing off one place in the answer to Exercise 21? Explain. 23 . Find the length of a root 3 rectangle whose width is 10 in. 24 . Express in a short form (\u00ab) xxxxxx, (6) y + y + y + y. 26 . Find to four significant figures the value of V6783 V6783. SUBTRACTION 429 Algebra as a Language Write in the shorthand of algebra: 26. The sum of two numbers, the difference of two numbers, the difference of the squares of two numbers, the sum of the cubes of two numbers. 27. Three times a number decreased by 13 28. Five times the sum of two numbers increased by 5 29. Translate these into English, referring to % as a number, etc.: {a) 2x – 1 (b) + 5 (c) x^ + x y Algebraic Numbers 30. What is meant by: A temperature of \u2014 20\u00b0 ? A profit of \u2014 $24? A height above sea level of — 240 ft.? Emily will be ll yrs. old in \u2014 4 yrs.? 31. Find the value of: 18 decreased by \u2014 5, the sum of \u2014 7 and 3. 32. Add: 5 X (b) \u2014 3 m’^n (c) 3 ac {d) – 7 ;ry2 -2x \u2014 8 m^n \u2014 4 ac 4 xyz -lx 9 m\u2018^n ac 5 xyz X 2 m^n \u2014 ^ ac \u2014 2 xyz 33. If I start at \u2014 7 and go 7 to the right, at what point do I arrive? 34. Virginia earns d dollars a week for 7 weeks and spends ci a day during that time. How many cents does she save? Subtraction 35. By how much does (\u00ab) 8 exceed 5? {h) a exceed 7? (c) 10 ex- ceed xl {d) m exceed w? {e) \u2014 3 exceed \u2014 7? (\/) 8 exceed \u2014 2? (g) – 3 exceed +5? 36. Caius was born in \u2014 18 and died in + 46. How many years did he live? 37. Remove parentheses, and collect terms: (<z) la \u2014 (3 <2 \u2014 4)-|-2<2 {b) 4 % – 2 + (3 – X) – 1 430 GENERAL REVIEW (c) – (7 – 5w) – (- 3\u00ab +4) (rf) 2 + 9 c – (3 – c) + (1 – c) 38. How many sharps are there in the key of E? Of Ab? OfF? OfB? Multiplication 39. Multiply: (a) -7(+5) (b) 2\u00ab(- 3\u00ab3) (c) \u2014 5 m^(2 mP) id) (- 1)^ (e) \u2014 4 x^(\u2014 3 x)(\u2014 2 x^) if) 3\u00abdW)(0) (g) – -2 kn-^ — 3 n^) (h) 7 abcia^b \u2014 5 ac^ \u2014 3 bh) 40. Factor: {a) 4 ax \u2014 3 bx^ (c) i Trr^ + |- -Krh ib) 9 \u2014 3jn\u2018^r (d) .3 _ 1.5 xy Division 41. Divide: (a) x^ 4 – x^ (c) \u2014 16 a\u00ae 2 (b) – 7 2 (d) – 13 cx^ ^ – 6 x) (e) (8 a\u2018^b^ – 6 ab^) 4- ( – 2 ab"^) if) (\u20149 m^x + 21 mx"^) -^ ( \u2014 3 mx) (g) (14 abc \u2014 lac) -i- 1 ac (h) + x^"^ \u2014 a:\u00ae) x^ Equations 42. Solve and check: (a) 3 a; – 7 = 13 – 2 ;t: (b) 8y + 11 = y – 3 (c) 1.3 m + 2 = m + 5.6 id) 3(4 -2 x) = 9 – (2 x + 1) ie) if) 3r 2r -^ = r + l 2 X -i~ 3 2x . ^ 3 5 X x + 4 3 = 0 – 1 PROBLEMS 431 Problems 43 . Find two supplementary angles in the ratio 3: 5. 44 . One of two vertical angles is 20\u00b0 more than twice a num- ber, and the other is 40\u00b0 less than 3 times that number. Find the number and the size of the angles. 45. If one of two vertical angles were 20\u00b0 less than it is, and the other were 20\u00b0 larger than it is, they would be in the ratio f. Find the angles. 46 . Find the number of degrees in the angles of a triangle if they are in the ratio 4:5:6. 47 . One angle of a triangle is twice another, and the third is 30\u00b0 less than their sum. Find the three angles. 48 . A broker makes on sales. How much must he sell to make a yearly salary of $3000? 49 . The expenses of a city are $32,000, and the city\u2019s as- sessed value is $2,048,000. Express the tax rate as a decimal, as a per cent. 50 . Mr. Smith has two investments totaling $30,000 that furnish him an income of $1320 a year. If the rates are 5% and 4%, how much did he invest at each rate? 51 . Mr. Washburn invested $5000 at 4%. How much must he invest at 7% to make his income 5% of his total investment? 52 . Fred leaves a town at 6 a.m. traveling 30 mi. an hr. At 9 A.M. George follows at 45 mi. an hr. In how many hours will George overtake Fred? 53 . In the last exercise, if George had started from another town 40 mi. from the point from which Fred started and through which Fred passed on his way, in how many hours would George overtake Fred? 54 . Two cars leave a town going in opposite directions at rates that differ by 5 mi. an hr. What are their rates if they are 330 mi. apart at the end of 6 hrs.? 55. To pass a period of 3 hrs. which he must spend in a strange town, Robert takes a ride out into the country at the rate of 20 mi. an hr. and walks back at 4 mi. an hr. How far can he go? 56 . How many pounds of peanuts worth 20^ a lb. should the H and K Company mix with 60 lbs. of Brazil nuts worth 50^ a lb. to make a mixture worth 32^ a lb.? 432 GENERAL REVIEW 57. An alloy contains 60% lead and 40% tin. How much tin should be added to 100 lbs. of it to make an alloy 40% lead? 58. Charles has a batting average of .280 for the first 25 times at bat. How many hits must he make in succession to raise his batting average to .400? 59. One of two alternate interior angles of parallel lines is represented by 4 + 30 and the other by 2 w + 60. What is the value of w? 60. Find each angle of an isosceles triangle if each of the two base angles is 15\u00b0 larger than the vertex angle. Sets of Equations 61. Make a graph of each of these equations: (c) y = 3x – 4 (d) X ^6 (a) 2x +3y ^ 12 (b) X -4y =S 62. Solve graphically: (a) 2 X + 3 y = 12 X \u2014 y = I 63. Solve algebraically: (a) 3 X y = 17 2x – y = S (b) 7 X + 5 y = 23 X + y = 3 (b) 3 X – 2 y = 12 2x + 5y = 8 (c) 3 X – 2 y = 3 X + 4y = 29 (d) 7 X + 2 y =34 2 X "b 3 y = 17 64. At what point does 5 x \u2014 2 y = 10 cut the x axis? The y axis? 65. Is the equation 3 x \u2014 y = 7 satisfied by the point (2, 1)? By (-2, 1)? By (2, -1)? 66. Solve for x and y: (d) X ~4y = 7 X + 5 , y – 2 4 + 3 (b) .4 X + .05 y = 5 .03 X + .15 y = 3.3 \u2022 (e) .2 X + .01 y = .8 X + 7 y \u2014 2 _ \u25a0V ' A X ^y 4 y = 2 X X \u2014 y = 1 FORMULAS 433 Problems in Two Letters 67. If the length of a rectangle is increased 2 and its width is decreased 1, its area remains the same, but if its length is in- creased 4 and its width decreased 3, its area is decreased 20. Find its length and width. 68. Mr. Banker loaned $4000 at one rate and $5000 at another rate. He received from them $480 a yr. interest. If he had loaned the $4000 at the second rate and the $5000 at the first rate, his yearly income would have been $465. What were the two rates? 69. $8000 is invested, part at 6% and part at 5%. How much is invested at each rate (a) If the income from the first part is twice that from the second part? (b) If the total income is $470? (c) If the income frorci the first part exceeds that from the second part by $40? 70. How many pounds of tea at 35^ a lb. and how many at 50^ a lb. must be taken to make a mixture of 200 lbs. worth 38^ a lb.? 71. In a race of 100 yds. Chester beat Tom by 2|- secs. In a second race Tom was given a start of 4 yds., but was still beaten by 2 secs. What were their rates? 72. The sum of the digits of a two-digit number is 11. If the digits were reversed, the number would be increased by 27. What is the number? Formulas 73. Solve I for m. Then find m when I 30, \/ = 600, and a = 8. 74. The temperature, volume, and pressure of a gas are T <^+m) related according to the formula P = y . Then find T when C = 12, F = 2, and P = 8. Solve for T. J ^ ^ 75. The formula ^ = Db \u2014 R image \/ of a body B, made by a lens. Solve for R. Find the radius if \/ = 3, P = 5, P = 7, and Db = 15. Check by sub- stituting in the original equation. 434 GENERAL REVIEW General Solution 76. (a) A man invests d dollars, part at rate p and part at rate r. If his income is i dollars a year, how much has he at each rate? {b) Using the solution to {a) as a formula, solve: A man invests $800, part at 4% and part at 5%. If his income is $36 a yr., how much has he at each rate? 77. (\u00ab) A merchant has p lbs. of candy worth a lb. How much candy worth a(^ a lb. must he use with it to make a mixture worth mi a lb.? {b) Using the solution to (\u00ab) as a formula, solve: A mer- chant has 300 lbs. of candy worth 60^ a lb. How much candy worth SOjzi a lb. must he use to make a mixture worth 40^ a lb.? 78. {a) A train leaves a station and travels m mi. an hr. After h hrs. a faster train follows at r mi. an hr. In how many hours will the second train overtake the first? {b) Using the solution to (c) as a formula, solve: A train leaves a station and travels 25 mi. an hr. After 4 hrs. a second train follows at 35 mi. an hr. In how many hours will the second train overtake the first? Business Problems 79. A merchant sells shoes that cost $3.50 a pair for $4.90. His profit is what per cent of the cost? Of the selling price? 80. A dealer buys ladies\u2019 hats at $3.60 each. At what price must he sell them to make 25% of the cost? 25% of the selling price? 81. Find the bank discount and the proceeds on: {a) $200 at .6% for 48 d. (c) $180 at 5% for 50 d. {b) $350 at 6% for 3 mo. {d) $480 at 4^% for 72 d. 82. Using the compound-interest table, find the amount of: {a) $500 at 6% compounded annually for 10 yrs. ib) $300 at 4% compounded semi-annually for 4-|- yrs. 83. Using logarithms, find the amount of: (fl) $200 at 4% compounded quarterly for 20 yrs. {b) $600 at 6% compounded semi-annually for 80 yrs. (c) $1095 at 6% compounded semi-annually for 25 yrs. {d) $24 at 6% compounded semi-annually for 350 yrs. RATIO AND PROPORTION 435 Ratio and Proportion 84. Solve for a:: L = ^ + 13 2x + 3 (e) (f) X \u2014 3 X 2 X -j- 2 X \u2014 5 _ 2x -5 2 .T + 6 X \u2014 4 ~ X – 51 85. (a) Find two numbers in the ratio a:b whose sum is s. (b) Using the answer to (a) as a formula, find two num- bers in the ratio 5:8 whose sum is 78. 86. Three partners invested $6000, $7000, and $8000 in an enterprise. If their profits are to be divided in the ratio of their investments, how should a profit of $5292 be divided? 87. When coal burns, a gas called carbon dioxide (C + O 2) is formed. Assuming that the coal is 80% carbon, how many tons of carbon dioxide are formed by the burning of 15 T. of coal? The atomic weights are C = 12 and O = 16. 88. What is the width of a root 3 rectangle whose length is 40 in.? 89. A low-pitched scale has vibration rates: C = 24, D = 27, E = 30, F = 32, G = 36, A = 40, B = 45, and C = 48. (a) Write the vibration rates of a scale in which C = 120. (b) Which is the worst discord: C and D, or E and F? (c) Do A, F, and C form a pleasing chord? (d) What vibration rate should E have, to the nearest tenth, so that the ratio E:D would equal that of D:C? 90. If the amount of water carried by a pipe varies as the square of the diameter, a 4-in. pipe will carry how many times as much water as a 2-in. pipe? As a 1-in. pipe? 91. How high is a tree that casts a shadow 27 ft. long when an 8-ft. vertical pole casts one 3 ft. long? 92. A line parallel to a side of a triangle cuts a second side into segments of 7 and 5. If the length of the third side is 18, what are the lengths of the segments into which the line cuts it? 93. On a map of Ethiopia, the distance from Adowa to the Eritrean border is 4 cm. and from Adowa to Addis Ababa is 6.9 cm. If Adowa is 345 mi. from Addis Ababa, how far is Adowa from the border- 436 GENERAL REVIEW 94 . Make a formula to fit these tables: t 1 2 3 4 5 d ~ 8 ~ 16 24 32 40 A 15000 16000 17000 C 8000 9000 10000 96 . Dorothy, who weighs 84 lbs., sits 6 ft. from the support ^ f a teeter board. How far from the support should Sarah, who weighs 72 lbs., sit? The Straight Line 96. At what point will 3 x + 2 y =8 cut the y axis? The X axis? 97. Does 4 X \u2014 y = 7 pass through (a) The point (1, 1)? (6) The point (2, 1)? (c) The point (3, 5)? 98. Give the slope and y intercept of: (\u00ab) y = 2 X + 3 (c) 3 X + y = 7 (&) y = X \u2014 4 (x\u2014 a, 113. 1 x’^ -?>x -12, ^x – 7 – xU -^x^ – X 114. M – AN, 2 N -5 M,M ^N,2M -2 N 116. 2 Xi \u2014 2 X2 Xz, 2 X2 \u2014 Xz \u2014 Xi, \u2014 Xz Ar A Xi \u2014 X2 116. 1.3 a – 2.7 b A- A c, 2.5 a + lA b – c, a + b 117. k \u2014 .02 m, m \u2014 .55 k, .5 k \u2014 m, 2 m \u2014 2.4 k 118. ^ a \u2014 ^b-‘c, ^b \u2014 ^ c A’ ^ a, ^ c \u2014 ^ a Ar Subtract and check: 119. Subtract 5 a \u2014 2b A- c from 2 a \u2014 2 b A- c 120. Subtract \u2014 3X + 2F \u2014 Z from X \u2014 F + 4 Z 121. From 2 m \u2014 2 n A- P, take 5 m A- n \u2014 2 p 122. From 7 Xi \u2014 2 X2 \u2014 3 Xs, take Xi + X2 \u2014 4 Xa 123. Take 3 c + 5 from 7 \u20142a 124. Take \u2014 x + 23; \u2014 5z from 4 x + z 125. From 3.7 x \u2014 2.9 y \u2014 z, take 4.7 x A- 2.1 y \u2014 .A z 126. Subtract ^x \u2014 ^y A- i z from ^x\u2014 ^yA-^z Multiply and check: 127. x2 + 2x-3by3x-4 128. 2 a^ \u2014 2a\u2018^-{-7a \u2014 5hY2a\u2014 2 129. (5 x2 – 3 xy – 2 (3 – 4 xy^) 130. (4 ^2 _ 2 ^ + 3 r)(2 ^ – 3 r) 131. x^ + X\u00ae + x2 + X + 1 by X \u2014 1 132. M2 – .2 M + .04 by .1 + 1.2 M – .3 133. -g- ^2 \u2014 \u2022\u00a7\u25a0 + 4- by ^ fl2 _ i ^ _j_ 1- 134. (x\u201d + i – 3x\u201d – 2x”-0(^ – 2) Divide and check: 135. 3×2-4x + lbyx-l 136. 8 x3 – 27 by 2 X – 3 y2 137. (6 A3 – 5 A2 – 4 A – 3) -f- (2 A – 3) FRACTIONS 138. (x^ + + y*) (x^ – xy y^) a* + 11 + 23 \u00ab2 _ 55 \u00ab _ 140 – 5 , M3 – 2.1 M2 + 2.18 M 1.32 139. M – 1.2 141. (\u00ab4 – .0016) -4- (\u00ab – .2) 142. (a:\u201d + 4 4_ 3;(;^ + i _ 10 a;”-2) -4- (a;3 2 ) Factoring Factor and check: 143. fl 2 _ 25 152. 4 Z)2 – 36 g 2 144. 9 ;c 2 _ 49 153. 6 ^2 – 5 ^ – 1 145. c2 – f 154. 2 ttT? \u2014 3 0x02 4\u201c ^ 2 * 146. – 16 62 155. 3v^ – 3v -6 147. M 2 – .0025 156. 16 m 2 4 – 1 \u20148m 148. 3;2 _ 1 157. 158. 2x^ \u20143 x”^ \u2014 ^ X \u2014 a ;2 4 – 24 ;i; \u2014 44 149. i – \u2014 a :2 3\/2 159. 160. x^ \u2014 10 a: 23\/2 4 – 93 \/^ \u2014 13 \u00ab3 4- 36 a 150. – 16 n* 161. Xn + \u2018i _ + 1 _ 20 ;C\u2019 151. -36 h 162. – 4.01 a ;2 -}- .04 Find the value of: 163. 172 – 162 166. 4432 – 4332 164. 3442 – 3422 167. 4512 – 4492 165. 7832 – 2172 168. 5013 _ 501 X 4992 Fractions Reduce to lowest terms: 169. 170. 171. 15 fl2&3 25 a^h – 14 Mm 173. 174. 21 MW2 – 22 – 33 175. 15X%2 – 5Xx 176. 8 x^y’^z 8 x^yH 9&2 \u2014 Z ah 5 x’^y 10 x^y”^ + 15 5-53\/2 43\/4-4 439 172 . 440 GENERAL REVIEW 177. 178. 179. 180. 6 m 2 \u2014 24 tB 181, 7×2 – 21 X 3{m \u2014 2 w )2 14 x 2 – 14 X – 84 C 4 1 CO 182. 2 \u00ab2 _ 5 a _ 3 .01 X + .02 j \u00ab2 + 4 fl – 21 x 2 – 16 183. 3 m2 \u2014 9 m \u2014 12 x 2 \u2014 8 X + 16 3 m2 + 15 m + 12 3 x2 – 75 184. x2 \u2014 ,1 X \u2014 1.1 2 x2 + 8 X – 10 x2 – 1.3 X + .22 Multiply: 185. 186. 187. 188. 189. 72 X 35 x^ _ 7×3 49×3 9 x2 8 X ^ 3b^ 4 \u00ab3 aP- 25 b (3 tyB B4p+ ‘2n y I 6^2 <3 py 4 ttB\/ 'A\u00b1y 4 ttB \u201c6T .2 7rr\/ 5 a\u2018^x^ 6 m^y 3 m'^y \u2014 10 ax^ Divide: 4A^B 195. 196. 5 AO 3M^ 5S 10 O -^2 MS 7 r2 197. Ix^y 2y 198. 199. 10 a^b 3^3 \u2014 5 a: 6 ab^ 9^4 Add: 205. 5 m 3 m 190. + 6 3k^ 21 7k^ + 21 (3x + 2yy ^ 2 X 3 x \u2014 2 y 6 x\u2018^ + 4 xy – 4 . 3 S + 9 S2 – 9 6 S – 12 193. 194. 200 . 201 . 202 . 203. 204. 206. 3 a :2 + 9 a: – 12 ^ 2 2 a; 2 + 8 a :-10 B – i -6 , B – 1 + 3 \/ + 2 * 22 – 4 \/ + 3 5 \/\/2 5 \/\/2 3i\/2_3 \u2022 2 H'^ -3H – b^ , a – b fl2 + ' 2 a2 4- 2 (x \u2014 v) 2 2 X \u2014 2 jy 2x+v \u20196 x2 4-3xv a- \u2014 b"^ _^b \u2014 a \u2014 y2 \u2019 y \u2014 X ax \u2014 2 a ^ \u2014a 4 – x2 ' (x + 2) 2a2^ 3\u00ab25 10 5 2 QUADRATIC EQUATIONS X +3^ _ 2 2 xy y 3 2 207. 5 4 _ 2 _ J_ ' 3 X X 3 X 211 . 208. 5 a 0 1 a X -3\u201c+3 212 . 209. CO| 1 213. 210 . a + b a \u2014 b 214. 2 3 p – q q – p 4 -2 -hi 3 + 5 m 1 \u2014 m m'^ \u2014 1 441 _ 3 2 V Quadratic Equations Soke and check: 215. 4- X -12 =0 216. m – N -2 =0 217. r2 + 3 r = 10 218. \/2 – 21 = 4\/ 219. X + 20 x^ 220. 2p + 2j – 2 =0 221 . 12 \u00a32 4 – 52 – 2=0 222. 12 + 17 \/? + 6 = 0 223. 6 g2 _ 23 e + 20 = 0 224. 10 4- .7 a: – .03 = 0 Solve, leaving the answer in radical form: 225. 3i?2_2i?-6=0 228. 5\/2-8\/ + 2= 0 226. 2p^ -4 = 3p 229. – 1.2 b + .1 = 0 227. 2 c2 4- 5 c = 4 230. .02 3^2 = jy 4. 4 Find the roots to the nearest tenth: 231. 3×2-4a:- 2=0 233. x2_5;^;4-3=o 232. 2:c2 4-8a: + 3 = 0 234. 4;c2-9:^+3=0 Solve and group your answers: 235. X + y = 7 239. 2 m; + 5 2 = 7 LO 11 1 2 m ;2 4 – 5 \u00a32 = 7 236. 2 k -m ^5 240. 1 i? – f r = 0 ^2 4 – m 2 = 10 .5 + .01 Rr = 8.12 237. a + 3 d =^7 241. rii^ + W 1 W 2 + ^ 2 ^ = 3 + ad = 3 nx + n^ + 1 =0 238. r =2v 242. m – P = 5 2r^ -3 rv = 13 77 + \/ = 5 442 GENERAL REVIEW Radicals Simplify: 243 . 5 Vi 248 . 261 . 6Vf_ 244 . 2 Vi 5V2 252 . 9 V^ 245 . 3V96 4 253 . 5v:^ 246 . 2V80 VT2 249 . ^ V6 3VI5 254 . 5V3i 247 . V3 260 . \u2014 ^ 5V12 255 . INDEX Abscissa, 193 Accounting formulas, 227 Acute angle, 69 Addition, 95 and subtraction as opposites, 217 of fractions, 375 of polynomials, 359 parentheses in, 122 Ahmes, 89 Algebra as a language, 84 Algebraic numbers, 95 addition of, 95, 104 division of, 148 in business, 107 in music, 126 in surveying, 101 multiplication of, 135 subtraction of, 116 Aliquot parts of 100%, 133 Alternate interior angles, 165 Analysis of a problem, 173 Angle, 67, 404 classification of, 67 degrees of, 69 how to bisect an, 74 how to copy an, 70 how to measure an, 69 included, 405 of depression, 323 of elevation, 323 right, 68 sides of an, 67, 404 sine of an, 321 straight, 68 tangent of an, 319 vertex of an, 67, 404 Angles, alternate interior, 165, 415 sum of, in a triangle, 168 supplementary, 71 vertical, 71, 163 Approximate measurement, 245 Arc, 403 Area, formulas, 30, 33, 34 of circles, 39, 286 of similar polygons, 285, 287 problems, 204 Art, ratio in, 277 square root in, 257 Assets, 47, 227 Astrolabe, 312 Astronomy, exponents in, 137 Athletics, trigonometry in, 330 Average, 19 deviation from the, 99 Axioms, 409 addition, 110 division, 151 multiplication, 141 subtraction, 121 Axis, of symmetry, 75 a:, y, 192 Bank discount, 232 Bankers\u2019 time table, 233 Bar graph, 1, 4 Beauty, symmetry and, 74 Binomial, 139, 383 Bisect, 73 Boyle\u2019s law, 299 Broken-line graph, 9 Business, algebraic numbers in, 107 equations in, 171 factoring in, 146 graphs in, 59 logarithms in, 350 marking goods in, 49 percentage in, 47 ratio in, 270 the formula in, 47, 225 Capital, 47, 227 Centigrade thermometer, 95 Characteristic, 347 rule for finding the, 348 Chemistry and medicine, formulas, 275 ratio in, 275 Chord, 403 Circle, area of, 39, 286 circumference of a, 39 definition of a, 403 graphs, 77 Circumferences are to each other as, 284 Clinometer, 313 Coefficient, 84 Coin problems, 398 Compound interest, 235 formula for, 237 graph of, 59 table for, 238 Cone, 40 444 INDEX Congruent triangles, 405, 412 Consecutive numbers, 154 Constant, 293 Construction, definition of, 66 how to bisect, an angle, 74 a line segment, 73 how to copy an angle, 70 of a line segment, 66 of a triangle, 406 Cooking, ratio in, 273 Coordinates of a point, 193 Corresponding angles, 166 Cosine of an angle, 321 Cotangent of an angle, 320 Cube, area of a, 39 the exponent, 37 volume of a, 38 Curve, normal frequency, 22 Cylinder, 39 Degrees, 69 Dependence, problems in, 293 Dependent equations, 197 Depression, angle of, 323 Depressions, graph of, 17 Deviation from average, 99 Diameter, 404 Difference of two squares, 366 Digit problems, 207 Diophantus, 102, 158 Discount, bank, 232 business (trade), 49 Distance on a graph, 255 Distribution graph, 77, 79 Dividend, 149 Division, 148 axiom for, 151 in equations, 152 of fractions, 374 on the slide rule, 352 polynomial by monomial, 151 polynomial by polynomial, 364 Divisor, 149 Drawing to scale, 313 for the Boy Scout, 315 Elevation, angle of, 323 Elimination, by addition and sub- traction, 197 by substitution, 200 Equations, 89, 121, 152 algebraic solution, 197 containing parentheses, 124 dependent, 197 for the merchant, 178 fractional, 142, 377 graphical solution of, 195 graphs of, 193 :n business, 171 in geometry, 164, 202 in investment problems, 174 in mixture problems, 178 in motion problems, 176 in puzzles. 111 inconsistent, 197 leading to quadratics, 393 literal, 215, 218 multiplication in, 141 of straight lines, 193, 303 quadratic, 383 radical, 258 roots of, 89 simple, 108, 155 substitution method in, 200 systems of, 188 Euclid, 389, 403 Exponents, 37, 85 fractional, 339 in astronomy, 137 in division, 149 in multiplication, 134 logarithms as, 341 of two, 338 zero, 340 Extremes in a proportion, 268 Face of a note, 232 Factoring, difference of squares, 366 in business, 146 monomial, 144 trinomial, 384 Factors, definition of, 37, 84 dividing by, 371 Farm problems, ratio in, 272 Formulas, 29 accounting, 227 compound interest, 237 for the circle, 39 for the cone, 40 for the cube, 38 for the cylinder, 39 for the parallelogram, 33 for the rectangle, 30 for the square, 37 for the trapezoid, 44 for the triangle, 34 from rules, 53 from tables, 54, 294 fun from, 50 graphs of, 56 in business, 47, 225 in geometry, 33 in medicine and pharmacy, 180 in science, 46, 221 in the home, 35, 45 installment payment, 228 lever, 51, 296 parentheses in, 44 percentage, 47, 171, 225 INDEX 445 Formulas \u2014 {Continued) Pythagorean, 252 quadratic, 389 simple interest, 227 subscripts in, 51 the general solution, 223 Fractional, equations, 142, 377 exponents, 339 Fractions, 371 ff. Frequency polygon, 21 Functions, variation of, 293 General, review, 427 solution, the, 223 Geography, ratio in, 291 Geometry, area problems in, 30, 204 demonstrative, 402 congruent triangles, 405 direct proof in, 411 indirect proof in, 423 isosceles triangles in, 419 parallel lines in, 415 similar triangles in, 421 equations in, 164, 202 formulas in, 33 measurement in, 64 ratio in, 281 sets of equations in, 202 square root in, 254 the quadratic equation in, 392 the sum of the angles of a triangle in, 168, 417 Graphs, 1 as a ready reckoner, 191 bar, 1 broken-line, 9 business, 59 circle, 77 depression, 17 distances on, 255 distribution, 77, 79 formula, 56 histogram, 21 in motion problems, 188 interest, 59 of equations, 193 pictogram, 7 rectangle, 79 smooth-curve, 13 statistical, 1 stock-market, 10 Hipparchus, 314 Histogram, 21 Home, formulas in the, 35, 45 ratio in the, 288 Hypotenuse, 252 Inconsistent equations, 197 Indirect, measurement, 310 proof, 423 Installment payments, 228 Insurance, life, 288 Interest, compound, 235, 237 graph of, 60 table for, 238 simple, formula for, 227 graph of, 59 sixty-day method, 230 Interpolation, trigonometric, 332 Inverse, variation, 296, 298 squares, law of, 300 Investment problems, 174 Isosceles triangles, 405, 419 Lagrange, 42 Language of algebra, 84 Least common denominator, 376 Lever problems, 51, 296 Life insurance, 288 Line, 403 how to bisect, 73 how to construct, 66 how to find slope of, 301 segment, 64, 403 straight, y = mx, 300 y = mx b, 303, 331 Lines, parallel, 415 Literal equations, 215, 218 Logarithms, 338 characteristic of, 347 how to use tables of, 342 in business, 350 in music, 350 tables of, 344 Mantissa, 347 Marking goods, 49 Maturity, date of, 232 Means in a proportion, 268 Measurement, approximate, 245 geometric, 64 indirect, 310 Measures, square and cubic, 41 Median in statistics, 19 Medicine, formulas in, 180 ratio in, 275 Minuend, 118 Mixture problems, 178 Mode, 20 Monomial, 138 factor, 144 Motion problems, 176 the graph in, 188 Multiplication, 132 and division as opposites, 217 exponents in, 134 446 INDEX Multiplication \u2014 {Continued) in equations, 141 of binomials, 383 of fractions, 372 of polynomials, 139, 362 of signed numbers, 135 on the slide rule, 353 rule of signs in, 136 terms and factors in, 139 with logarithms, 343 Music, algebraic numbers in, 126 logarithms in, 350 ratio in, 278 the black keys in, 280 the modified scale in, 279 Negative numbers, 96 Newton, 203 Normal frequency curve, 22 Note, promissory, 232 Numbers, algebraic, 95 approximate, 245 consecutive, 154 system, 207 Obtuse angle, 69 Order of operations, 43 Ordinate and origin, 192 Parallel lines, 165, 415 Parallelograms, 33 Parentheses, 122, 124 in formulas, 44 Percentage, 171 formulas in, 47, 225 Perimeter, 32 Perpendicular, 69, 404 Pictogram, 7 Planning the home, ratio in, 288 Polygon, 169 Polynomials, 138, 359 Positive numbers, 96 Power of a number, 37 Problems, analysis of, 173 area, 204 bar graph, 4 broken-line graph, 11 business, 227 coin, 398 consecutive number, 154 digit, 209 for the druggist, 179 geometry, 400 in two letters, 202 in two unknowns, 162, 210 investment, 174 lever, 51, 296 mixture, 178 motion, 176 number, 154 parallel line, 165 percentage, 171, 225 ratio, 266 review, 431 smooth-curve graph, 15 sum of angles of triangle, 168 vertical angle, 71, 163 work, 380 Proceeds of a note, 234 Proof, arrangement of, 412 direct, 411 indirect, 423 in life situations, 425 Proportion, definition of, 268 extremies and means in a, 268 Protractor, use of a, 69 Puzzles, equations in. 111 Pythagorean theorem, 251, 422 Quadratic equations, 383 in geometry, 392 in science, 393 solving, 385, 388, 389 Quadrilateral, 416 Quotient, 149 Radical equations 258 in science, 260 Radicals, 395 Radius, 403 Ratio, and proportion, 263 applied, 267 , for the Boy or Girl Scout, 289 in art, 277 in business, 270 in chemistry and medicine, 275 in geography, 291 in geometry, 281 in life insurance, 288 in music, 278 in planning the home, 288 in science, 274 in sewing and cooking, 273 in trigonom.etry, 319 on' the farm, 272 terms of a, 263 Rectangles, areas of similar, 285 formula for area of, 30 having equal altitudes, 284 problems, 204 Regiomontanus, 314 Remainder, 118 Review exercises, 24, 60, 80, 92, 112, 129, 157, 181, 212, 239, 260, 305, 335, 357, 427 Right angle, 68 Right triangle, 251 INDEX 447 Right triangle \u2014 {Continued) Pythagorean formula for, 252, 422 Root, of an equation, 89 rectangles, 257 square, 243 tables, 254 Round numbers, 247 Scale drawing, 313 Science and engineering, 46 inverse variation in, 299 quadratic equations in, 393 radical equations for, 260 ratio in, 274 slide rule in, 355 square root in, 256 Scout, Boy or Girl, ratio for, 289 problems for the, 315 trigonometry for the, 326 Segment, line, 64 how to bisect a, 73 how to copy a, 66 Sewing, ratio in, 273 Sextant, 313 Shorthand, algebraic, 31, 88, 90, 91, 112, 126, 130, 154, 161, 186, 203 Sides of an angle, 67 Significant figures, 246 Signs, plus or minus, 386 rules of, 136, 149 Similar, figures, 281 areas of, 285, 287 theorems on, 421 Simple interest, formula, 227 graph, 59 Simultaneous equations, 188 leading to quadratics, 393 Sine of an angle, 321 Sixty-day method in interest, 230 Slide rule, how to divide on, 352 how to find square root on, 354 how to make a, 351 how to multiply on, 353 in science and engineering, 355 Slope of a line, 301 Smooth-curve graph, 13 Solving equations, by arithmetic, 89 by axioms, 109 by formula, 389 by transposing, 155 Square, completing the, 388 measure, 41 Square root, 243 and the formula, 249 applied, 256 in art, 257 in geometry, 254 in science, 256 on the slide rule, 354 table. 254 Squares, law of inverse, 300 perfect trinomial, 386 Statistics, 17 average and median in, 19 graphs of, 1 mode in, 20 Stock-market graphs, 10 Straight angle, 68 Straight line, equation of, 300 slope of, 302 Subscripts, 51 Substitution, axiom, 109, 410 in quadratic equations, 393 in solving equations, 200 Subtraction, 116 axiom for, 121 making change in, 116 parentheses in, 122 polynomial, 360 Subtrahend, 118 Sum. of angles of triangle, 168, 417 Supplementary angles, 71, 404 Surveying, algebraic numbers in, 101 instruments, 310 problems, 326 Symmetry, and beauty, 74 axis of, 75 Systems of equations, 188 in geometry, 202 leading to quadratics, 393 Tables, bankers\u2019 time, 233 compound-interest, 238 of exponents of two, 338 of logarithms, 344 of square measure, 41 of squares and square root, 254 of trigonometric functions, 324 Tangent of an angle, 302, 319 Term of a note, 232 Terms, like, 85 multiplying, 139 of a ratio, 263 Tests, 27, 63, 82, 93, 115, 131, 159, 186, 214, 242, 309, 337, 358 Theorem, 409 Thermometer, 95 Transit, 310 Transposing, 155 Trapezoid, 44 Triangles, area formula for, 34 congruent, 405 exercises on, 412 facts about, 405 isosceles, 405, 419, 420 how to copy, 406 perimeters of, 33 right, 251 similar, 421 sum of angles of, 168, 417 448 INDEX Trigonometric functions, 319 table of, 324 Trigonometry, for boys and girls, 334 for the practical man, 329 for the Scout, 326 in athletics, 330 Vertex of an angle, 67 Vertical angles, 71, 163, 404 are equal, 164 Vieta, 102, 158, 389 Volume formulas, 35, 39 ratios for finding distances, 319 Trinomial, 139 factoring the, 384 squares, 386 Work problems, 380 X and y axes, 192 Variables, 293 Variation, direct, 293 in science, 299 inverse, 296 the lever in, 51, 296 Zero exponent, 340 Date Due 1 1 1 .-iouqatipn 9676 historical COLLECTION OA 39 Ml 3 C. 2 McCormack f Joseph Patrick* Mathematics tor aodern life, 39673877 CURS 'CURRICULUM EDUCx^TION LIBRARY
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