THE UNIVERSITY OF CHICAGO MATHEMATICAL SERIES ELIAKIM HASTINGS MOORE GENERAL EDITOR SCHOOL OF EDUCATION TEXTS AND MANUALS GEORGE WILLIAM MYERS EDITOR FIRST-YEAR MATHEMATICS FOR SECONDARY SCHOOLS THE UNIVERSITY OF CHICAGO CHICAGO, ILLINOIS agents THE BAKER & TAYLOR COMPANY NEW YORK CAMBRIDGE UNIVERSITY PRESS LONDON AND EDINBURGH First -Year Mathematics For Secondary Schools Professor of the Teaching of Mathematics and Astronomy, College of Education of the University of Chicago and WILLIAM R. WICKES HARRIS F. MAcNEISH ERNST R. BRESLICH ERNEST A. WREIDT ERNEST L. CALDWELL ARNOLD DRESDEN Instructors in Mathematics in the University High School of the University of Chicago 2 / 2. g 4 SCHOOL OF EDUCATION MANUALS SECONDARY TEXTS THE UNIVERSITY OF CHICAGO PRESS CHICAGO, ILLINOIS COPYRIGHT 1906 BY GEO. W. MYERS COPYRIGHT 1909 BY THE UNIVERSITY OF CHICAGO First edition privately printed October, 1906 Second impression published April, 1907 Second edition August, 1907 Third edition September, 1W9 Second impression December, 1910 Composed and Printed By The University of Chicago Preii Chicago, Illinois, U.S.A. QA 2>.\ 3 1 TABLE OF CONTENTS CHAPTER PAGE PREFACE ix I. GENERAL USES OF THE EQUATION .... i II. USES OF THE EQUATION WITH PERIMETERS AND AREAS 14 III. THE EQUATION APPLIED TO ANGLES ... 30 IV. POSITIVE AND NEGATIVE NUMBERS .... 62 V. BEAM PROBLEMS IN ONE AND Two UNKNOWNS 89 VI. PROBLEMS IN PROPORTION AND SIMILARITY. . 107 VII. PROBLEMS ON PARALLEL LINES. GEOMETRIC CONSTRUCTIONS 137 VIII. THE FUNDAMENTAL OPERATIONS APPLIED TO INTEGRAL ALGEBRAIC EXPRESSIONS . . . . 153 IX. PRACTICE IN ALGEBRAIC LANGUAGE. GENERAL ARITHMETIC 181 X. THE SIMPLE EQUATION IN ONE UNKNOWN. . 210 XI. LINEAR EQUATIONS CONTAINING Two OR MORE UNKNOWN NUMBERS. GRAPHIC SOLUTION. OF EQUATIONS AND PROBLEMS 234 XII. FRACTIONS 262 XIII. FACTORING. QUADRATICS. RADICALS . . . 275 XIV. POLYGONS. CONGRUENT TRIANGLES. RADICALS 323 SUMMARIES at the ends of the first thirteen chapters, see pp. 13, 28-29, 59-61, 88, 106, 136, 151-52, 180, 207-9, 232-33, 260-61, 273-74, 321-22. PREFACE This volume aims to present in teachable form an inter- weaving of the more concrete and the easier portions of the first courses in both algebra and geometry. It is intended as a text for students of the first year of secondary schools. The type of work begun here will be continued by a second- year text, to appear soon and to include not less than the rest of the subject-matters that make up the customary two years of required secondary work in algebra and plane geometry. The present book places chief emphasis upon algebra, but weaves in a considerable body of related fundamental notions and principles of geometry, even advancing somewhat in one direction or another upon the most concrete, graphic, and practical aspects of elementary geometry. Geometrical treat- ments are at first intuitive, inductive, and experimental, fol- lowed in a few instances by the quasi-experimental methods of superposition. A like informality as to method characterizes the first half of the algebra. The algebraic treatments feature distinctly: (i) the liberal use of inductions from arithmetic; (2) the early and persistent use of the equation, both as an instrument for problem-solving, and as a means of suggesting and of treating topics; (3) the early, varied, and systematic use of pictured and graphic modes of rendering algebraic truths vivid and appealing to beginners. The algebra of the last half of the book is more formally deductive in character. Even as early as p. 26 the four pro- cess-axioms are given as simple statements of the four ways in which equations have been changed in solving arithmetical and mensurational problems. They are stated as laws for ix x Preface using the equation, and are given only as much generality as the antecedent work justifies. Transformations of equations are then made on the basis of these laws and of others brought out from time to time as the work proceeds, until p. 216 is reached, where a fifth general axiom is added, and the five are here given final statement. These five axioms are hence- forth made the reasons for all changes in equations. The root-axiom is added later when an occasion to use it arises. Thus the transition from the informal procedure of the earlier part of the algebra to the formal procedure of the later part is gradual, while throughout the latter half the reasoning becomes more and more highly deductive. This is in accord with what both classroom practice and a-priori reasoning show to be the natural order for secondary mathematical education. The subject-matter of the two years of work of this volume and its companion, Second-Year Mathematics, fully meets collegiate entrance requirements of one unit of algebra and oie unit of plane geometry. There is gain in keeping the twofold point of view continually before the learner, habituat- ing him to face his problems as mathematical problems rather than as algebra problems, or geometry problems. This re- quires him to keep both kinds of subject-matter longer before him, and brings to his help continually the reflex light that each subject throws upon the other. Thus the result must be a superior quality of collegiate preparation. The college teacher of pupils prepared on this plan, no less than the high- school teacher, will be gratified by the results the work of such students will show. Colleges do not usually require, and they should not re- quire, that the desired units of credit be made by the study of algebra and geometry as separate and distinct domains of mathematical truth. Much is lost educationally by this plan, and the authors are confident that college teachers will be Preface xi the first, the most numerous, and the wannest friends of the plan of these two books. That such material as this book contains furnishes a better fitting for the mathematical needs of daily life than a year of formal algebra will hardly be denied. Accordingly it has no apologies to make for its appearance. It works an almost virgin soil in a field in which it elbows no competitors. On the other hand, this book should not be looked upon in any sense as a protest against algebra as algebra, or geom- etry as geometry. Some teachers are succeeding measurably in making high-school algebra grip the motives of high-school boys and girls, and this is the main thing in mathematical teaching. For most teachers, however, it is much easier to motivate a school subject in a broad than in a narrow field. This is particularly true when in the narrow field the ideas are highly sophisticated and abstract, as is the case with school algebra. Greater freedom and more informality in getting at the beginner's field of ideas and fund of experiences are desired by many of our best mathematical teachers, and their number is increasing. This text has been thoroughly seasoned through four years of classroom use in the University High School and purged by the ordeals of severe criticism. Six high-school teachers, with experience in high-school work ranging from five to twenty- five years, have participated in its authorship and in the practical tests of the classroom. The present form of the book is a thoroughgoing revision in the light of all this use and criticism. The conditions under which the material has been tried in every way are precisely the conditions of the public high school. The marked improvement that has been wrought in the mathematical attitude and tone of the early classes of the University High School mainly through the agency of this text justifies strong claims for its suitability and worth for first- xii Preface year classes. Finality is, however, not claimed for it. It is claimed that this book will work now, and will work easily and effectively, under average conditions in the public high schools of the nation. Actual experience under such conditions proves beyond the power of a-priori argument to disturb, that without loss of mathematical solidity this book contributes in a superior degree to the much-neglected virtues of mathematical interest, earnestness, and spirit, and to an early and genuine faith in the worth of mathematical study. The second-year text will make geometry the central theme and the algebra will take second rank. Algebraic matters will be related to kindred parts of geometry, and will, when needful, be given parallel or independent topical treatments. The methodology of last treatments of this year will be increasingly deductive and logical, though even here austere ideals are not attempted. The secondary school can do prof- itably little more than rough out the bolder outlines of the sciences of algebra and geometry. The hold on the algebra learned the first year will be maintained, and the customary topics well along into simultaneous quadratics will be treated with customary fulness. Plane geometry will be completed, with careful regard to a smaller number than is customary of very well-worked-out propositions. In conclusion the authors desire to render full acknowl- edgement to Dr. William B. Owen, dean of the University High School, for his interest and very substantial aid in furthering the practical side of this educational experiment. Without his assistance and continued interest, the task would have been difficult, -if not impossible. He was among the first to recognize the importance of the function of the University High School as an experimental laboratory for secondary edu- cational problems. Any success that may come to this book is due largely to him. THE AUTHORS CHAPTER I . / %/ * GENERAL USJi-o OF THE EQI/ATION Problems and Exercises on the Balance i. Problems in weighing are readily solved by the aid of the equation. i. A bag of grain of unknown weight, w lb., together with an 8-lb. weight just balances an i8-lb. weight. How much does the bag of grain weigh ? The problem may be stated in an equa- tion, thus: u; + 8 = i8, find w. Suppose 8 lb. taken from each pan, giving The bag of grain weighs 10 pounds. FIG. i 2. Two equal, but unknown, weights together with a i-lb. weight just balance a i6-lb. and a i-lb. turnrr'{('M 7 c s 9 2 - o = 7 " z = , i 20. W=i 5 12. ^X = IO 7 3- =6 4- 7 -=8 13- 14. 7^=49 8^=32 27. s=6 12 9 15- 6^=48 28. I2/ = I32 5- r 16. 5^=45 29. IIX=12 9 i7- 9* =72 3- 63^ = 9 6. r = 9 1 8. 85 = 72 31. 72y=S 7 # 19. 7^=63 32- 56^ = 7 7- r 7 20. 9^ = 63 33- 48r=8 a; 21. 122=84 34- 35^ = 7 . 6 =9 22. n<2=66 35- 6od = 50 y 23- 12^ = 72 36. I.2W = I 9. =1 12 37. Helen had 8 cents and her father gave her 19 cents more. many cents had she then ? 38. A boy had 15 cents and was given c cents more. How many cents had he then ? 3. The sum of 15 and c is written thus: 15+^, or The first form is read "fifteen plus c" and the second form, "c plus fifteen." General Uses of the Equation 5 The difference of 15 subtracted from c is written c 15, and is read "c minus fifteen." 1. A man had 22 acres and sold 8 acres. How many acres had he left ? 2. A man had 22 acres and sold a acres. How many acres had he left ? 3. A man had a acres and sold b acres. How many acres had he left ? 4. A man had 50 acres and sold 20 acres. How many acres had he left ? Problems to be Solved by Arithmetic or Algebra 4. Many problems may be solved either by arithmetic, 01 by the use of the equation. When the solution of a problem is made by the use of the equation, it is commonly called an algebraic solution. i. Divide a pole 20 ft. long into two parts so that one part shall be 4 times as long as the other. ARITHMETIC SOLUTION The shorter part is A certain length. The longer part is four times this length The whole pole is then five times as long as the shorter part. The pole is 20 ft. long. The shorter part is of 20 ft, or 4 ft. The longer part is 4-4 ft., or 16 ft. Hence, the parts are 4 ft. and 16 ft. long. ALGEBRAIC SOLUTION Let x be the number of feet in the shorter part, ^ then 4X is the number of feet in the longer part, and xi-^x, or 5^=20, *= 4, teet in Hence, the parts are 4 ft. and 16 ft. long. 2. A farmer wishes to inclose a rectangular pen with 80 ft 6 First- Year Mathematics of wire fencing. He wishes it to be three times as long as it is wide. How long shall he make each side ? ALGEBRAIC METHOD Let * be the number of feet in the smaller side, then 3* is the number of feet in the longer side, and * + 33;, or 4* is the number of feet half-way round the pen. 4X =40, x =10, 3* = 3. Hence, the sides are 10 feet and 30 feet long. 3. A boy sold a certain number of newspapers on Monday, twice as many on Tuesday, 10 more on Wednesday than on Monday, and 24 on Thursday. He sold 94 in the four days. How many did he sell on each day ? 4. A man divides up his i6o-acre farm as follows: He takes a certain number of acres for lots, 4 times as much for pasture, 4 times as much for corn as for pasture, ^ as much for wheat as for corn, and 15 acres for meadow. How many acres does he assign to each purpose ? 5. A book dealer has in stock twice as many Readers as Arithmetics, four times as many Readers as Histories. In all he has 70 Readers, Arithmetics, and Histories. How many of each has he ? m 6. A may-pole 22 ft. high breaks into two pieces so that the top piece, hanging beside the lower piece, lacks 6 ft. of reaching the ground. How long is e^f h piece ? 7. A pony, a saddle, and a bridle together cost $120. The bridle costs as much as the saddjflfeid the pony costs $12 less than 12* times as much as the s^He. What was the cost of each ? 4 ^^^^^ 8. A bicyclist rode a certain number of miles on Monday, f as many miles on Tuesday, as many on Wednesday, f as General Uses of the Equation 7 many on Thursday, as many on Friday as on Monday, and 20 miles on Saturday. On the six days he rode 152 miles. How many miles did he ride each day ? 9. James has 3 times as many cents as Charles, and 4 times as many as William. All together they have 57 cents. How many cents has each ? 10. The area of a triangular piece of ground is 315 sq. rd. One side is 30 rods. How long is a fence at right angles to this side from the opposite corner ? (Use principle below.) PRINCIPLE. The area of a triangle is equal to \ the product of its base and altitude. 11. If this fence divides the side (30 rods) so that one part is twice as long as the other, what are the areas of the two lots? 12. If the fence divides the side (30 rods) so that one part is five times the other, what are the areas of the two lots ? 13. A local train goes at the rate of 30 miles an hour. An express starts two hours fater and goes at the rate of 50 miles an hour. In how many hours and how far from the starting- point will the second train overtake the first ? 1.4. If 75 dollars are to be divided between two persons so that one shall have 27 dollars more than the other, how much must each one have ? 15. Three men, A, B, and C, wish to divide 1,584 shares of stock among themselves so that A shall have 25 more than B, and C shall have 50 more than B. How many shares must each receive ? 16. At an election in the Fxeshman class of the University High School 84 votes were cast. Candidates A and B each received a certain number of votes; candidates C and D each received three times as many as candidate A; and candidate 8 First-Year Mathematics E received four times as many as D, plus 4. How many votes did each candidate receive ? 17. A regulation foot-ball field is 56$ yards longer than it is wide and the sum of its length and width is 163$ yards. Find its dimensions. . 18. The manager of a high-school foot-ball team paid the manager of a visiting team $63 for expenses. If the payment was made in $2 and $5 bills, the same number of each, how many bills of each kind were paid ? 19. A line 40 ft. long is divided so that the longer part is 12 ft. more than 3 times as long as the shorter part. Find the length of each part. Draw a line 4 inches long to represent the 40 ft. line, and mark on it the lengths of the two parts. 20. Two men, A and B, wish to divide $5,247 between them so that B shall receive $324 more than twice as much as A. How much should each receive ? 21. The length of a rectangle is 26 yd. greater than the width. If the perimeter (distance around) is 432 yd., find the dimensions. 22. Find two consecutive numbers whose sum is 203. (Suggestion: Let x be one number, and x+i the other.) 23. Find three consecutive numbers whose sum is 474. 24. Find two consecutive odd numbers whose sum is 156. (Suggestion- Let x be one number and x+2 the other.) 25. Find two consecutive even numbers whose sum is 378. 26. Find three consecutive even numbers whose sum is 372. 27. A college student spent during his Freshman year $620 for tuition, room and board, and books. For tuition he spent four times as much as he did for books, plus $20, For room General Uses of the Equation 9 and board together he spent ten times as much as for books. How much did he spend for each item ? 28. The tuition at the University High School is $150 a year. If this tuition is paid in twenty-dollar bills, and if the same number of five-dollar bills is received in change, how many twenty-dollar bills are paid ? 29. Divide 341 into three parts so that the second part shall be five times the first part, and the third part shall be two times the first part plus 5. 30. A farmer wishes to lay out a field in .the shape of a rectangle whose width is f of its length, and whose perimeter is 320. How long and how wide must he make the field ? EXERCISE III The following exercises test and review the addition and multiplication facts of arithmetic, and give exercise in alge- braic equations. Find the value of the letter, x, doing all you can orally: 1. 2X+x=<) 16. 8# 8=64 2. $x x=8 17. 8x2X=42 3. 2X + IO = 22 l8. 8x + X = 8l 4. 2# 7 = 13 19. 8x+2x=7o 5. 7^ 8=48 20. 8x 2X = 'J2 6. 7^+1=64 21. 7- 7^ + 6=62 22. 8. 7# 9=40 23. 9. > jx-\-2x=2'j 24. 8x+4X 7#= 10. jx x=$6 25. n. jx2X=6o 26. 12. 4#+5#=8i 27. nx 13. 8x+2=66 28. 5^+6^ + 7^=3 14. 8^4=62 29. 7^ + 2^8=46 15. 8^+9=81 30. 7# 2^+6=41 io First- Year Mathematics 31. i2x 4#+#=9o 41. 32. sxx+3x=2o 42. 2#+7# 33; 6=24 33. 6x+x-$x = iS 43. \x=6 34. 7#-#+5#=77 44. *=3 35. i2x+Sx-$x=4$ 45. %x=8 36. i&x 9#+2#=66 46. f#=2o 37. 19^8^-3^=56 47. x+%x=6 . 38. 2ox 8x+x=26 48. ## = 7 39. 14^+4^-9^=63 49. $x+$x- : bx=i& 40. 12^+9^3^=63 50. 6x$x #+7 = 129. Problems and Exercises 1. A man bought a certain number of ducks, three times as many turkeys, and five times as many chickens as turkeys. In all he bought seventy-six head of poultry. How many ducks did he buy ? How many turkeys ? How many chickens ? 2. A thermometer showed a certain number of degrees rise one hour, and 2 degrees more than 3 times as many degrees rise the next hour. The rise for both hours was 14 degrees. Find the rise for each hour. 3. The rise of a mercury column in two hours amounted to 7 degrees, and the rise the second hour was 3 degrees less than 4 times the rise the first hour. Find the rise for each hour. 4. In two hours the total rise of the mercury was 3 degrees, and the rise the second hour was 5 degrees more than the change the first hour. Find the change for each hour. What do the results mean on the thermometer ? 5. In two hours the total rise of the mercury was 2 degrees and the rise the second hour was 8 degrees more than twice the change the first hour. Find the change in the reading for each hour. Interpret the results by means of the ther- mometer. General Uses oj the Equation ii The following exercises test and review the laws of arith- metical calculation, and give exercise in algebraic equations. Find the values of the letters, doing all you can orally: 6. 2X+i=6 37. 4.5^1.25^+12.7=38.7 7. 33; 2 = 12 38. 16.5^+15.82.3^ = 186.2 8 - 4J-3 = 12 39- 6. 15^-1. 65^+7. 8=57. 3 9- 5J+3 = I 5 40. 8^+6.875+2^=46.875 4i- 3+5. 37Z-8. 73=61. 34 42. i 3 /-8. 75^+6.87=57. 87 43-' I 5*+3-73*-9- 2 3 =6 5- 6 9 45. 404^-304^ + 12^=560 x 10. 8^-7=53 41. z- II. 8545 = 26 42. 13 12. 9^ + 11 =92 43-' 15 1 3- lot 11 = 104 44. 32 14. 12^+8=98 45- 4C '5- i2/ 6 = 74 x 16. 126 + 16=82 46. 17- 12^-9=43 t 18. 12^ + 7=95 47- 3~ 19. i2# 15=30 20. 16^ + 13 = 73 48. - 21. 18^-12^=33 5 22. 212+15 = 102 49- y. 2 3- 25217 = 113 4 24. 28^+14 = 158 5- 2 25- 28^9=251 2 26. 2 O^ + 2 X I oX ^ = 2 2 S 27. 17^ 3^+16^ = 105 5- 4 28. 175 + 75-135 = 88 Ii 29. l6/ + 2/-I 3 / = 22i S 2 - ~a 3- 3.4^ 1.2^ + 4.8^ = 70 4 31- 3.5^ + 7.6^-8.6^ = 15 53- x 32. 5- 8 >'-3-93' + I2 - 6 >' = 5 8 33- 65-3.55+5.55=68 54- z 34- 75 3 .5^ "I" 1 " 2 == 54- 7 35- 6.825 + 1.1853.55=45 ee. 9 t 36. 8* -4. 5* +5. 2^=87 =6. 12 First-Year Mathematics Problems on Percentage and Interest 1. Find the percentage of $120 at 4%; at 6%; at 7$%; at r%. 2. Find the percentage at 8% of $25; of $250; of $1,527; of $6. 3. Find the percentage at r% of $20; of $80; of $b. 4. Calling p the percentage, b the base, and r the rate, show that and show that, by one of the laws of multiplication of frac- tions this may be written bXr p = - . (2) 100 5. Read both (i) and (2) of problem 4 and translate them into words. 6. Find the interest on $175 at 4% for 2 years; for 5 years; for of a year; for 2$ years; for t years. 7. Find the interest on $600 for 5 years at 3%; at 5%; at 8%; at 6J%; at r%. 8. Find the interest on $160 for / years at 6%; at 3$%- at r%. 9. Find the interest at 6% for 3 years on $200; on $360; on $756; on $p. 10. Find the interest at 5% for t years on p dollars. 11. Find the interest i, at r% for t years on p dollars. 12. State in words the meaning of i=pX Xt. 100 13. State in words the meaning of ._pXrXt v - 100 General Uses o] the Equation 13 IMPORTANT CONCLUSIONS 1. An equation is an expression of balance of values, or of numbers. 2. The equation is a convenient tool for problem-solving. 3. To solve a problem by the use of the equation, one equation is derived from another (1) by adding the same number to both sides, (2) by subtracting the same number from both sides, (3) by multiplying both sides by the same number, or (4) by dividing both sides by the same number. 4. Many problems may be more easily and simply solved by the equation than by arithmetical methods. 5. All the arithmetical processes of addition, subtraction, multiplication, and division, of both integers and fractions, must be well known in algebra. 6. Solving problems by the equation sometimes leads to subtracting larger from smaller numbers and necessitates using negative numbers, or "numbers below zero" on the thermometer scale. 7. Percentage and interest problems may be easily and cklv solved bv the eauation. quickly solved by the equation. CHAPTER II USES OF THE EQUATION WITH PERIMETERS AND AREAS Indicating Arithmetical Operations Algebraically 1. A boy rides on his bicycle 8 mi. in one hour and 5 mi. the next hour; how far does he ride in the two hours ? NOTE. Write the sum of 8 and 5, not 13, but 8 + 5. The form 8+5 is just as truly a sum as is 13. It will sometimes be desirable to refer to the form 8 + 5 as the indicated sum. 2. If the boy rides a mi. the first, and 7 mi. the second hour, how far does he ride in the two hours ? 3. A boy has m marbles and buys p more; how many has he then ? 4. A boy had 18 marbles and lost 7 ; how many had he then ? NOTE. Give the indicated difference. 5. A boy had m marbles and lost 8 of them; how many had he left ? 6. A boy had m marbles and lost of them; how many had he left ? 7. Show the sums of these pairs of numbers and the differ- ences, the first of the given numbers being the minuend, and the second, the subtrahend: (1) x and 7 (5) $ and / (9) re and a (2) a " 12 (6) 15 " n (10) d " c (3) y " 10 (7) r s (ii) c " b (4) x " y (8) a " x (12) t " m. 8. How many yards of cloth are 12 yd. and 10 yd. ? 9. How many dozens of eggs are 8 doz. and 4 doz. ? 10. How many i2's are 5 i2's and 4 12*3? 11. How many times 12 are 9X12 and 4X12 ? 14 Uses 0} the Equation 15 12. How many half-dozens are 8 half-dozens and 6 half- dozens ? 13. How many times 6 are 8X6 and 6X6 ? 14. How many times 3 are 5X3 and 7X3 and 10X3 ? 15. How many times x are 2 times x and 3 times x? x and x ? x and 3 times x ? 5. It is customary in algebra to write x+x, x+x+x, x+x+x+x, etc., thus, 2X, $x, ^x, etc., and to read them "two x," "three x," "four x," and so forth. But x times x, x times x times x, and x times x times x times x are written x 2 , x 3 , x 4 , etc., and are read, "x square," "x cube," "x fourth power," etc. x square may be read "x 2d power" or "x 2d;" x cube may be read "x 3d power," or "x 3d;" and "x fourth power" may be called "x 4th," etc. The dot is often used instead of the times sign to indicate multiplication. Thus, xXxXx may be written x-x-x t meaning x times x times x. 6. In 2X, $x, ^x, ax, etc., the 2, 3, 4, a, etc., are called coefficients of x. 7. In x 2 , x 3 , x 4 , x n , etc., the 2, 3, 4, n, etc., are called exponents of x. What would be the written form of "x 5th?" "x 6th?" "x 7th?" "x loth?" "x nth?" What would be the mean- ing of each of these forms ? 8. Notice that 4* means 4 x, or that x is to be used as an addend 4 times, while x 4 means x x x x, or that x is to be used as a factor 4 times, and similarly, for the other forms, as $x and x 3 , and $x and x 5 , etc. Letting x=6, give the values of 1. 2X 4. x 3 7. $x 10. 3# 2 2. # 2 5. 4^c 8. x$ ii. 5# 3 3. 3# 6. # 4 9. 2# a 12. 2X*. i6 First- Year Mathematics 13. Describe briefly the meaning of factor and of addend as they are used in arithmetic. Expressing Perimeters and Areas i. What is the sum of the sides of a triangle (Fig. 5) whose sides are 2X ft., 2x ft., and 3* ft. long ? Express the sum as a certain num- ber of times x. FIG. 5 FIG. 6 2. What is the sum of the three sides 20, 5a, and 6a of a triangle ? 3. A lot has the form of an equal-sided (equilateral) triangle (Fig. 6), each side being x rd. long. How many rods of fence will be needed to inclose it ? p. Any figure whose sides are all equal is called an equilateral figure; as equilateral triangle, equilateral penta- gon, etc. The sum of all the sides of any closed figure is called the perimeter of the figure. y i. In Fig. 7, called a parallelogram, what part of the perimeter does x+y equal? What is the whole perimeter? F 10 - 7 2. Show (i) that the perimeter, p, of a square whose side is 8 is given by p=4 8 and (2) that the area, A, is given by A =8 2 . 3. Show by equations the perimeter, p, and the area, A, of a square whose side is (1) 12 (4) a (7) 10+2 (2) * (5) y (8) *+7 (3) x (6) 2b (9) c+d. 4. What is the perimeter of a square of which the area is 16 sq. ft. ? a 2 sq. ft. ? \a? sq. ft. ? x 2 sq. ft. ? L, ' : d c.d d* c c* t-d 10 2 FIG. 8 c d FIG. 9 Uses of the Equation 17 5. What is the area of a square whose perimeter is 20 ft. ? 4* ft. ? 80 ft. ? 6. Draw figures to illustrate your solutions of 5. 7. The dimensions of a rectangle are a and b, what is the perimeter? The half -peri meter ? The sum of a pair of op- posite sides ? The sum of the other pair? 8. What is the area of the rectangle of problem 7 ? 10. Numbers denoted by letters are literal numbers. The product of two different literal numbers, as x and y, is shown by writing the letters, or factors, side by side, as xy, with no sign between. We are familiar with the form xXy from arithmetic. The form xy is most used in algebra. It is often convenient to use the form x y. 1. Recalling that the exponent of a number shows how many times the number is to be used as a factor, give the products of the following pairs of factors: (1) 4 2 4 2 (5) 2 a (9) b 2 - bs (2) 8 83 (6) a* 03 ( I0 ) c - c* (3) io 2 io 4 (7) a - 05 (n) C 3 . C 6 (4) 123 . I2 4 (8) 04 07 (12) X* X s . 2. Show from Fig. io, (i) that the per- imeter, p, of a square of $x is given by P = i2x, and (2) that the area, A, is given by 3. Express by an equation the area, A, of a rectangle that is 8 in. long and 5 in. wide; of the same length and 4 in. wide; 3^ in. wide; 6\ in. wide. 4. Express by an equation the area, A, of a rectangle 12 in. long and of the following widths: 6 in.; 8J in.; 9^ in.; lof in.; x in.; y in. i8 First- Year Mathematics 5. Express by equations the areas of rectangles / in. long and of the following widths: 12 in.; 9 in.; h in.; n in.; x in.; a in. 6. Express by equations the areas of rectangles of width w and of the following lengths: 8; 10; 12 J; x; a; I; b; z. 7. Express by equations the products of the following pairs of factors: (5) b by b 3 (9) x* by x (6) a 2 by a (10) a by x 3 (7) a 2 by a* (u) a * by x (8) * by *s (12) g by /'. 8. Show from Fig. n, that the area, A, of a rectangle ix by 3* is given by A=6x\ 9. Show that the area, A, of a rectangle 30 by 50 is given by A = i$a a . 10. Express by an equation the , - " "> N area, A, of parallelograms of altitude a, and of the following lengths: 10; 12 ; 16.8; 6; c; 07 2; 52. (I) (') (3) (4) a b b a by x by c by & by a X ** i i i X i i FIG. ir b FIG. 12 ir. Express by an equation the 7 area, ^4, of triangles of altitude a, and of ' the bases, 12; i8j; 20.25; b; d; x; zy; 42; 5 C - F 10 - X 3 12. Show that the area, A, of the trapezoid, in the accompanying Fig. 14, 6+8 is given by A =4 . 13. Show that the area, A, of the FIG. 14 Uses of the Equation trapezoid, in the accompanying Fig. 15, is given by A=T ) 9 14. Show that the line EF= (6 +c), FIG. 15 c meaning \ of the sum of b and c of Fig. 16. 15. Show from Fig. 16 that the area, A, of the trapezoid is given by b+c a(b+c) A=a = . h FIG. 1 6 16. Express by an equation the area, A, of a rectangle of dimensions (1) 6 and #+3 (3) a and x 2 (2) #anda+5 (4) a+2 and x+i. 17. Indicate the area of a rectangle of dimensions a+b and x+y. NOTE. The product of m+n and c+d is written 18. Express in terms of its base c and altitude the area of the rect- angle (i) of Fig. 17; of (2); of (3); d of (4). 19. How then may you express (2) (*> (1) (3) FIG. 17 (m+n) (c+d) by using the truth that any whole equals the sum of all its parts ? 20. State in words the value of (1) ( x +y)(a+b) (3) (a+x)(b+y) (2) ( x +y}(m+n) (4) (r+s)(a+x). 21. Show by a figure the value of (1) (a+b)(a+b) or (a+6) a (3) (x+yY (2) (c+d)' (4) (m+n)'. 2O First-Year Matliematics 22. Each of the following expressions is the product of what two equal numbers ? (1) a 2 +2ax+x 2 (3) k 2 + 2kb+b 2 (5) x 2 +6x+g (2) b 2 + 2bc +c 2 (4) s 2 + 2st +t 2 (6) c 2 +8c-fi6. 23. The base of a rectangle is 8 yd. and the area is 40 sq. yd. What is the altitude, a< 24. What is the altitude, a, of a rectangle having a base =8 in. and an area =32 sq. in. ? =8 in. " " =i6sq. in. ? " =8 in. " " =12 sq. in. ? =5 ft. = 7isq. ft. ? 25. Show by an equation the base, b, of a rectangle having an altitude =9 ft. and an area = 27 sq. ft. =9 ft. " " = i8sq.fi. =9 ft. " " =15 sq. ft. =9 ft. " =12 sq.ft. =9 ft. " " =A sq. ft. 26. Show by an equation the other dimension of a triangle having a base =6 ft. and an area =24 sq. ft. " =6 ft. " " =12 sq. ft. ' =6 ft. " " =9 sq. ft. " =6 ft. " " =3 sq. ft. an altitude =4 yd. and an area = i6 sq. yd. =4 yd. " " = 8 sq. yd. =4 yd. " " =4 sq. yd. =a ft. " " = a sq. ft. = /trd. " " = sq. ft. =6 in. " " =a&sq. in. = 6 in. " " =by sq. in. =c in. " " = a sq. in. =ain. " " = 6sq. in. Uses of the Equation 21 3C 11. The quotient of x divided by y is written - , or x-^-y y X and is read "# divided by y" - is also read "x over y." Show by an equation the quotient, q, of the first of the. c numbers divided by the second: 1. m and n 5. a+b and c 9. a 2 b 2 and a b 2. a " b 6. a " x+y 10. x 2 " a + b 3. b "a 7. a+b " c+d n. (a+b) 3 " a + b 4. 4* " $y 8. a 2 -b 2 * " a+b 12. (a-b) 2 " a-b *NoxE. a 2 b 2 is read "a square minus b square;" (ab) 2 is read "the square of ab," and (a+b) 2 is read "the square of a + b." Definitions 12. In an expression like ^xy+^axgb^, the numbers $xy, fax, gb, and 4, that are connected by addition or subtrac- tion, are called terms of the expression. 13. A one-term expression, like 8, yx, abc 2 , etc., is called a monomial. 14. The coefficient of any factor, or number in a term is the product of ah 1 the other factors of the term. Thus, in 4axy, the coefficient of axy is 4, of xy is 40, of ax is 4y; in the coefficient of c is - , of b is - , etc. 3 33 When the coefficient of the term is spoken of the arith- metical factor is usually meant. Thus, it is common to say the coefficient of the term ^axy is 4; of is , etc. If no o coefficient is written as in a, x 4 , i is understood, as though i a, ix 4 were written. 15. The exponent of a number is the small figure or letter written to the right and a little above the number symbol, to 22 First-Year MatJtematics denote the number 0} equal factors in a product. Thus, in 63, meaning the product 6X6X6, the 3 is the exponent. The number 6, itself, is called the base, and the product 6 3 is the power. Thus, 216 (=6 3 ) is the 3d power of 6, because using 6 as a factor 3 times gives 216, and so with other numbers. When no exponent is written, as in ax, the exponent is understood to be i for each letter, as though the number were written a 1 x 1 . 1 6. A number as 40^+10, denoted by an expression o] two terms is a binomial; a three-term expression is a trinomial. 17. Number expressions consisting of two, three, four, etc., terms, are called polynomials. The area of a rectangle whose altitude is 5 and whose base is the sum of two lines, one a inches and the other b inches long, may be written thus: 5(a+6). If the altitude were 7 and the base equal to the difference, a b, the area would be written 7 (a 6). 1 8. The sign, ( ), called the parenthesis, means that the terms within, as the a and the & in 17, are first to be added, or subtracted, and then the sum or difference is to be multiplied Give the meaning of the following: 1. a 3 4. 8(40+2) 7. i6(a+&) 2. 4a 2 b 5. 9(603) 8. 2o(xy+i) 3. 6(10-3) 6. 7(80 + 1) 9. 25(#-a). Finding Values of Expressions Of what kind of figures may the following equations ex press the perimeters, p ? x FIG. 1 8 Uses o) the Equation 23 2. p=4X 6. p=8x 10. p=2ox 3- P=5 X 7- P=9 X ii- p=nx 4. p=6x 8. p=iox 12. p=ax. 13 Give the coefficient of x in the second members of problems i to 12. * Show, by a sketch, figures whose peri- meters, p, are given by 14. p=22X+2 17. p= 15 ^ = 2(3^+2) 18. p=6x 12 16. p=6x+4 19. ; = i2# 24 23. Find the values of the perimeters in problems i to 12 for x=2 in.; for x=$ ft.; for x=6 yd. 24. Find the values of the perimeters, p, of problems 14 to 22, for x=$ and y=2; for x=i2 and y=4; for x=a and =6. <--"- -*3-- Of what plane figures do the following equations give the areas, A ? 25. A=x* 30. A=a\ .^ 26. A=$ax 31. A=x(y2) 32. = 28. ^=5(^+3) d _^+i 2 /6+c\ !fl 2 ; _ 29. ^4=^(a+2) 2 36. ^4=^ 2 4X 37. Find the value of A in problems 25 to 36 for x=$, y=2, a=4, 6 = 1, c = i; for #=5, ^=3, a=6, 6=2, c=4. 19. An expression like (#+5)(# y) means that a and 5 are first to be added, then y is to be subtracted from x, and the sum, a +5, is to be multiplied by the difference, x y. Give the meanings of the following: 1. (*+;y)(a + io) 3. (x+y)(a 6) 2. (o+6)(c+rf) 4. (a-&)(c-rf). First-Year Mathematics Sketch and show dimensions of rectangles whose areas are given by 5. 4 = 6. ,4= 7. A=(a 8. A = (a- 9. ^ = (a- 10. A = (a- 11. A = (a 1)(6 12. ^4 = a 13. X = 14. A = 15. 4 = (a 4)(a+4) 16. A = a FIG. 23 20. The signs, V and V , called radical signs, have the same meaning as in arithmetic. Give the values of the following for a =3, 6=3, c = i, t=8, x=io: 2. 8(a+#) 7- 3. a(b+c) 8. 4. t(abc) 9. I/a 2 13. x*> 5. a(x + b c) 10. Vab 14. 25^3 n. The Use of the Fundamental Laws in Equations i. The width of a rectangle is #3, the length x, and the perimeter is 66 yards. Find the width and the length in vards. Uses of the Equation 25 The perimeter being denoted by both 66 yards and 4* 6, we write the equation 4X 6 = 66. Adding 6 to both sides 4# = 72. Dividing both sides by 4 # = 18. Subtracting 3 from both sides *-3 = I 5- (4) The width is 15 yards and the length 18 yards. CHECK: 15 + 18 + 15 + 18 = 66, or from equation (i) 4i8 6 = 72 6 = 66. 2. The width of a rectangle is x ft., the length 00 + 12 ft., and the perimeter is 144 ft. Find the width and length. Both 144 and 4^+24 denoting the perimeter, we write 4^+24 = 144. (i) Subtracting 24 4X = 1 20. (2) Dividing by 4 x= 30, the width. (3) Adding 12 x + i2= 42, the length. (4) CHECK: 30+42 + 30 + 42 = 144, or from equation (1)4^+24=4.30 + 24 = 120 + 24 = 144. 3. One-fifth of the perimeter of a field plus 3 J rods is 10 rods. Find the perimeter. Call p the perimeter, then ^ + 3i = io. (i) Multiply both sides by 10 2/> + 35 = ioo. ( 2 ) Subtract 35 2^ = 65. (3) Divided by 2 />= 32*. (4) CHECK: +3$ = 6i + 3^ = 10. 4. In the solution of problem i show where each of the following laws is used : a6 First-Year Mathematics (1) If the same number be added to equal numbers, the sums are equal; (2) If equal numbers be divided by the same number, the quotients are equal; (3) If the same number be subtracted from equal numbers, the remainders are equal. 5. Show where the laws of problem 4 are used in the solu- tion of problem 2. 6. In the solution of problem 3, show where the law is used: "if equal numbers be multiplied by the same number the products are equal. 7. In the same solution show where the following are used : "If the same number be subtracted from equals the re- mainders are equal." "If equal numbers be divided by the same number the quotients are equal." 21. The four laws thus far used are I. // the same number be added to equals the sums are equal, II. // the same number be subtracted from equals the re- mainders are equal, III. // equal numbers be multiplied by the same number, the products are equal, IV. // equal numbers be divided by the same number, the quotients are equgl. Solve the following, using the equation and showing which fundamental law is used in each step. 1. The perimeter of an equilateral triangle is 27 rods. Find a side. 2. The perimeter of an equilateral quadrilateral is 48 ft. Find a side. 3. The perimeter of an equilateral hexagon (6-side) is 186. Find a side. Uses of the Equation 27 4. The perimeter of an equilateral decagon (ic-side) is 255. Find a side. 5. The perimeter of an equilateral dodecagon (12 -side) is 294. Find a side. 6. The area of a rectangle is 96 yards, the base is #+8 and the altitude is 8. Find the base. Find x. 7. The altitudes, bases, and areas of parallelograms are as given. Find the value of the unknown dimension in arith- metical numbers. Altitudes Bases Areas * 3 4 12 3 *+2 8 56 10 3X + 2 IIO 16 5*~7 288 2x+i 15 105 25 600+40 2,500 28^17 30 840 Application of the Fundamental Laws 22. The four fundamental laws stated above may be re- ferred to thus: I is the addition-law, II, the subtraction -law, III, the multiplication-law, and IV, the division-law. These laws are also called axioms. EXERCISE IV Solve the following equations, i. e., find the number for which the letter stands, and show where each of the four laws just stated is used : < 1. x 5 = 12 6. 3# 4 = 14 ii. i2x=']2 2. 2# 6 = 18 7. 4#+i=2i 12. ii#=9o 3. 3#-fi=i9 8. 4# 6 = 18 13. f# 2 = 10 4. 3# I=2O 9. 4# + 2=26 14. f# + I4 = 20 5. 3^+4=25 10. 4^-16=32 15. |#-io=2 28 First- Year Mathematics 16. ^-+3=17 + i*=5* 2 7 7-~-S 23. +1=50 28> * + * =2 o 2 C 18. 2# 3^=3^ 24. (-2=26 7 # # 19. 4^ + 2^=31 2 (*+2) 29. - =3 20. 4 j-2i = i 3 f 2 5- ~ 21. -i=3$ 26. - =3 30. - - = i. SUMMARY Chapter II has exemplified or taught the following: 1. Numbers may be denoted by letters. Numbers thus de- noted are called literal numbers. 2. A sum is indicated by writing the plus (+) sign between the numbers to be added. 3. A difference is shown as in arithmetic by writing the minus ( ) sign between the minuend and the subtrahend. 4. A product of literal numbers, or of arithmetical num- bers by literal numbers, is usually shown by writing the factors side by side with no sign between them. The multiplication dot ( ) and the times (X) sign may also be used between factors to indicate their product. 5. A quotient is indicated as in arithmetic, or by means of the fractional form, the dividend being written above and the divisor below a horizontal line. 6. Powers are denoted by writing small figures, or letters to the right and above the base, the number of units in the figure, or letter, showing how many times the base is a factor in the power. 7. Roots are shown by the radical sign, as in arithmetic. Uses of the Equation 29 8. An equal-sided figure is an equilateral figure. 9. How to denote perimeters and areas of geometrical figures by algebraic expressions, and equations. 10. How to find the product of binomials by monomials, and of two binomial sums, both algebraically and geometri- cally. 11. A single number, or a product or quotient of numbers is a term. 12. A one-term number is a monomial ; a two-term number is a binomial ; a three-term number is a trinomial ; and a num- ber expressed by any number o) terms is a polynomial. 13. The coefficient of any factor in a term is the product of all other factors of the term. 14. The coefficient of a term usually means the arithmeti- cal factor of the term. 15. The exponent of a power is the number that shows how many times the base is a factor in the power. 1 6. The value of a letter is the number for which the letter stands. 17. To find the value of an expression means to substitute values for the letters, and to reduce the result of all operations indicated to the simplest form. 1 8. Changes in an equation can be made only in accordance with a few fundamental laws, called axioms. CHAPTER III . THE EQUATION APPLIED TO ANGLES The Measurement of Angles 23. What part of a complete turn does the hand of a clock or watch make (Fig. 25) when it rotates (turns) about the center post from 12 to 3 ? 12 to 6? 12 to 9? 12 to 4? 12 to 8? 12 to 2? 12 to 10? 24. If a line, OA (Figures 26, 27, 28, 29), be imagined to rotate (turn) in a plane about a fixed point, O, in the direction indicated by the arrow-heads, until it reaches FIG. 25 the position O B, it is said to turn through the angle x (also FIG. 26 FIG. 28 FIG. 29 The Eqwtion Applied to Angles 31 called the angle A OB). In Fig. 30 the rotating line has made a complete turn about the point O. The word "angle" means "the amount of turning" of a rotating line. The original meaning of the word was "corner," from the Latin angulus. Draw an angle made by a line which has rotated \ of a complete turn; of a complete turn; of a complete turn; i J turns; if turns. 25. If a line, rotating in a plane about one of its points, makes i of a complete turn, the angle it makes is called a right angte (Fig. 31); if the line makes a half -turn the angle is called a straight angle (Fig. 32). FIG. 31 * FIG. 32 1. Draw an angle equal (1) to two right angles (2) to three right angles (3) to four right angles (4) to five right angles (5) to two and a half right angles (6) to two straight angles (7) to one and a half straight angles (8) to two and a half straight angles (9) to three-fourths of a straight angle (10) to one and three-fourths straight angles. 2. How many right angles are there in an angle made by First- Year Mathematics a complete turn of a rotating line ? By a half turn ? By 7 turns? By turns ? Bv turns? By t turns? turns ? By - turns? ' 2 3. How many straight angles are there in an angle made by a complete turn of a rotating line ? By i^ turns ? By f of a turn ? By J of a turn ? By 9 turns ? By 3/ turns ? By turns ? 4 By 3(2^5) turns? 4. How many right angles are there in a straight angle ? In 5 straight angles? In 7^ straight angles? In 5 straight 5 I "ZS angles? In - straight angles? In straight angles? In 6 8 95+8 straight angles ? In 35 2 straight angles ? 5. How many straight angles are there in 4 right angles? In 2 right angles ? In 3 right angles ? In 7 right angles ? In r right angles ? In right angles ? In right angles ? O J In 6r+8 right angles ? In 4^6 right angles ? 6. Fold a crease (A O B, Fig. 33) in a piece of paper. Fold again so that the edge, O A, falls along the edge, O B. Unfold the paper. The two creases, A O B and C O D (Fig. 34), form four angles, x, y, z, w. Show that these angles are equal. Are they right angles ? Why? 26. On the protractor (angle meas- urer) shown in Fig. 35, the right angle, XOY, is divided into 90 equal angles. Each of these angles is a degree. FIG. 33 The Equation Applied to Angles 33 35 degrees is written briefly, 35. The symbol for angle is Z ; for angles, is Z*. i. How many degrees are there in ZX O A (Fig. 35) ? In ZXOB? InZAOB? InZXOC? InZAOC? FIG. 35 2. Express by an equation the number of degrees in ZXOA+ZAOB (Fig. 35); in ZAOB + ZBOY; in ZXOB-ZAOB; in ZXOB-ZXOA. 3. How many degrees are there in 2 straight angles ? In 4 right angles ? In of a right angle ? In | of a right angle ? 9f In -- of a right angle ? In r right angles ? In -^ right angles ? In r +7 right angles ? In - - right angles ? 27. The protractor is commonly a semicircle (Fig. 36). If a straight line were drawn from each mark on the circular 34 First-Year Mathematics rim to the center, O, 180 equal angles would be formed at O each of which would be an. angle of one degree (i). FIG. 36 1. Suppose a pointer to rotate about the fixed point, O (Fig. 36), in the direction indicated by the arrow. In rotating from the position O X to the position O Z through how many right angles would it turn? Through how many straight angles ? Through how many degrees ? 2. Find the number of degrees and the number of right angles in each of the following angles of Fig. 37: XOA; XOB; XOY; X O C; XOD; A OB; A O Y; B O C; BOD; COZ; A O Z; X O Z. 3. Read the angle in Fig. 37 which is the sum of angles X O A and A O B; of angles A O B and B O C; of angles X O B and B O Y. 4. Angle X O B is the sum of what two angles of Fig. 37 ? Answer by an equation. 5. Show by an equation that the sum of two angles of Fig. 37 equals ZAOC; ZXOC. The Equation Applied to Angles 35 6. Show by an equation that the sum of three angles of Fig. 37 equals ZX O C; ZA O C. FIG. 37 7. If ZXOA is taken from ZXOB (Fig. 37), what angle is left ? Answer by an equation. 8. Show by an equation the angle that is the difference between ZX O Y and ZXOA (Fig. 37); ZA O Y and ZAOB; ZXOY and ZB O Y. ' 9. From a point, O, on a straight line X Y (Fig. 38) draw three lines as in the figure. Find the number of degrees in each angle, first, by estimating, then by measuring with the protractor. Tabulate results as in Fig. 39, and find the sum. FIG. 38 Angle Estimated Measured I 40 39 2 90 3 4 Sum FIG. 39 10. What should be the sum of all the angles about a point, on one side of a straight line ? Check the sum in exercise 9. NOTE. To check results means to show that they are correct. Students should habitually check results, because correct results show that the work is probably correct. First- Year Mathematics FIG. 40 11. Draw four lines from a point (Fig. 40). Find the number of degrees in each angle, first, by estimating, then by measuring with a pro- ti actor. Tabulate results as in exercise 9, and check. 12. What is the sum of the angles that just fill the plane about a point ? The Sum of the Angles about a Point i. Find the number of degrees in each angle of Fig. 41. FIG. 41 We may write X+2X+2X + &+X+22=$6o. Why? This equation may be written x+2x + 2x + x + 8 + 22 = 36o. Why? Combining like terms 6^ + 30 = 360. Subtracting 30 6^ = 330. What axiom is used here ? Dividing by 6 x== 55. Whence 2# = iio. What axiom ? Why? Why? Why? (i) (2) (3) (4) (5) (6) (7) (8) .V+22 = 77 CHECK: 55 + 110 + 118 + 77=360. 2. With a protractor measure the angles of Fig. 41 to see if the figure in the book is accurately drawn. 3. To obtain the second equation, problem i, from the first equation, the order of the terms was changed on the left side of the first equation. Why may this be done ? The Equation Applied to Angles 37 4. Perform the following additions and subtractions: (1) 8 + 7+3+6 (5) 8 + 7-3+6 (9) a + 7 (2) 8+3+6 + 7 (6) 8 + 7+6-3 (10) 7+a (3) 7+6+8+3 (7) 7-3+8+6 (n) 3 a + 7 +a (4) 6+8+3+7 (8) 6-3+8 + 7 (12) 3 a+a + 7 . 5. In problem 4 compare the results of (i), (2), (3), and (4); of (5), (6), (7), and (8); of (9) and (10); of (n) and (12). 6. In the following problems perform the additions and subtractions in different orders. Compare the results of each problem for the different orders, each term as it changes posi- tion carrying with it the sign next before it : (1) 12+7-4+6 (3) (2) 7^+5-3* (4) 7. What conclusion do you draw from problems 4, 5, and 6? 28. The order of terms of an expression may be changed, without changing the value of the expression, provided each term as it changes position carries with it the sign next before it. i. Find the number of degrees in each angle of Fig. 42, writing the solu- tion as in the foregoing problem i. All the angular space about a point in a plane is divided into angles repre- sented by the following expressions; find x and each angle in degrees. With pro- tractor draw figures for 2, 3, and 4: 2. 2X, x, 4^+40, 1 80 $x 3- 4- 5. 6. 7. 7^+24, FIG. 42 38 First-Year Mathematics All the angular space about a point in a plane, on one side of a straight line, is divided into angles represented by the following expressions. Find x and each angle in degrees. Draw figures for 8, 9, and 10: 8. x, s#, 7* -2 9. gx, x, 37 2X, 5^-26 10. 8x, 4833:, 5# 22, 4# 14 11. 25+5#, 8#+8, 6x, 9^-2* 12. $x, 2(x+g), x, 42 x 13. 2x, 2(# + io), #-18, 3(36-*) 15. 2.8^+39.33, i.2 32.09, ^+7.16 16. 6.93^, 4.&2X, 1.27^ + 5.09, 138.91 9. 02.T. EXERCISE V Solve the following, doing all you can mentally: V2. 5^-6+^=24 \ 3. 8-^+3^=32 V4- 8/-IO-2/ = 5 > i5 2 6. 6^- 9 . ,,10. a; 4 12 3 " ~+5=9 18. 7a^ ax+a = ia. Adjacent Angles 29. The point at which the sides of an angle meet is called the vertex of the angle. Vertex is a Latin word meaning "turning- point." The Equation Applied to Angles 39 i. On tracing paper make a trace of Z.x (Fig. 43), and fit this trace on /_y. How do the angles compare in size? Does the size of an angle depend on the lengths of its sides ? Measure /_x and /_y with a protractor. FIG. 43 2. Read the vertex and the sides of ZA O B (Fig. 44); of ZBOC. 3. Do the angles of Fig. 44 have a side in common? A common vertex ? 30. Two angles that have the same vertex and a common side between them are adjacent angles. The sides which are not common are called the exterior sides. 1. Are Z& and Zc (Fig. 45) adjacent? Zc and Zd? Z&andZa? Z&andZd? 2. Read the exterior sides of Z# and Zy (Fig 44); the common side. FIG. 46 FIG. 45 31. The angles x and y (Fig. 46) may be added without measuring them, by placing them adjacent. . First- Year Mathematics i. Express by an equation that ZA O C (Fig. 47) equals the sum of two angles. FIG. 47 FIG. 48 2. If the angle y (Fig. 47) is turned over O B as a hinge, so as to make O C lie along the dotted line O C' (Fig. 48), express the value of angle A O C' in terms of Z.x and Z.y. 3. Draw two angles, a and b, and show how to find the sum without measuring them. Show how to find the differ- ence without measuring. 4. Show the sum and the difference of angles by folding or cutting paper. 32. With a protractor draw an angle of 60. Examine Figures 49 and 50 and construct on a line, A B, an angle of 60, like B O C, Fig. 51. FIG. 49 o FIG. 50 The Fquation Applied to Angles 41 1. With a protractor draw the following angles, marking on each angle the num- ber of degrees, as in / c Fig.5i:8o ; 9 2i;i7o . 2. With a protractor draw adjacent angles of 75 and 85; of 103^ and 57^; of 31^ and 2i. Check the work, A B first, by finding the sum FlG- 5I arithmetically, then, by measuring the sum with a protractor. 3. Draw adjacent angles of 56 and 124; of 19^ and 160^; of 92 and 88. With a ruler, or straight edge, see if the exterior sides of each pair of angles form a straight line. What term is applied to each angle-sum ? 4. Draw two straight lines cutting each other so as to make a pair of adjacent angles equal (with the aid of a pro- tractor), and show that the angles are right angles. 33. If two straight lines intersect (cut each other) making a pair of adjacent angles equal, the angles are right angles, and either line is perpendicular to the other. 34. An angle less than a right angle is called an acute (sharp) angle. An angle greater than a right angle and less than two right angles is called an obtuse (blunt) angle. 1. Draw two adjacent acute angles whose sum is a right angle. Point out two perpendicular lines. 2. Draw two adjacent obtuse angles whose sum is 3 right angles. 3. Draw two adjacent angles, one obtuse and the other acute, whose sum is two right angles. First-Year Mathematics 4. Draw a triangle of which all angles are acute. From the vertex of each angle draw a line at right angles to the opposite side. 5. Draw a triangle having an obtuse angle, and draw per- pendiculars as in problem 4. 6. Draw a triangle having a right angle, and draw per- pendiculars as in problem 4. 7. How many altitudes does a triangle have ? How many bases ? Supplementary Angles 35. Two angles whose sum is a straight angle (180) are supplementary angles. Either angle is said to be the supple- ment of the other. If the supplementary angles are adjacent they are called supplementary adjacent angles. 1. In Fig. 51 read the angle which is the supplement of ZBOC; of ZCOA. 2. Draw two supplementary adjacent angles. 3. On tracing paper make a trace of Z.x and of Z_y (Fig. 52) so that the angles on the tracing paper are adjacent. Show with a ruler, or straight edge, whether the angles are supplementary. In the same way show whether Z_a and Z6 are supplementary; /_c and Z.d. FIG. 52 4. Are 50 and 130 sup- plementary? 37 and 133? 60 and 120? 90 and 90? The Equation Applied to Angles 43 5. How many degrees are there in the supplement of an angle of 45? Of 120^? Of 90? Of a? Of 180-*? 6. Write the supplement of an angle of x degrees. V 7. Write the supplement of a; of 6; of 5 i9+^ 39- 4 3 x x 40. ,4. 87+-, - + 69 4i> 4 6 First- Year Mathematics EXERCISE VI Solve the following: r . 4. - -15+^=8 3 4 5- 6. 2(X- 7- 5( x 8.^!- 2 x+8 +8 = 12 3X-2 10. - -~i=f II. 14. ii +8 = 13 12. 9*---i= j i -~ 16. - _ . _ ,, Vertical Angles 36. Two angles having a common vertex, and having sides in the same straight line, but in opposite direc- tions, are called opposite or vertical angles (as x and z, Fig. 53). 1. Read both pairs of vertical angles in Fig- 53- 2. On tracing paper make a trace of angles 'y and z (Fig. 53). Put this trace on angles x and w and see whether Z.z coincides with (fits on) Z_x, and Z;y with /_w. How do the ver- tical angles compare in size? The Equation Applied to Angles 47 3. Test your conclusion in problem 2 by drawing two intersecting straight lines and measuring both pairs of vertical angles with a protractor. 4. Are Z* x and y (Fig. 53) adjacent ? Are they supple- mentary ? What is their sum ? 5. Are Z. s y and z (Fig. 53) supplementary adjacent ? What is their sum ? 6. From problems 4 and 5 show that x+y=y+z. 7. Show how to get the equation xz from the equation in problem 6. What axiom must be used ? 8. Using the equation y+x=x+w, show as in problems 4, 5, 6, and 7 that y=w. 37. The problems of section 36 show the truth of the following theorem or proposition: THEOREM. // two lines intersect, the -vertical angles are equal. 1. Show that the above theorem is true for any two inter- secting straight lines, no matter what angles they make. 2. State the three ways in which the truth of the foregoing theorem was shown in 36. By which of these ways was the truth of the theorem shown for any pair of vertical angles ? 3. Find the values of the four angles of Fig. 54. FI G- 54 First-Year Mathematics We may write Why? (i) Equation (i) may be written 7* + 3i*-3f* + 9 2: Subtracting 92 Multiplying by 3 Dividing by 22 7* + 3$* = 7f ' I2 + 3$- 12 = 92+42 = 134. 92-3!*= 92 3f 12 = 92-46= 46. The other two angles are also 134 and 46. Why? CHECK: 134+46 + 134+46 = 360. f 92 = 180. Why? (2) f 92 = 180. Why? (3) 7$*= 88. What axiom? (4) 223; =264. What axiom? (5) x= 12. What axiom? (6) 4. Find the four angles made by two intersecting straight lines, if two adjacent angles are 9^+41 and 5^29. Check results by drawing a figure. 5. Find x and the four angles 3#+37 and Since the given angles are vertical angles, made by two intersecting straight lines, if two vertical angles (Fig. 55) are Subtracting 7 Subtracting 3* Dividing by 2 5* 2^ 3^ + 30. 30. What axiom? What axiom? What axiom? (i) (2) (3) (4) 5*+ 7 = 5- 15+ 7 = 82. 3* + 37=3- i5 + 37= 82 - Each of the other two angles is 180-82 = 98. Why ? CHECK: 82 + 98 + 82 + 98 = 360. Find x and each of the following angles made by two intersecting straight lines. Draw figures for 6, 7, 16, and 17: The Equation Applied to Angles 49 VERTICAL PAIRS 6. ix +27 and 4^+87 7. 3^17 and # + 103 8. f*+i6 and "9. 2 T 4 r #-i3 and 10. $x+$x and 11. a:+|# 28 and x 12. $x+ and - 4 2 7* * J 2X I A 12. - --- and -- h o 46 3 x . x , x 14. -+F an d H l8 3 6 4 15 . _ an d ADJACENT PAIRS ,xx t , 3# x 16. -+-+172^ and * --- 52 10 4 x j 3 X , 17. x and - l-go 7 4 . 2X x , 18. -- h- and 154*- o o 19. 1-4# and 87 -- 4 3 20. 6%x-2$x and 4!^ 365 x , x 21. -+2# and -7+35 4 22. T V(26^-i43) and fx x\ I x 23- 2 ^ + 6/ /3C \ 24. 2 -+#) and 50 First-Year Mathematics EXERCISE VII Solve the following, doing all you can mentally. 14. 8Z -- = 2. 5# 6=2x 3 3. 4*+3=/ + i2 52 __ L = * 4. 35 2=25 + 7 8 16 2 5- 7:v-7 i = 6- *~ 7. 22*+i=3#+5 J 7- 4(*~7 8. 6(z-7)=z+8 * * 18. --- = 17-* 9- 7(3z-2) = 5z+2 3 5 10. *n. 3(^+i)=8(^-4) 12 20. ax+a= 12. -+4 5 21. z- 3 =5 4* , ^ , t 7^ 13. -- hio=3C+4 22. -^ + 0=^. 7 35 *Suggestion. After multiplying as indicated, subtract 3* from both sides of the equation, and add 32. It is sometimes convenient, in solving equations, to add or to subtract such a number or such numbers that the term containing the number whose value is required may stand alone on the right side. Complementary Angles 38. Two angles whose sum is a right angle are comple- mentary angles. Either angle is said to be the complement of the other. 1. What is the complement of /.a (Fig. 56)? Of Z&? 2. Draw two adjacent complementary an- _ gles. May either angle be obtuse ? Point out FIG. 56 two perpendicular lines. The Equation Applied to Angles 3. On tracing paper make a trace of /_x and of (Fig. 57) so that the angles on the tracing paper are adjacent. With a protractor show whether the angles are complementary. In the same way test whether Z# and /_b are complementary; /_c and 4. Show whether 22 and 68 are complementary; 43 and 46$; 89! and |. 5. What is the comple- ment of 60 ? Of 30? Of lof? Of 4 sf? Ofo? Of 90-*? 6. Write the complement of n degrees. 5 1 ^y FIG. 57 7. Write the complement of dP; of 3^; of; of x+y; of ; of 7 (a +6) degrees; of $x 2 degrees; of jy 3 degrees; 5 of $x 2 5;y 4 degrees. 8. If angles of 40 and d? are complementary, how many degrees does d stand for ? 9. If y + 7o=9o , what is the complement of y ? Why ? What is the value of y ? 10. In the equation, c +d=9o , what is the complement ofc? Ofd? Why? 11. State by equations that the following pairs of angles are complementary: ^ 2 First- Year Mathematics (i) y and 50 (2) 30 and z (3) w and * (4) a+3o and a-2o (5) 2* + 7 and 5* -2 (6) 3(^+7) degrees and 5(2^8) degrees (7) ^ X ~ I 5^ degrees and 26^+43! degrees. 12. If 1-55 is the complement of - , find n, h55, 2 342 , n n and . 3 4 We may write - + 55H = 90. Why? (i) Rearranging the terms of equation (i) ^+--- + 55 = 9- Why? (2) Combining the w-terms 7^ + 55 = 90- (3) Subtracting 55 ^= 35. Why? (4) Multiplying by 12 7^=420. Why? (5) Dividing by 7 n= 60. Why? (6) n 60 2 2 n n 60 60 3434 CHECK: 85 + 5 = 90. 13. x is the complement of .-v+48. Find the angles. 14. One of two complementary angles is 24 larger than the other. Find the angles. 15. One of two complementary angles is 28 smaller than the other. Find the angles. 1 6. The difference of two complementary angles is 83. Find them. 17. Find two complementary angles whose difference is 21; 36*; 73i; <*. The Equation Applied to Angles 53 1 8. The "difference between an angle and its complement is 27. Find the angle. 19. How many degrees are there in the angle, x, which is the complement of 4* ? Of 6x ? Of 5^ ? 20. How many degrees are there in an angle that is the complement of 3 times itself ? Of 7 times itself ? Of 6 times itself? Of 3J times itself? Of of itself? Of of itself? 21. Write in symbols the double of the angle y 17 added to 3 times the angle 31 subtracted from 5 times the angle 6 times the sum of the angle and 19 f of the sum of the angle and 22 5 times the angle minus 16. 22. Write in symbols (1) the complement of an angle y (2) 6 times the complement (3) 4 times the complement (4) 19 added to 4 times the complement (5) 17 subtracted from 4 times the complement (6) the complement of y increased by 12 (7) the complement diminished by 25 (8) the complement divided by 7 (9) one-fifth of the complement (10) 12 added to the complement (n) 13 added to one-fourth of the complement (12) 18 subtracted from f of the complement. 23. If an angle is doubled, and its complement is increased by 40, the sum of the angles obtained is 160. Find the complementary angles. 24. If an angle is trebled, and its complement is diminished by 40, the sum of the angles obtained is 130. Find the complementary angles. 54 First-Year Mathematics 25. The sum of an angle and of the complement is 75. Find the angle. 26. If an angle is increased by 15, and the complement is divided by 3, the sum of the angles obtained is 75. Find the complementary angles. 27. If 20 is added to 3 times an angle, y, and 6 is sub- tracted from of the complement of y, the sum of the angles obtained is 102. Find the complementary angles. The Sum of the Angles of a Triangle i. Measure the angles of a triangle with a protractor and find the sum. 2. Draw and cut out a triangle. Tear off the corners, and place them as in Fig. 58. What seems to be the sum of the three angles of the triangle ? Test with a ruler, or straight edge. 3. Draw two triangles different i n si ze ar >d shape from the triangle usec * m P r bl em 2 > an d find the sum of the angles as in prob- FIG. 58 lem 2. 4. Draw a triangle. From a vertex draw a line (B O, Fig. 59) at right angles to the opposite side. Cut out the triangle and fold so that the vertices come together at the foot of the per- pendicular (at O, Fig. 59). What seems to be the sum of the three angles of the triangle? 5. Draw two triangles different in size and shape from the triangle FIG. 59 The Equation Applied, to Angles 55 used in problem 4, and find the sum of the angles as in problem 4. 6. Draw a triangle. Place a pencil or stick in the position i (Fig. 60), noting the direction it is pointing. Rotate the pen- cil through the Z J x, y, and z, successively, as indicated in the figure. Through what part of a com- plete turn has the pencil rotated ? Through how many right angles ? How many degrees ? FIG. 60 7. State by an equation the number of degrees in the sum of Z J x, y, and z (Fig. 60). 39. If one side of the triangle ABC (Fig. 61) is prolonged at each vertex, the angles w, s, and / are called the exterior (outside) angles of the triangle ABC. / Angles x, y, and z are called the interior (inside) angles of the B^ $ triangle ABC. 40. If one angle of a triangle is a right angle, the triangle is called a right triangle. i. State by an equation the sum of the interior angles of FIG. 61 triangle ABC (Fig. 61). 2. How many degrees in x+iu (Fig. 61)? In y+s? In 3. Show that 56 First- Year Mathematics 4. From the equation in problem 3 show that The equation in problem 3 may be written *+y+z+w+s-M = 540. Why? (i) But *+y+z = i8o. Why? (2) If the left side of equation (2) is subtracted from the left side of equa- tion (i), and the right side of equation (2) from the right side of equation (i), the result is z6o. Why? (3) 5. Translate equation (3) of problem 4 into words. 41. We have considered the following theorems (proposi- tions) about the angles of triangles: THEOREM I. The sum oj the interior angles of a triangle is 180. THEOREM II. The sum of the exterior angles of a triangle, taking one at each vertex, is 360. i i. Prove Theorem II, 41, by rotating a pencil as indicated in Fig. 62. 2. The interior an- gles of a triangle are $x, x, 6x. Find their values. 3. Find the value ^ of each angle of a tri- angle, in degrees, if the first angle is twice the / second, and the third is FlG 6 2 three times the first. 4. Find the angles of a triangle if the first angle is 6 times he second, and the third is \ of the first. The Equation Applied to Angles 57 5. Solve a problem like 4, supposing the third angle to be J of the first; of the first; of the first; f of the first. 6. Find the angles of a triangle if the first is of the second, and the third is of the first. 7. Find the angles of a triangle if the first angle is 18 more than the second, and the third is 12 less than the second. 8. The difference between two angles of a triangle is 20, and the third angle is 36. Find the unknown angles. 9. Find the angles of a triangle if the first is 25 more than the second, and the third is 3 times the first. 10. Find the angles of a triangle if the first angle is double the second, and the third is 3 times the first, less 9. 11. Find the angles of a triangle if the first is J of the second, and the third is \ of the first, plus 18. 12. Find the angles of a triangle if the first is 3^ times the second, minus 8, and the third is of the second. 13. Find the angles of a triangle if the first is 6 times the second, plus 18, and the third is J of the first, minus 7. 14. If one angle of a triangle is a right angle, what is the sum of the other two angles ? What are such angles called ? 15. How many angles of a triangle may be right angles ? Obtuse angles ? Acute angles ? 1 6. Find the values of the acute angles of a right triangle if one angle is (i) 3 times the other (2) 5 times the other (3) of the other (4) i of the other (5) 6 more than 7 times the other (6) \ of the other, diminished by 33. n n 17. The acute angles of a right triangle are -and -. Find the values of n and of the acute angles. First- Year Mathematics 1 8. The acute angles of a right triangle are equal. Find them. 19. Find the unknown interior angles of the triangles of figures 63, 64, 65, 66. (Suggestion: Use the theorems on vertical angles, and on the angle-sum of a triangle.) 20. The three interior angles of a triangle are equal. Find them. 21. The three exterior angles of a triangle are equal. Find the value of each exterior and interior angle. EXERCISE VIII Solve the following: 1. 6# 2=26 2. 9# S=x ,3. 52+6 = 11 4. 2(2-3) =4 5- 3(y+2)=S 6. 2-= The Equation Applied to Angles 59 7. 6r 5=4 r +7 > 6a i 18. \-a=4 2a r r $ 8. -+-=5 2 3 9- f ~p7 10. i-| = i / O T r^'V* /T O -V O/T * JL UvV u- ^v "*(* 11. ^ -Sl-8 ^ 22. S(x-a)=S+4a a T,a 2 2 2X + 21 . 13. 4X 19 = rt 24. x 2a+^ = i 4 14. 2 *-i= l6 8 2 72 Z 16. 72-8=62+- 27. 5 8 ~\a 17. 5<* + i= ^-+5 28. SUMMARY 1. An angle is formed by a line rotating in a plane about a point called the vertex. An angle may also be formed by two intersecting lines. 2. A straight angle is the angle made by a half-turn of a rotating line. 3. A right angle is half of a straight angle, or the angle made by a quarter-turn of a rotating line. 4. An acute angle is less than a right angle. An obtuse angle is greater than a right angle and less than a straight angle. 60 First-Year Mathematics 5. A degree is one-nintieth of a right angle. 6. Two angles are adjacent if they have the same vertex, and a common side between them. 7. If two straight lines intersect so as to make the adjacent angles eq-ual, the angles are right angles, and either line is perpendicular to the other. 8. Angles may be added and subtracted without measuring. 9. The sum of the angles having a common vertex, and just covering the plane about the vertex, is 360, or four right angles. 10. The sum of the angles having a common vertex, and just covering the part of the plane on one side of a straight line is 180, or two right angles. 11. Two angles are supplementary when their sum is two right angles, or 180. If the supplementary angles are adjacent they are called supplementary adjacent angles. 12. The exterior sides of supplementary adjacent angles lie in the same straight line. *3- Opposite or vertical angles are angles that have a common vertex, and have sides in the same straight lines, but in opposite directions. 14. Two angles are complementary when their sum is a right angle, or 90. 15. The truth of the following theorems has been shown: THEOREM I. // two straight lines intersect, the opposite or vertical angles are equal. THEOREM II. The sum of the interior angles of a triangle is 180, or two right angles. THEOREM III. The sum of the exterior angles of a triangle, taking one at each vertex, is 360, or jour right angles. The Equation Applied, to Angles 61 1 6. A triangle having a right angle is a right triangle. 17. At least two angles of a triangle must be acute. 1 8. The acute angles of a right triangle are complementary. 19. The equation may be used to express the relations between angles, and to solve problems on angles. 20. The order of terms of a polynomial may be changed, without changing the value of the polynomial, provided each term as it changes position carries with it the sign next before it. CHAPTER IV POSITIVE AND NEGATIVE NUMBERS Uses of Positive and Negative Numbers 42. The plus (+) and minus ( ) signs are used to denote numbers supposed to be used in certain directions, called directed numbers. 1. The top of the mercury column of a thermometer stands at o at the beginning of an hour. The next hour it rises 5 and the next 3. What does the thermometer read ? 2. If the mercury stands at o, and rises 8, then falls 5, what does the thermometer read ? 3. Denoting a rise of 10, or of x on the thermometer by R 10, or by R x, and a fall of 10, or of x, by F 10, or F *, give the readings of the thermometer after the following changes, if the top of the column reads o at the start : (1) R 8 followed by R 5 (4) R 13 followed by F 18 (2) R 12 " by F 9 (5) F . 5 " by R * (3) R 16 " by F 12 (6) R a " by F 6. 4. If the change in the mercury column is a rise, a positive or plus (+) sign will be written before the number that de- notes the amount of the change. If the change is a fall, a negative or minus ( ) sign will be written before the number. If the reading at the start is o, give the readings after these changes: (1) +10 followed by + 2 (5) +x followed by +;y (2) +10 " by - 2 (6) +a " by -x (3) +20 " by -18 (7) +a " by -a (4) +9 " by -12 (8) -a " by -x. 62 Positive and Negative Numbers 5. On a winter day the thermometer was read at 9 A. M. and every hour afterward until 5 o'clock. The hourly read- ings were -5, o, +2, +8, +10, +10, +5, o, -5. 1AM FIG. 67 On squared paper the readings were marked off from hour to hour, calling one vertical space 5. The points were con- nected as shown. How did the mercury change from 9 to 10 o'clock? From 10 to n o'clock? From n to 12? From 12 to i ? From i to 2 ? From 2 to 3 ? From 3 to 4 ? From 4 to 5? Tell for each hour whether the change was a rise, or a fall. 6. Draw a line to show the following hourly readings, be- ginning at 8 A. M. Indicate the hourly changes in amount by figures, and in direction by the + and signs : +2, -2, -4, -2, +2, +4, +4, +4, +8, +10. 7. The average monthly temperatures for a northern town are Jan. - 4 May +42 Sept. +48 Feb. - 7 June +52 Oct. +37 Mar. +14 July +62 Nov. +25 April +26 Aug. +60 Dec. + 2. Using a convenient scale, draw the temperature-line. 64 First-Year Mathematics 43. Drawing temperature-lines is called graphing, or plotting. 1. The daily average temperatures for 14 days at a certain place were +8, o, -10, +12, -6, +14, +15, + 2 , -5, +15, +20, o, o, +10. Graph these readings to ITconvenient scale. 2. A bicyclist starts from a point and rides r8 miles due northward (+18 mi.) then 10 mi. due southward (10 mi.); how far is he then from the starting-point ? 3. State how far and in what direction from the starting- point a bicyclist would be after rides indicated by each of these pairs of records: (1) +10 mi. then 8 mi. (3) +100 mi. then +50 mi. (2) 20 mi. " +20 mi. (4) + a mi. " + b mi. 4. How far and in what direction from the starting-point is a traveler who goes eastward (+) or westward ( ) as shown by these pairs of numbers: (1) +16 mi. then 6 mi. (4) +a mi. then +c mi. (2) 18 mi. " +28 mi. (5) -\-m mi. " n mi. (3) ~ m nu- " +3 roi- (6) m mi. " + mi. 5. A car in the middle of a moving train is drawn forward with a force of 8 tons and at the same time it is pulled back- ward with a force of 7$ tons. The two forces together are equal to what single force ? 6. Denoting a forward pulling force by F and a backward by B, give amount and direction of a single force equal to each of these pairs of forces: (1) F 14 oz. with B 6 oz. (3) F 25 tons with B 15 tons (2) F 20 Ib. " B 12 Ib. (4) B 25 tons " F 40 tons. 7. Denoting forward-pulling forces by the positive or plus (+) sign and back-pulling forces by the negative or minus Positive and Negative Numbers 65 ( ) sign, give the single force which is equal to each of these pairs of forces: (1) +20 and 12 (4) 15 and + 8 (7) +a and +b (2) +20 " -20 (5) -12 " -12 (8) +a " -b (3) -15 " - 8 (6) + x " -12 (9) -a -b. 8. A toy balloon pulls upward with a force of 9 oz. If a weight of 6 oz. is attached, will the balloon rise or fall ? With what force ? 9. Call upward forces positive, or plus (+), and down- ward forces negative, or minus ( ). State what single force will have the same effect as these pairs: (1) +17 Ib. and 7 Ib. (4) 23 Ib. and +10 Ib. (2) +17 Ib. " -10 Ib. (5) + xlb. " + ylb. (3) -23 Ib. " -10 Ib. (6) + xlb. " -#lb. 10. Denoting motion northward by the positive or plus (+) sign and motion southward by the negative or minus ( ) sign, and supposing a ship to start on the equator and sail as indi- cated, tell the latitude of the ship in both amount and sign for each pair of sailings: (1) +28 then + 2 (5) +*then -10 (2) + 2 " -18 (6) -x -10 (3) +12 " -12 (7) +x " - f (4) +12 " -24 (8) -x " - y. n. A boy starts work with no money. He earns 50^ (+50^) and spends 40^ (40^). How much money has he then? 12. If a man's debts be indicated by writing D before their amount and his possessions (assets) by P before their amount, what is the condition of a man's affairs if his debts and possessions are indicated by P $1,200 and D $1,000? by P $73 and D $50? D $75 and P $60? D $300 and P $1,000? 66 First-Year Mathematics 13. If water pushes (buoys) a floating body upward with a force of 18 lb., and the body's weight pulls it downward with a force of 10 lb., the two forces together equal what single force ? 14. If a man was born 40 B. c. and died 45 A. D., how old was he when he died ? 15. Denoting a date A. D. by + and B. c. by , give the length of time between these pairs of dates : (1) - 5 to +10 (3) - 52 to -50 (5) - 150 to + 150 (2) + i6to +86 (4) loo to +50 (6) +1600 to +1900. 16. Virgil was born 70 and died 19; how old was he at death ? 17. The first Punic War lasted from 264 to -241; how long did it last ? 18. Egypt was a Roman Province from 30 to +616; how many years was this ? 19. Augustus was Emperor of Rome from 50 to +14; how many years was he Emperor ? 20. What will denote the distance and direction from your school house to your home, if the distance and direction from your home to your school house are denoted by +60 rd. ? + imi. ? +# rd. ? 80 rd. ? ij mi. ? ami.? 21. While a freight train is moving at the rate of 10 mi. an hour toward the south ( + 10 mi. an hr.) a brakeman walks along the top of the cars toward the north at the rate of 4 mi. an hour (4 mi. an hr.). How fast and in what direction does the brakeman move over the ground ? Answer with the aid of the plus, or minus, sign. 22. The conductor of a passenger train walks from the front toward the rear of the train at the rate of 3 mi. an hour while the train is running at the rate of 12 mi. an hour. How Positive and Negative Numbers 67 fast does the conductor move over the ground ? Answer with the aid of the (+), or ( ), sign, supposing that + means toward the north and first, that the train is running north; then second, that the train is running south. 23. What does the sign ( ) denote if the sign (+) de- notes: (i) above? (2) forward? (3) upward? (4) to the right? (5) after? (6) east? (7) north? (8) possessions ? 44. The foregoing problems show the need for distinguish- ing numbers 0} opposite nature. Positive and negative signs afford a convenient means of making this distinction. 45. A number that denotes both magnitude and direction, or both size and quality, is called a directed number, as 16 units east, or +16, 10 units west, or 10, ax units downward, or ax. If the direction or quality is denoted by the sign (+) or ( ), the number is called a signed number. Either of the two opposite directions, or qualities, may be denoted by the plus (+) sign, whereupon the opposite direction, or quality, is denoted by the minus ( ) sign. 46. The plus and minus signs are also used, as in arith- metic, to denote addition and subtraction. 47. It is sometimes desirable to indicate definitely that the sign is to denote direction, or quality, when used with a num- ber. This is done by inclosing the number together with the sign of quality in a parenthesis, thus, (+6), (8), For example, the expression, 6 + ( + 2), means that +2 is to be added to 6; the expression, 7 (+4), means that +4 is to be subtracted from 7. Give the meaning of the following: 1. 6-(+3) 3- -5 + C + 7) 5- + (-&) 2. 8 + (-4) 4- -6-(-9) 6. m-(-x). First-Year Mathematics 48. The positive sign (+) need not always be written. It is generally omitted from the first number in an expression. The negative sign is never omitted. An expression like +xa, where the first number, x, is positive, would com- monly be written x a, and is read "x minus a." The posi- tive sign, when omitted, is said to be "understood." Graphing Data i. Using a convenient scale, and calling the verticals age- lines, graph these average heights of boys and girls: Girls 4. 5 ft. 4.8 5-2 5-3 5-4- Age Boys Girls Age Boys 2 yr. i. 6 ft. i. 6 ft. 12 yr. 4.8ft. 4 2.6 2.6 14 5.2 6 3.0 3.0 16 5.5 8' 3-5 3-5 18 5-6 10 4.0 3.9 20 5.7 At what age do boys grow most rapidly ? Girls ? H 4 2 j u u + 1 ' I a 1 I 3 ' / j i u a 3 / y f // x^^ 8 6 4 2 2J / / _^ FIG. 68 2. The populations, in millions, of the United States for each 10 years beginning 1790, are 3.9, 5.3, 7.2, 12.9, 17.1, 23.2, 31.4,38.6, 50.2, 62.6, 76.3. Positive and Negative Numbers 69 Graph these numbers, drawing a smooth, free-hand curve through the points and predict the population for 1910. 3. The standings of the champion batters from 1900-1907, inclusive, are here given in percents, for The National League 38.4 38.2 36.7 35.5 34.9 37.7 33.9 35. The American League' 38.7 42.2 37.6 35.5 38.1 32.9 35.8 35 Graph these percents for each league to a convenient scale, both on the same sheet. Tell what the percent-line shows. 4. The monthly average rainfall or snowfall, in inches, at a certain place for 30 years is as follows: Jan. 2.8 May 3.59 Sept. 2.91 Feb. 2.30 June 3.79 Oct. 2.63 March 2. 56 July 3.61 Nov. 2.66 April 2.70 Aug. 2.83 Dec. 2.71. Graph these data and tell what the connecting line shows. 5. Graph these average lengths of day from sunrise to sunset in latitude 42. Hr. Hr. Hr. Jan. 16 9-5 May 1 6 14-5 Sept. 15 12-5 Feb. 15 10-5 June 15 i5-o Oct. 16 II. 2 March 16 II.9 July 1 6 14.9 Nov. 15 9 .6 April 15 13-3 Aug. 1 6 13-9 Dec. 16 9.1, 6. Using the same sheet, scale, and dates as in problem 5, graph the average day's lengths 14.4 14.9 14.9 15.6 What differences in the change of the day's length in dif- ferent latitudes do the three graphs show ? in latitude 38: 9 7 10 .8 12 .0 13 -3 M .6 13 7 12 5 ii .2 IO 5 9 -5 in latitude 45: 9 .1 10 4 II 9 I 3' 5 15 3 14 .1 12 .6 II . I 9 .6 8 8. 7 First-Year Mathematics 7. A ship's latitude from week to week was +4 2 , +3 8 > +30, +20, +12, +2, -i, -6, -3, +12. Graph these latitudes and tell when the ship crossed the equator. Graphing Precise Laws i. Rectangles are 3 inches wide and their lengths are, 4" S", 6", 7", 8", 9", 10", n", 12", 13". Calculate the areas of these rectangles and plot them to a convenient scale. How does the graph show the area to vary with the base ? * 4 S 6 7 * 9 10 11 12 U 14 FIG. 69 2. Write the equation for the area, y, of a rectangle having an altitude 3 and a base x. 3. The base of a rectangle is 4 and the altitude is x. Write an equation showing what the area, y, must be. Positive and Negative Numbers 71 4. In the equation of problem 3, let x have the values in the first line just below, and calculate the values of y. x=i, 2, 3, 4, 5, 6, 7, 8, 9, 10 y=4, etc. Graph these values of x and y and state how the area, y, varies with the altitude, if the base is 4, a fixed number. 5. The side of a square is x. Write an equation showing how to find its area, y. 6. In the equation of problem 5, suppose x to have the successive values of the first line shown here, and calculate the areas, y. * = i,2, 3, 4, 5,6, 7 y=i, 4, etc. 7. Graph the values of y (problem 6) on the i, 2, 3, 4, etc., lines and draw, free-hand, a smooth curve through the points. 8. The altitude of a triangle is 4 and the base is x. Write an equation to show its area, y. 9. Plot the values of y (problem 8) for the values of x=i, 2 > 3> 4> 5> etc -> an( i tell how the area of a triangle depends on its base if the altitude is a fixed number. 10. Write an equation to show the area, y, of a triangle having a base 6, and an altitude x. 11. Graph the values of y (problem 10) for x = i, 2, 3, 4, 5, 6, etc., and show how the area of a triangle depends on its altitude, if its base is a fixed number. 49. We have graphed the laws y=$x, y=4x, y=x 2 , and yzx. Other equations may be graphed just as these laws were. Graph the following equations: 1. y=x 4. y=2x i 7. y=x 3 +2 2. y=x+i 5. y=2x + i 8. y=$x 3. y=x i 6. y = 2#+3 9. y = $x 6. 72 First- Year Mathematics 50. It is not necessary to have the whole equation to make the graph. Only the second member is necessary. 1. Graph #+3. This means take for x the successive values, i, 2, 3, 4, 5, etc., cal- culate the values of # + 3; viz., 4, 5, 6, 7, 8, etc., and plot as before. 2. Graph the following expressions: (1) 2X (4) 2#+I (7) 2X 1 (2) x+2 (5) x+i (8) x' (3) x-2 (6) x-i (9) 4*+2. Adding Positive and Negative Numbers 1. Denoting distances traveled northward by positive num- bers, and distances traveled southward by negative numbers, find for each of the following cases the distances and the direc- tion of the stopping-point from the starting-point. When the stopping-point is north of the starting-point mark the result + ; when south, mark the result . An automobile goes: (1) +15 mi., then 10 mi. (6) 12 mi., then 10 mi. (2) +15 mi., " 14 mi. (7) 20 mi., " +15 mi. (3) +15 mi., " 20 mi. (8) 15 mi., " +22 mi. (4) +25 mi., -35 mi. (9) -20 mi., " +21 mi. (5) 18 mi., " +24 mi. (10) 22 mi., " +22 mi. 2. In the following problems the numbers indicate distances traveled northward, if negative; and southward, if positive. The sum in all cases must denote the distance and direction of the stopping-point from the starting-point. Write the sums with their proper signs: 00 00 (3) (4) (5) (6) (7) (8) + 15 -15 +15 -15 +38 -38 +38 -38 + 8 8 8 + 8 +19 19 19 +19 (9) (10) (n) (12) (13) + 4 - 4 +n 4 +12 + 26 26 26 +26 12. Positive and Negative Numbers 73 3. Examine (i), (2), (5), (6), (10) of problem 2 and make a rule for adding two numbers having like signs. 4. From (3), (4), (8), (n), (12), and (13) of problem 2 make a rule for adding two numbers having unlike signs. 51. Sums, with their proper signs, of positive and negative numbers, are called algebraic sums. The sums and differ- ences of numbers regardless of sign, are called arithmetical sums and differences. 52. The algebraic sum of two numbers with like signs is their arithmetical sum, with the common sign prefixed. 53. The algebraic sum of two numbers with unlike signs is their arithmetical difference, with the sign of the larger num- ber prefixed. 1. In the following problems the positive numbers indicate gains and the negative numbers indicate losses. The sums indicate the net change in the man's capital, and whether the net change is an increase or a decrease. Find the sums and tell their meaning: (i) (2) (3) (4) (5) (6) (7) + 5 +35 -45 +75 2 3 6 + 8 * 14 +25 38 20 +13 +780 6x 460 18 +24 +60 86 95 4X +770 I_? ~ 15 +55 +_8_ + 45 +7* - 5 a - 2. State a way of adding any number of positive and negative numbers. 3. A force of 12 Ib. pulling toward the right ( + 12 Ib.) together with a force of 9 Ib. pulling toward the left give a combined pull equal to what force ? 74 rirst-Year Mathetmtics 4. What single force has the same effect in pulling the ring R as the following pairs of forces acting together ? D FIG. 70 (1) + 12 Ib. and - 8 Ib. (9) -16 Ib. and - 8 Ib. (2) -12 Ib. " + 81b. (10) + 3lb. " -12 Ib. (3) -10 Ib. " +10 Ib. (n) + x\b. " + ylb. (4) -i61b. " +13 Ib. (i2)+*lb. " -ylb. (5) +14 Ib. " -17 Ib. (13) - xlb. " + ylb. (6) + 9 Ib. " -20 Ib. (14) - x Ib. " - y Ib. (7)+nlb. " +15 Ib. (is)+#lb. " -*lb. (8) -i61b. " +12 Ib. (16) -sxlb. " + xlb. 5. A man draws a pail of brick, weighing 60 Ib., to a house-top by pulling on a rope which runs over a pulley, with a force of 65 Ib. What single force equals the sum of the two forces acting on the handle of the pail ? 6. A balloon pulls upward on a stone, weighing 6 oz., with a force of 8 oz. What is the sum of the forces ? 7. A piece of iron weighing 18 Ib., when placed under water, is pushed (buoyed) upward with a force of 2 A pounds. What is the sum (combined effect) of the two forces together ? 8. An elevator starts at a certain floor, goes up 65 ft., down 91 ft., up 52 ft., down 13 ft., and up 65 ft., and stops. How far and in what direction is the stopping-point from the starting-point? Give your answer in the form of an alge- braic sum. Positive and Negative Numbers 75 9. A vessel starting in latitude +20 sails +13 in lati- tude, then 60, then +40, then 10. What is its latitude after the sailings ? What is the latitude of a ship starting in latitude 50 after these changes of latitude: +10, 5, + 18, -7, +38, -12, +60 ? 10. A boatman rows, at a rate that would carry him 3 miles an hour through still water, down a river whose current is 2 mi. an hour. What is his rate per hour? What is his rate per hour, if he rows up the river ? EXERCISE IX Add the following, doing all you can orally: 1. +f 7- 6| 13. 13^ 19. +S2b 2 c +f +8f[- +23! 2&b 2 c 2. % 8. + 7$- 14. 6.69 20. +f -7f +8.04 3- +T6- 9- ~ I2 i I 5- + 8.95 21. +S(xa) 11.25 6(x a) 4- +3i I0 - -"I'Sf 16. i6r 22. i2(x+y) + iSr - 71 5. 5^ n. 2.12 17. 3.25 23. 6S(mr) -2f -1.88 -6.85 -75(m-r) 6. +4^ 12. +3.16 18. +7f# 24. 6%(c d) -6\ -4.08 Snbtracting Positive and Negative Numbers i. In this problem positive numbers indicate the readings above zero and negative numbers, readings below zero. The difference means the number of degrees the top of the mercury column, must rise, or fall, to change from the second reading -j 6 First-Year Mathematics to the first. If the change is a rise, mark it positive ( + ), if a jail, mark it negative ( ). (i) (2) (3) (4) (5) First reading: +68 -98 -30 - 7 + i Second reading: +42 -18 +65 +32 -28 (6) (7) (8) (9) (10) First reading: -8 +6x -40 - 2y Second reading: + 78 +8 -3* -90 +6oy Define minuend. Define subtrahend. 2. By sections 52 and 53 find the sums in the following problems and compare the exercises and your results with those of the like numbered exercises of problem i : (i) (2) (3) (4) ( 5 ) (6) (7) (8) (9) (10) +68 98 30 -7+1 08 +6x 40 - 2y 42 +18 65 32 +28 78 8 +3* +ga 6oy 3. Show by comparing the problems of i and 2 that the difference of any two numbers can be found by changing the sign of the subtrahend and then adding. 4. Find the differences of the following: (i) ( 2 ) (3) (4) (5) (6) (7) + 19 -60 -75 + 8 30 + 18* i2y 10 25 +25 16 20. 6x 7y (8) (9) (10) (n) +a +3# 18 + a+2b c 120+33; b +2X+ 6 +30 5&+3C + >ja2x 54. To find the difference of two numbers change the sign of the subtrahend mentally, then add. Positive and Negative Numbers 77 EXERCISE X Subtract the lower from the upper number of the following, doing all you can mentally: I. +i 9- + i2 17. 9(c s) +1 16$ +3(^5) 2. -1 10. 11^- 18. -\-ac 4 d 3 s -*! -f- 8|- ac*d 3 s 3- + ii. 6av 2 19. +6.7502 -* + a?; 2 7.250^ 4- H 12. -i 3 bc 3 20. - 3 .i6c 2 ^ -i + Sbc 3 o.8c)c 2 d 5- -A 13- gcxy 21. 4.76(/+s) + i -15^3; +9.67(^+5) 6. +4* 14. + jab 2 c 22. +o.82(a 2 ^ 2 ) -51 gab 2 c ~3-T5( a *~ x2 } 7- -8| 15- 8m 3 r 2 s 23. o.75(c+J 2 ) + 2^ +3W 3 f 2 ^ o.qo(c+d 2 ) 8. -9t 16. +6(^+z) 24. +o.9i6c6 2 c 3 -3i 7(#+z) i.8o3fl6 2 c 3 Multiplying Positive and Negative Numbers Suppose the short spaces on the line east-west, E W, repre- sent a mile. 1. Show what space starting from o in each case repre- sents + 2 mi.; +3 mi.; +5 mi.; +8 mi.; +10 mi. 1 , . . vr E FIG. 71 2. Starting again from o, show the space that represents i mi.; 2 mi.; 5 mi.; 7 mi.; 10 mi. 78 First-Year Mathematics 3. Show the spaces from o a man goes if he travels +2 mi. a day for rday; 2 days; 3 days; 5 days. 4. What is 2 times 2 mi. ? 3 times 2 mi. ? 4 times 2 mi.? 5 times 2 mi.? 5. What is 5 times 4 mi. ? 10 mi. ? 20 mi. ? -25 mi.? loo mi:? 6. What is 1 6 times 2 mi. ? 5 mi. ? 10 mi. ? 20 mi. ? loo mi. ? 7. What is 5oX(-2)? 25X(-io)? 4oX( 12)? 8X(-i2o)? 4X(-a)? ioX(-*)? 8. The value of a man's property changes by +$1000 a year. How much does it change in 2 yr. ? 5 yr. ? 8 yr. ? 10 yr. ? In each case tell whether the change is an increase, or a decrease. 9. If the value of a man's property changes by $500 a year, how much does it change in 3 yr. ? 5 yr. ? 7 yr. ? 8 yr. ? 10 yr. ? 12 yr. ? In each case tell whether the change is an increase, or a decrease. 10. How much and in what way does a man's property change in 12 yr. at the rate of +$50 a yr. ? $100 a yr. ? +$400 a yr. ? +$1,800 a yr. ? $2,000 a year ? 11. How much and in what direction does the height of a mercury column of a thermometer change in 6 hours at the rate of +10 an hr. ? -8 an hr. ? -7 an hr. ? +5^ an hr - ? ~3i an hr. ? +2^ an hr. ? a an hour ? 12. State a way of multiplying a negative number by an arithmetical number. 55. A light bar (Fig. 72) supplied with equally spaced pegs is balanced about its middle point, M. With a number of equal weights ; w, the following experiments are performed. Positive and Negative Numbers 79 1. Hang a weight of 2W on the peg l x . This weight tends to turn the left end of the bar downward. How much weight must be attached to the ,/,. .....,..,,..,,,,, hook, H, to balance this turning-tendency ? Now hang the same weight, 2w, on peg 1 2 , and measure its downward turning-ten- dency by attaching to the , u hook, H, a weight sufficient to balance the bar. 2. In a similar manner find the turning-tendency caused by the weight 2w on the peg 1 3 ; on 1 4 ; on l s . 3. Using a weight of $w, find the turning-ten- dency when placed on peg FlG - 7 2 U; on 1 2 ; on 1 3 ; on 1 4 ; on l s . 4. Perform experiment 3, using a weight qw. 56. Experiments i, 2, 3, and 4 show that when a weight (as 2w) is hung on peg l s , the turning-tendency caused by this weight (2w) is five times as great as when the same weight is hung on peg 1 T ; on 1 4 its turning-tendency is four times as great as on l x ; on 1 3 it is three times, and on 1 2 it is two times as great as on 1,. 57. The same facts hold when $w, ^w, or any other weight is used. In other words, with an apparatus like that in Fig. 72, the turning-tendency, or leverage, caused by a weight is measured by the product of the weight by the distance from the turning-point, M, to the peg where the weight hangs. go First-Year Mathematics 1. What is the turning-tendency caused by a weight of jw hung on peg 1 5 ? on 1 3 ? a weight of gw on 1 4 ? on l x ? 2. If the bar had 12 pegs on each side of M, what would be the turning-tendency caused by a weight of 8w on peg 1 I2 ? TW on 1 9 ? w on 1 XI ? 3. Attach the cord, C (Fig. 72), to peg r r and perform the experiments of 55 on the right side of the bar. 58. Experiment 3, 57, shows that the same facts are true on the right side as on the left, but the bar turns, or tends to turn, in the opposite direction. To avoid confusion, some simple method of distinguishing between these two directions of turning is desired. Suppose a watch laid upon the page of the book with its face up. When the bar turns, or tends to turn with the hands of the watch, the turning-tendency will be called negative and designated ; if it turns, or tends to turn, against (oppo- site to) the watch-hands, the turning-tendency will be called positive and designated +. 59. In all the experiments thus far performed the weights which caused the bar to turn were downward-pulling weights, or forces. By arranging an apparatus as in Fig. 73, p. 82, forces can also be made to pull upward. Downward-pulling weights or forces will be designated by , and upward-pulling forces by +. 60. The distance from the turning-point, M, to the peg where the weight, or force, acts will be called the lever-arm, or arm, of the force. Lever-arms measured from the turning- point toward the right will be marked +; those toward the left, -. For example, if the distance from M to peg r x be repre- sented by +i, then the distance from M to the peg r 4 will be represented by +4; from M to 1 3 , by 3, and so on. Positive and Negative Numbers 81 61. If a force of 2 acts on peg 1 3 , its turning-tendency, or leverage, is the product of 2 and 3. Since the bar tends to turn against the watch- hands, the turning-tendency is written +6. If T represents the turning-tendency, then If a force of 2 acts on r 3 , its turning-tendency is the product of 2 and +3, or 6, since the bar tends to turn with the watch-hands. In this case i. In the following experiments write (as in 61) the leverages, or turning-tendencies, for the forces and arms in- dicated : Forces Arms Leverages Forces Arms Leverages I ~3 2 VII -6 + I II 4 + 2 VIII -8 I III -7 -3 IX a + b IV -6 -4 X m z V i + 5 XI yiv 2X VI -8 +9 XII -6c I2y 2. If loadings I and VII in the foregoing tables were on the apparatus at the same time, would the bar balance or turn ? If it turns, in what direction would it turn ? Answer similar questions for II and VIII; V and VII; VI and VIII. 62. By the apparatus in Fig. 73 (p. 82) forces can be made to pull upward on either side of the bar. First-Year Mathematics If a force of 2 pulls upward on peg r 4 , its turning-tendency is the product of +2 and +4- Since the bar tends to turn against the watch-hands, the ,, ,..,,, turning-tendency is written +8. Using T for the turning- tendency, r=(+ 2 )(+ 4 ) = +8. On peg 1 4 the turning- ^ t tendency is the prcduct of ' +2 and 4, or 8, since the bar tends to turn with the watch hands. In this case r=(+2)(- 4 )=-8. i. In the following experi- ments write the leverages for the indicated arms and forces : FIG. 73 Forces Arms Leverages Forces Arms Leverages I +3 2 V +6 + I II +7 -3 VI + m + Id III + i +3 VII +3 2 IV +8 +9 VIII +6w I2Z 2. If III and VII in the foregoing tables are on the appara- tus at the same time, will the bar balance or turn ? If it turns, in which direction will it turn ? Answer similar questions foi landV; II and VII; IV and VII. 3. Write the leverages for the following loadings: Positive and Negative Numbers 8.3 j Forces Arms Forces Arms I +8 -9 VII +7 +8 II +8 +9 VIII -7 +8 III -8 +9 IX +7 -8 IV -8 -9 X -7 -8 V + x W XI a +66 VI +4X +3? XII -$ -2b 4. What sign (direction) has the turning-tendency of a plus force with a plus arm ? A plus force with a minus. arm ? A minus force with a plus arm ? A minus force with a minus arm? 63. If 3 (4) means that four is to be measured, or laid -12 -8 -4 FIG. 74 off, 3 times from o in the negative direction (the direction of 4), what is the value of the product, 3 (4), or, what is the same thing, (+3)(~4)? The product means the distance and direction from o, evidently 12. 1. Show on a figure the meaning and value of the follow- ing products: (1) a(+3) (3) 3(-2) (5) 3(-5) (7) *(+6) (2) a(-3) (4) 3( + 2) (6) i(+5) (8) a(-6). 2. Show on a figure the products: ( 3)(+4); ( 3)( 4-)- ( 3) ( + 4) means that four is to be laid off 3 times in the direction opposite to the direction of +4, i. e., in the negative direction. ( 3) ( 4) means that four is to be laid off 3 times in the direction opposite to the direction of 4, i. e., in the positive direction. 84 First-Year Mathematics 3. Interpret these products on the same principle: (1) (-2)(-3) (4) ( + 2)(+8) (7) (- (2) (+3)(-2) (5) (-3X-5) (8) (-O(-S) (3) (~2)(+4) (6) (+3)(-5) (9) (-3X+6). 4. Show on Fig. 75 the value of 2 times +a; 2( a), -4a -3a -2a -a a 2a 3a 4a FIG. 75 5. Interpret the meanings of: (0 (+3)(+) (5) (+)(+&) (9) (2) (+ 3 )(-a) (6) (+a)(-6) (10) (3) (~3)(+*) (7) (-)(+&) ( JI ) (4) (-3)(-*) () (- fl )(~ & ) ( I2 ) SUMMARY In digits: In letters: 6. Examine the eight products of the summary and make a rule for obtaining the algebraic sign of a product of two numbers from the signs of the factors. Compare your rule with 64. Law of Signs for Multiplication: // two factors have like signs, the product is positive; ij two factors have unlike signs, the product is negative. EXERCISE XI Find the products of the following, doing all you can men- tally: Positive and Negative Numbers 85 3- (+*)(-!) 18. (- 4. (-!)(+!) 19- (-^c}(-*} 5- (-JX-A) 20. (+7i*)(-7bO 6. (+T 7 r)(-f) 21. (+ 5 )(-^ 2 ) 7- (-6f)(-6f) 22. (- 7 )(-r/) 8. (+6f)(-6f) 23. ( 9- (-6f)(+6f) 24. ( 10. (+6f)( + 6f) 25. ( 12 - (-3)(- fl;X; ) 2 7- 13- (~7)(+) 28. 14. (-a)(+#?) 29. IS- (-0(-^) 30. Dividing Positive and Negative Numbers 65. The product of two factors and either factor being known, the other factor is the quotient arising, as in arithme- tic, from dividing the known product by the known factor. 66. The quotient of ab divided by b is indicated thus, ab + b, or ; of ab divided by a, thus, ab-r-( a), or . o a i. Since ( + 2)( 5) = 10, what must io-i-(+2), or io , be? - ? + 2 2. Looking at the first four equations in the foregoing summary, 63 and 64, answer the following questions, giving reasons for answers: (2) ( + I2)-r-(+4)= ? (6) ( I2)-i-(+4)= ? (3) (-i 2 )-(+3)=? (7) ( + i2)-K- 3 )=? 36 First- Year Mathematics 3. From the second list of four equations in the foregoing summary answer, with reasons, the following: (1) (+ab) + (+a)=? (5) (-a&)-s-(-a)= (2) (+a&)-K+&)=? (6) (-aft)- (3) (~a&)-K+a)=? (7) (+*&) (4) (-aft)-K-6)=? (8) (+a&) 4. Answer the following, with reasons: . 18 18 -vy .> (?) - =? (7) - =? (n) - -=? v<5 ' +2 +9 x 5. Examine the answers to problems 2 and 3 and state what the sign of the quotient is (1) if the sign of both dividend and divisor is plus, (+); (2) if the sign of both dividend and divisor is minus, ( ); (3) if the sign of the dividend is plus, ( + ), and of the di- visor, minus, ( ); (4) if the sign of the dividend is minus, ( ), and of the divisor, plus, (+); (5) if the signs of dividend and divisor are alike (i. e., both +, or both ); (6) if the signs of dividend and divisor are unlike (i. e., one and the other +). 6. State the law of signs for division and compare the statement with 67. Law of Signs for Division: // dividend and divisor have like signs the quotient -is posi- tive; i} dividend and divisor have unlike signs the quotient is negative. Positive and Negative Numbers 87 Answer the following: + 18 a 6a '^6 =? 4 'a =? ? 6 =? -5 -a 3 .^33= ? 6.=^=? o. + 10 m i2a EXERCISE XII Find the quotients of the following, doing all you can men- tally: 2. (-!)-( + !) 22. (-a3)-(-fa) 3- (+D-K-I) 23. (+a-)--(-a) 4- (+4)-( + f) 24. 5- (+f)-(-f) 25. 6- (+!)-(-) 26. 7- (-D-(+t) 27. 8- (+4)-*-(-I) 28. 9- (-f)-(-f) 29. 10. (-*)-*-( + 30. 11. (-f)-(-f) 31. 12. (+f)-(+f) 32. i3- (-l)-(-l) 33- 14. (-|)-(+f) 34- 15- (+B)-K-t) 35- 16. ( 2a)-=-(+2) 36. 17. (-2a)H-(-a) 37. 18. (-5&)-( + 5) 38. 19. ( + 12*)-*- (-4*) 39. 20. ( a)-J-( ia) 40. (+6.82fl0)-H( 310). 88 First-Year Mathematics SUMMARY Positive and negative numbers are needed to record tem- perature, directed distances, opposing forces, opposite latitudes, debts and assets, dates before and after the beginning of the Christian era, motions in opposite directions, and directed magnitudes. Graphs may be made to represent thermometer readings, average temperatures, ages and heights of persons, batting averages, populations, rain and snowfall, lengths of day, areas of figures, equations, and general expressions of number. The algebraic sum of numbers with like signs is the arith- metical sum, with the common sign prefixed. The algebraic sum of two numbers with unlike signs is the arithmetical difference, with the sign of the larger number prefixed. The algebraic sum of any number of numbers may be found (i) by adding all the numbers in order, or (2) by adding all the positive numbers, then all the negative numbers, then adding the sums. The algebraic difference of two numbers is found by add- ing to the minuend the subtrahend with sign changed. The turning-tendency, or leverage, of a force on a bar or lever, is the product of the force by the lever-arm. If two factors have like signs the product is positive. If two factors have unlike signs the product is negative. If dividend and divisor have like signs the quotient is positive. If dividend and divisor have unlike signs the quotient is negative. CHAPTER V BEAM PROBLEMS IN ONE AND TWO UNKNOWNS Problems in One Unknown Number 68. In this chapter some practical problems arising out of the common uses of forces will be solved by means of the equation. It is necessary first to discover a law of these forces.* Mill Mill FIG. 76 FIG. 77 i. A bar or lever (Fig. 76) has loadings as in Fig. 78. Draw a diagram (Fig. 77) and find the turning-tendency, or leverage, for each loading: No. Force Arm No. Force Arm I +3 -6 VII - 4 ~9 II + 2 -3 VIII - 3 2 III +3 + 2 IX 12 +3 IV X +3 X X -3 V X 2 XI X + 2 VI +3 X XII 2 X FIG. 78 2. Draw a diagram showing I and VIII (Fig. 78) on the bar at the same time; I and VII; II and VIII; II and IX; * See pages 78-83, on turning-tendencies. 89 go First- Year Mathematics I, III, and VIII. In each case state whether the bar balances or turns, and if it turns, in which direction it turns. 69. In problem 2, if the turning-tendencies for I and VIII are added, the total turning-tendency is which says in mathematical language that the bar does not balance, but turns in the negative direction. If the turning-tendencies for II and VIII are added, the total turning-tendency is which shows that the bar does not turn. If the turning-tendencies for III, VII, and IX are added, the total turning-tendency is (+3)(+2)( 4)( 9) + ( 12) ( + 3) = +6+36 36 = +6, which shows that the bar does not balance, but turns in the positive direction. 70. If two or more forces are acting on the bar at the same time, the total turning-tendency is found by adding alge- braically the separate turning-tendencies. If the algebraic sum is zero, the bar balances. If the sum is not zero, the bar turns in the direction indicated by the sign of the sum. i. Find the total turning-tendency, and interpret it, as in 69, when the following loadings of Fig. 78 are on the bar at the same time. Draw a diagram for each case: (1) I and IX (3) I, HI, and IX (2) II and VII ( 4 ) I, III, and VII. * * ~* ._ 2. If a force of 4 on an arm of 9 (Fig. 79), and a -* +/ force of +/ on an arm of 2, are FlG - 79 on the bar, what must / be, for balance ? Beam Problems in One and Two Unknowns If the bar is balanced, the sum of the turning-tendencies must be zero. We may then write (i) (2) (3) (4) -2) + (-4)(-9)=o. Multiplying 27+36 = o. Subtracting 36 2/= 36. What axiom ? Dividing by 2 /= 18. What axiom? CHECK: From equation (i), ( + i8)(-2) + (-4)(-9) = (-36) + ( + 3 6)=o. 3. Write the equation of total turning-tendency, and find what the unknown force or arm must be, for balance, for each of the five loadings of Fig. 80. No. Force Arm Force Arm Force Arm t / +3 +3 -6 II +3 -6 w 2 III r 2 r -3 -4 -9 IV 2 ' d d +3 + 2 -3 V 2 I +3 I + 3 + 2 FIG. 80 71. Law of Turning-Tendencies or Leverages. For balance, the algebraic sum of all the turning-tendencies must equal zero. i. A bar (Fig. 81) is balanced , 5 V < jj>- >< *--> by a force of +10 on an arm of 6, and a force of 5 + 3 on an arm of +5. Find the values of s and 5+3. +10 aJ-is Tt + 12 FIG. 82 +3. Find the values of w and 31^ FIG. 81 2. A force of 37^15 (Fig. 82) on an arm of 4 is balanced by a force of -f 12 on an arm of pa First- Year Mathematics 3. A bar is balanced by each of the five loadings of Fig. 83. Draw a sketch and find the value of the unknown forces and arms for each loading. No. Force Arm Force Arm Force Arm I w+ 5 -3 + 15 +4 II '- 3 + 7 -/ -8 + 13 -3 III + 3 m-s + 15 -4 IV + 3 -8 + 3* +4 2 +4* V +39f -4 5* + 4 -13* -4 FIG. 83 Practical Applications i . A B (Fig. 84) is a crowbar, 6 \ ft. long, supported at F, i ft. from A. A stone presses down at A with a force of i ,800 pounds. How many pounds of force must be exerted by a man pressing down at B to raise the stone ? A * B <_ --- 6%' ------ > FIG. 84 2. With other conditions as in problem i, what would be the pressure at B if the fulcrum F (point of support) were 3 in. from A ? 3. With the fulcrum \ ft. from A (Fig. 84) what weight would be held in balance by a pressure of 200 Ib. at B ? Beam Problems in One and Two Unknowns 93 4. A suction pump (Fig. 85) is a device for raising water from wells. The handle, O B, works against a pin at A, so that when the hand pushes downward at B, the point O rises, and by the aid of a piston on the lower end, C, raises a mass of water. If O A = 2 in. and OB =3 ft., what load at O will be raised by a force of 20 Ib. pushing downward at B? 5. With other conditions as in 4, what force will be exerted at O by a downward force of 68 Ib. at B ? FIG. 85 6. A stone slab S (Fig. 86), weighing 2,400 Ib., rests with its edge on a point B, 6 in. from the fulcrum F of a crow- bar F A, 6 ft. long. How many pounds of force must be exerted at A to raise the slab ? !4' -> L FIG. 86 -966 FIG. 87 7. A steel beam 24 ft. long and weighing 966 Ib. (Fig. 87) is being moved by placing under it an axle borne by a pair of wheels, as shown at A, the end B being carried. If the axle is 2 ft. from the middle of the beam, what is the weight at B ? The weight of the beam itself may be treated as a load of 966 Ib. hanging to the bar at the middle point. 94 First- Year Mathematics 8. If the supporting point in problem 7 had been n ft. from O, what force at B would have balanced the rail ? 9. How far must the axle in problem 7 be placed from the middle M, that the weight at B may be 241 J pounds (Fig. 88) ? -9tt FIG. 88 10. With the fulcrum 2 ft. to the left of M, what would be the weight of the rail if it is balanced by an upward force of 140 Ib. at B ? 11. How may a steel rail weighing more than a ton be weighed with a pair of balances reading only to 60 pounds? 12. A dry goods box (Fig. 89) weighing 360 Ib. is being moved along the floor by the aid of a roller. If the box is 6 ft. long, what force at C is needed to hold the box horizontal when the roller is i ft. from B ? 2 ft. from B ? /%' 1 1 FIG. 89 FIG. 90 13. A wheelbarrow (Fig. 90) is loaded with 45 bricks, averaging 6 Ib. a piece. What lifting force will be needed at A to raise the load if the bar O A is 4} ft. and the distance Beam Problems in One and Two Unknowns 95 from the center of the wheel, O, to the point, B, where the vertical line through the center of the load crosses A O, is 2 feet ? 14. With the same load and length of bar, O A, as in problem 13, how far is it from O to the crossing-point B of the vertical center line of the load, if 90 Ib. at A just raises it ? EXERCISE XIII Simplify the following products and quotients: 1. (-I5-OC+3) -36*? 2. (io-0(-8 5 ) 3- (30(7-0 IS . 4. ,- 4 5- (-i5-70(-6) 16. ^ 6. (-i)(-s*) 7- C-r)(-5*+7) V' : 8. ( + i)(-5* + 7) 12. (o)(-763) o / I.W7 , \ 2O - 13. ( 2R)(l-{-X) 7 72. It is often convenient to measure lever-arms from a point at which the bar is not actually supported. i. A basket weighing 56 Ib. hangs on a stick 8 ft. long (Fig. 91) at a point i ft. from the middle, while it is being . 4 - K 4 A M B 1 J^ I FIG. 91 carried by two boys, one at each end. The boys lift 53; and 3* Ib., respectively. Find the values of x, $x, and $x. 96 first-Year Mathematics The bar or stick is balanced, no matter what point on the bar is regarded as the turning-point, or fulcrum. The lever- arms may, therefore, be measured from any point on the bar, whether the bar is actually supported at that point, or not. Taking M as the turning-point, the lever-arms are -4, -i, +4, and the turning-tendencies are ( + 5*)(-4), (-56)(-i), ( + 3*)( + 4). Since the bar does not turn, the sum of these turning-tendencies must equal zero. Solve the equation. It should be kept in mind that once a certain point is selected from which to measure lever-arms, all lever-arms must be measured from this point throughout the solution. J -i T or FIG. 92 FIG. 93 2. A bar is balanced by the forces shown in Figs. 92 and 93. Find the values of the unknown forces, measuring all lever-arms from M. * * 800 5/ ,], FIG. 94 FIG. 95 3. Find the values of the unknown forces on a bar balanced as shown in Figs. 94 and 95. Measure all lever-arms from the end-point L. 4. Solve problem 3, measuring all lever-arms from the end- point R. Beam Problems in One and Two Unknowns 97 Problems in Two Unknowns 73. Beam or lever problems in which two numbers are unknown also may be solved by the aid of the law of lever- ages. A * A/ 3 " -24 FIG. 96 1. A basket weighing 24 Ib. (Fig. 96) hangs on a stick 6 ft. long, at a point i ft. from the middle, while it is being carried by two boys, A and B. How much does each boy lift ? Using M as turning-point, 3^ 24 3^ = 0. Why? (i) y=8+x. Why? (2) 2. Can we find from the foregoing equation (2) the values of x and y that satisfy problem i ? To study this question, copy and fill in the following: In the equation y=8+# (2) if*= i, then y=8 + i =9 if x = . 2, then y =8 + 2 = 10 if#= 5, then y =8 + 5 = 13 \ix= 8, then y =8 + 8 = 16 if x = io, then y=8 + if a? = 11, then y = etc. FIG. 97 3. Show that each pair of values of x and y in Fig. 97 satisfies equation (2) of problem i. X y I 9 2 IO 5 J 3 8 16 IO II 14 19 08 First-Year Mathematics 4. Show that many other pairs of values of x and y can be found as those in Fig. 97 were found. From 2, 3, and 4 it follows that we cannot tell from equa- tion (2), alone, which pair of values in Fig. 97 gives the weights that A and B lift. If B lifts 9 times as much as A, then the pair 9 and i (Fig. 97) satisfies problem i. If B lifts 5 times as much as A, then the pair 10 and 2 satisfies problem i. To find the weights we must, therefore, know a second relation between them. 5. From Fig. 96, p. 97, show that x+y = 24. (3) Make a table (Fig. 98) of ten pairs of values of x and y, that satisfy equation (3). For conven- ience, solve equation (3) for y, thus 1 6 FIG. 98 y = 24-x. What axiom? (4) 6. Which pair of numbers in Fig. 98 is also in Fig. 97 ? Show that these numbers satisfy equation (3), problem 5, and equation (2), problem i, and that they are the numbers of pounds that A and B lift. Does any other pair of numbers in either table satisfy both equations ? 7. Make a table of pairs of values of R and S that satisfy the equation -25=8. (i) For convenience, solve equation (i) for R, thus R=S + 2 S. (2) 8. Make a table of pairs of values of R and 5 that satisfy the equation i*. (3) Beam Problems in One and Two Unknowns 99 9. Observe that many pairs of values of R and .S can be found that satisfy equation (i), alone, and equation (3), alone. Find a pair that satisfies both equations. 10. Find as in problems 7, 8, and 9 the pair of values of H and K that satisfies both of the equations: ii. Find the pair of values of M and L that satisfies the equations MzL= 3 12. A weight of 84 lb., hanging on a stick 8 ft. long at a point 2 ft. from the middle, is raised by two boys who lift at the ends of the stick. How much does each boy lift? 74. From the foregoing problems it follows (i) that an equation containing two unknowns is satisfied by many pairs of numbers, (2) that two different equations in the same two unknowns may be satisfied by a single pair of numbers. 75. The single pair of values that satisfies the conditions of some problems can be found from two equations in two unknowns directly as follows. i. The sum of two numbers is 24 \, and the difference is 8J. What are the numbers ? Let s represent the smaller number, and I the larger. Then by the conditions of the problem / + * = 2 4 J (I) l-s= 8J. (2) Adding (i) and (2) 2/ = 33 (3) J = i6J. (4) Subtracting (2) from (i) 25 = 16 (5) 5= 8. (6) CHECK: The sum of i6J and 8 is 24; the difference (8 taken from i6i) is 8J. Therefore the smaller number is 8 and the larger, i6J. ioo First- Year Mathematics 2. The sum of two numbers is 24 \, and the difference is nj. What are the numbers ? 3. The sum of two numbers is 12. If 3 times the smaller is subtracted from 4 times the larger, the result is 13. Find the numbers. Letting / represent the larger number, and s the smaller, 1 + 5 = 12 (l) 4/- 3 s = i3. (2) To find /, multiply both sides of equation (i) by 3, and add the resulting equation to equation (2). To find s, multiply both sides of equation (i) by 4, and subtract equation (2) from the resulting equation. CHECK. 4. Solve for w and t: 76. To solve beam or lever problems in two unknowns it is sometimes convenient to use, besides the law of leverages used thus far, another law which is explained in the following experiments. i. Putting a weight, x, at 1 3 (Fig. 99), and an equal weight, x, at r 3 , it will be found that two weights, each equal to x, hung to the pan, S, will hold the apparatus in balance. Begin- ning on the left, record the relations for balance thus: FIG. 99 x + ix x=o. 2. With two weights, each equal to x at 1 3 , and two weights, each equal to x, at r 3 , 4 weights, each equal to x, must be put at S for balance. But 3 or 5 weights, x, at S will be found not to balance. Make the record and interpretation thus: Beam Problems in One and Two Unknowns 101 Record Interpretation 2X+4X 2XQ Balance 2X-\-T > x 2X= x Movement downward 2X + $x 2X = -\-x Movement upward. 3. If 2 weights, x, be put at 1 3 , i weight, x, at 1 2 , i weight, x, at r 2 , and 2 weights, x, at r 3 , 6 weights, each equal to x, at S will balance the apparatus, but neither 5 nor 7 weights, x, will balance it. The results will run thus: Record Interpretation 2Xx -\-6x-x 2X=o Balance 2X x + 5# 2X x = x Bar moves downward 2X x + yx x 2X=+x Bar moves upward. 4. Write the equations and state the results as shown by a beam for each loading of the following table: No. u 1 3 i, i, s TI r 2 r 3 r 4 I x x X x 8* # # X X II x X 3* X X III y o y 6y y o o y IV 2y o 3? *sy Sy o 2y o V 2 y y zy y iSy 9 y o $y o -I. 5. Use the same bar, supported as shown in Fig. 100, and balance it. If a weight, x, is placed at both L and R, two weights, each equal to x at M balance the bar. The results are: Record Interpretation ;=o Balance. Try the same weights at L, and R and 3* at M; * at M. FIG. ioo 102 First- Year Mathematics 6. Eight weights, x, at l x , 3 weights, x, at R, and 5 at L, will balance the bar. If the 8 weights, x, are at r,, then 3 weights, x, at L and 5 weights, x, at R are needed for balance. The results are: Record Interpretation =o f =o f 7. Write the equations and state whether or not there is balance for these experiments: No. L u I* u I, M TI r a rj r 4 R I 6* o 8* o o O o 23f II 8* o 8* o 4* o o o 4* III 8* # 6x 4* o o X 5* IV 9* # 8x o 4* o o X 5* 8. When a bar is supported in two places (A and B, Fig. 100), it is called a beam. In the preceding experiments what is the test as to whether the bar, or beam, balances ? Observe that in all these experiments the forces are parallel to each other. 77- The two equations used in the following problems are derived from the following laws: Law of Forces. The algebraic sum of all the forces acting upon the material object (bar or beam) must equal zero, fi-r balance. Law of Turning -Tendencies or Leverages. For balance, the algebraic sum of all the turning-tendencies must equal zero. Beam Problems in One and Two Unknowns 103 Problems Applying Two Unknowns i. A box 12 ft. long of a three-horse coal wagon is loaded with 6 tons of coal. If the box extends 2 ft. in front of the front axle and 4 ft. back of the rear axle, what are the weights on the front and rear axles? See Fig. 101. -- 1-4* t - -4' -- - -*-- - -4'- - FIG. ioi FIG. 102 2. A lumber wagon is coupled out to a distance of 9 ft. between the axles and loaded with a pile of lumber 3^ ft. X4 ft. Xi8 ft. The load extends 3 ft. in front of the front axle and the material averages 48 Ib. per cubic foot. What are the pressures on the axles due to this load ? Fig. 102. 3. A wagon box, EFGO (Fig. 103), 10 ft. long and loaded with 40 bu. of wheat weighing 60 Ib. per bu., extends i^ ft. in front of the front axle, B, and 2\ ft. behind the rear axle, A. What is the load on each axle ? jr.?-?- --#- -> tx -no -jo +j FIG. 104 FIG. 103 4. Suppose a bar 10 ft. long, weighing 30 Ib., is used by two men, one grasping it at each end, to carry a load of 170 pounds. How many pounds must each man carry, if the load is attached 2 ft. from the left end ? f,. Measuring lever-arms from the middle point of the bar show that the equations are x + y = 2oo (i) -5* + 5io-o + 5;y=o. (2) The zero term in (2) arises from the leverage of the weight of the bar, which is 30*0, zero being the lever-arm. But 30*0 = 0, for mani- 104 First- Year Mathematics festly, a weight hanging at the middle point can have no tendency to turn the bar around this point, or, what amounts to the same thing, the turning-tendency about this point equals zero. Solve the equations. 5. A foot-bridge 15 ft. long between supports (15 ft. span) rests on timbers at L and R (Fig. 105). The bridge weighs -&o -looo FIG. 105 1,000 pounds. Two men, whose combined weight is 450 lb., stand just over A, 5 ft. from the end L. Find the pressures on the supports at L and R, using M as turning-point. 6. A bridge 20 ft. long weighs 2,400 lb. and supports two loads; one of 600 lb., 4 ft. from the left end, and the other, 800 lb., 15 ft. from the left. What are the loads borne by the supports ? 7. If, with the bridge of problem 6, four loads of 450 lb. each are placed, one 2 ft. from the left support, the second 6 ft., the third 9 ft., and the fourth 16 ft. from the left end, what are the upward forces exerted against the ends of the bridge by the supports? 8. A wagon standing on a culvert, A B (Fig. 106), has on the front axle a load of 2,500 lb., and on the rear axle 3,000 * u FT I -ls0 -MOO -1000 FIG. 106 pounds. The front wheels are 4 ft. and the rear wheels 10 ft. from the left end of the culvert. If the culvert is 20 ft. long, and weighs 2,000 lb., what are the pressures at the supports? 9. What would the pressures be if the front wheels stood 9ft. from the left support (wagon coupled to 6ft. between the axles) ? Beam Problems in One and Two Unknowns 105 10. Two men, lifting at the ends of a stick 8 ft. long, raise a certain weight. What is the weight, and at what point does it hang, if one man lifts 25 lb., and the other 75 pounds? (Use M as turning-point, Fig. 107.) 1 ~ 75 -*w> -j-l's +75 FIG. 107 f ii. Three boys desire to carry a i2-ft. log, weighing j 240 pounds (Fig. 108). Two *' of the boys lift at the ends of a hand-spike placed crosswise underneath the log and the third boy carries the rear end of the log. Where must the hand-spike be placed that all may lift equally ? (Use O as turning-point.) 12. Solve problem n using A as turning-point. Observe that the distance AB is 6 d. EXERCISE XIV Find the values of the unknown numbers, and check: I. 5 14+/=0 S 1 + 2 =O 2 . w n+3/=o 3- 2k IO+6l=0 4. 3(10 w) + ( 2)(+9i w 5. 5w zt ii =o 6. -i*- io6 First-Year Mathematics 7. i/ + J. r + i=o V + 7-$r=o 2(*+5) sy_ "3 1 4 SUMMARY 1. Beam problems in which one number is unknown may be solved by using the following law: Law of Turning-Tendencies. For balance, the algebraic sum of all the turning-tendencies must equal zero. 2. Beam problems in which two numbers are unknown may be solved by using the law of turning-tendencies, and the following law: Law of Forces. The algebraic sum of all the forces must equal zero, for balance. When only one number is unknown the Law of Forces may be used alone. 3. An equation containing two unknowns is satisfied by many pairs of numbers. Two different equations in the same two unknowns may be satisfied by a single pair of numbers. 4. To solve a problem leading to two equations in two unknowns, a third equation is derived which contains only one of the unknowns. 5. The product of two factors is zero, if one of the factors is zero. CHAPTER VI PROBLEMS IN PROPORTION AND SIMILARITY Drawing to Scale 78. Problems in finding distances may be solved by drawings made to scale. i. A man starting at O, Fig. 109, walks 45 y d - east and then 60 yd. north. What is the direct distance from the E FIG. 109 stopping-point to the starting-point, if i cm. on the drawing represents 10 yards? 2. Find the direct distance from the starting-point to stopping-point in problem i from a drawing in which i cm. represents 15 yards. 107 io8 First- Year Mathematics 3. A man walks 80 yd. south, then 144 yd. east, and then 120 yd. north. Find, by a diagram, his distance from the starting-point, letting i cm. on squared paper represent 1 2 yards. 4. Two men start from the same point. One walks 5 mi. west, and then 3 mi. north; the other walks 4 mi. south, and then 5 mi. east. How far apart are they ? 5. Draw a line 3 in. long, and let it represent a distance of 48 feet. What distance is represented by i inch ? By 2 inches? By 6 inches? By \\ inches? By 2$ inches? By i^ inches? In problem 5, the drawing is said to be made to a scale of i inch to 1 6 feet. 6. Draw to the same scale: 8 feet; 12 feet; 24 feet; 28 feet. 7. If a line 5 in. long represents a distance of 75 mi., what is the scale ? 8. If .7 of an inch on a map represents a distance of. 21 mi. on the earth, how many inches represent 87 miles? What scale is used ? 9. Draw a plan of a rectangular field 16 rods long and 12 rods wide, using the scale of i in. to 4 rods (i in. =4 rods), and find the distance in rods, diagonally across the field. Indicate the scale on all scale-drawings. 10. Draw to the scale, cm. to i ft., a plan of a room 24 ft. by 18 ft., and find the distance diagonally across the floor. 11. Draw to the scale, 2 cm. to 5 ft., a plan of the end of the house in Fig. iio, and find the height of the top of the roof from the ground. Use FIG. no a protractor to draw the angle 42. Problems in Proportion and Similarity 109 Ratio 79. The ratio of 6 to 3 is f, or 2; of 3 to 4 is f ; of a to b is 7 . The ratio of 6 to 3 is sometimes written 6:3; of 3 to '4, b 3 : 4; and of a' to b, a : b. 80. The ratio of any number to another number is the quotient found by dividing the first number by the second. Thus f is the ratio of 2 to 3. Any fraction may be regarded as an expression of the ratio of the numerator to the denomi- nator. 1. Write in two other ways the following ratios: (1) 5 to 20 (7) a+b to. c (2) 18 to 25 (8) x+ytoc+d (3) 251018 (9) a b to c+d (4) x to y (10) ax+ay to a (5) c to d (n) 3^ + 2 to a& (6) d to c (12) a+b to x. 2. How do two numbers whose ratio is i compare in size ? 3. What is the ratio of the cost of 5 yd. of silk at $1.50 to that of 50 yd. of cotton at 12^ cents? 4. What is the ratio of the length of the field (problem 9, p. 108) to the width ? Of the length of the plan to the width ? 5. What is the ratio of the length of the room (problem 10, p. 108) to the width ? Of the length of the plan to the width ? 6. What is the ratio of i yd. to i foot ? Of i yd. to i inch ? Of i yd. to 6 inches ? Of 3 yd. to 3 inches ? Of 3 yd. to 3 feet ? 7. What is the ratio of i Ib. to i ounce ? Of i oz. to 5 pounds ? Of i ton to 500 pounds ? Of 5 Ib. to 5 ounces ? 8. What is the ratio of i mi. to i yard? Of i mi. to i foot ? Of i mi. to 880 feet ? Of i mi. to 880 miles ? ! I0 First- Year Mathematics Si. Problems 6, 7, and 8 show that magnitudes must be expressed in the same unit before the ratio can be expressed as a single number. 1. Draw three triangles in each of which the angles are respectively 35, 65, and 80. Are all the triangles neces- sarily of the same size ? Do all have the same shape ? 2. Draw three triangles of different sizes in each of which the angles are 30, 60, and 90. Compare the triangles as to shape. Measure all sides of the triangles. Find the ratios of the sides of the first triangle to the corresponding sides of the second; to the corresponding sides of the third triangle. 3. Draw triangles of different sizes having the angles 52, 112^, and 15^. Compare the triangles as to shape. Find the ratio of the sides of one of the triangles to the corresponding sides of another. 4. Draw three triangles each of which has angles 90, 25, and 65. Are all of the triangles necessarily of the same size and shape ? Compare the ratios of corresponding sides. 5. Draw a triangle. Draw another having the angles equal respectively to the angles of the first triangle. Are the two triangles necessarily of the same size and shape ? Com- pare the ratios of the corresponding sides. Similar Triangles 82. Triangles having the same shape are called similar triangles. Similar triangles are not necessarily of the same size. i. Draw a triangle with sides 4 in. and 5 in., respectively, and an angle of 50 included between them. First draw it actual size and then to the scale of f in. = i inch. Are the tri- angles similar ? Why ? Measure with a protractor the pairs of corresponding angles. Find the ratios of the corresponding sides. Problems in Proportion and Similarity 1 1 1 2. Two sides of a triangle are 2$ cm. and 5.5 centimeters. The included angle is 60. Two sides of another triangle are 4! cm. and n cm., and the included angle is 60. Draw the triangles. Are they similar? Give reason for the answer. Measure the pairs of corresponding angles and find the ratios of the corresponding sides. 3. Draw a triangle. Draw another with sides respectively double the lengths of the first. Compare the triangles as to shape. Measure the corresponding angles. What is the ratio of the corresponding sides ? 4. Two sides of a triangle are 4.5 cm. and 9.5 cm.; the included angle is 70. Two sides of another triangle are 5! cm. and 13 \ cm. and the included angle is 70. Are the triangles similar? Compare the corresponding angles. Find the ratios of the corresponding sides. 83. The triangles of problems i, 2, 3, 82, though differing in size, have the same shape and have the corresponding angles equal. Notice also that each side of the smaller triangle (problem i) is f of the length of the corresponding side of the larger triangle. All similar triangles may be regarded as the same triangle drawn to different scales. They may be regarded as the same triangle magnified, or minified to a definite scale. 1. Draw two triangles having the same shape but different sizes. Measure two corresponding pairs of sides in each and compare the ratios. Measure another pair of corresponding sides and compare the ratios. 2. According to problems i, 2, 3, 82, and problem i, 83, what seems to be true of the ratios of corresponding pairs of sides of triangles having the same shape (similar triangles) ? 112 First- Year Mathematics 84. Two triangles are similar when the corresponding angles are equal and when the ratios of the corresponding sides are equal. i. In the two similar triangles of Fig. in, (i), if a =4 in., 4=i2 in., and 6=4 in., how long is B? FIG. in 2. In the similar triangles of Fig. in, (2), If a=4 in., A = 12 in., and 6 = 5 in., how long is B? If a=3 in., 6=8 in., and B =32 in., how long is A ? If a=x in., 6=8 in., and .6=32 in., how long is A ? 3. In the similar triangles in Fig. in, (3), If A =21 in., 6=9 in., and = 27 in., how long is a? If = 5^ in., A =22 in., and -6=30 in., how long is 6 ? 4. In the similar triangles in Fig. in, (4), If a =3 in., A =8 in., and C = 5 in., how long is c? If A =24 in., c=4 in., and C = 7 in., how long is a? If A=y in., 0=4 in., and C = 7 in., how long is a? 5. In Fig. 112, if the stake 3 ft. high casts, a shadow 8 ft. Problems in Proportion and Similarity long, and the tree, at the same time, casts a shadow 80 ft. long, how high is the tree ? 6. Are triangles O A B and O H K (Fig. 113) similar ? Give reason for answer. How long is x ? X FIG. 113 7. A boy holds a pencil, A B (Fig. 114), 2 ft. from his eye, so that it covers a flag-pole 360 ft. distant. To make the triangles E A B and E F K similar, how must the pencil be held ? If the pencil is 6 in. long, how high is the pole ? FIG. 114 8. A lumberman who is 5 ft. tall wishes to find a tree 60 ft. to the first limbs. He drives a stake in the ground and places ii4 First-Year Mathematics his feet against it as in Fig. 115. If the stake is 4 ft. high, how far must it be placed from the foot of the tree, that he may determine whether or not the trunk is 60 ft. to the limbs ? FIG. 115 9. The gables of a house and of a porch have the same shape. The' sides of the porch-gable are 7 ft., 7 ft., and 10 feet. The longest side of the house-gable is 25 feet. What is the ratio of the corresponding sides? How long are the other two sides of the house-gable ? 10. The sides of a triangle are 8, 10, and 13. The shortest side of a similar triangle is n. What is the ratio of the cor- responding sides ? Find the other sides. 11. The sides of a triangle are 4.6 cm., 5.4 cm., and 6 centimeters. The corresponding sides of a similar triangle are x cm., y cm., and 15 centimeters. Find x and y. 12. The sides of a triangle are 1,2, and 3, and the longest side of a similar triangle is 20. Find the other sides of the second triangle. 13. Draw a triangle (Fig. 116) having two of the sides A B and A C equal to 10 in. and 12 in., including any con- FIG. 116 Problems in Proportion and Similarity 115 venient angle between them. Call the third side the base. Through a point on A B, 5 in. from the vertex A, draw DE parallel to the base. Measure the distance A E. How does the ratio of the corresponding parts of the sides A B and A C, cut off by the parallel D E, compare with the ratio of the sides A B and A C ? 14. Draw a parallel to the base (Fig. 116) through a point of the jo-in. side, 2^ in. from the vertex, and measure the distance from the vertex to the crossing-point of the 2^ in. parallel with the i2-in. side. Compare the ratio of the cor- responding parts of the sides with the ratio of the sides them- selves. 15. Compare the ratios of the corresponding parts of the sides made by parallels to the base, through a point of the 10 in.- side 3 in. from the vertex; 6 in. from the vertex; i J in. from the vertex; 7^ in. from the vertex. 85. A line drawn parallel to one side of a triangle divides the other two sides into corresponding parts having the same ratio as the sides themselves. Any number of parallels to the base of a triangle divide the other two sides into parts having the same ratio. 1. In Fig. 117, AB = 2i, AC = 35, and A D = 3. D E is parallel to B C. Find the value of x. 2. Show that triangles A D E and ABC, Fig. 116, are similar. 86. If a line is drawn parallel to one side of a given triangle, meeting the other two sides, a triangle is formed which is sim- ilar to the given triangle. i. AB (Fig. 1 1 8) is parallel to C D. Find C D, the distance across the lake. FIG. n8 n6 First- Year Mathematics 2. Draw a triangle, as A B C, Fig. 119. Divide A B and A C into parts having the ratio i : 3. Connect the points of FIG. 119 division, D and E, by a straight line. Divide A B and A C in other ratios, as 2:3, 1:1, 3:7, 1:9, 1:4, and connect the corresponding points of division by straight lines. Notice that these lines are parallel, that is, they will not meet, however far extended. 87. // two sides of a triangle are divided into parts having the same ratio, the line joining the points of division is parallel to the third side of the triangle. Problems in Surveying 88. Problems in finding directions and distances may be solved by drawings made to scale. In Fig. 121 the direction shown by the arrow is read "30 east of north;" in Fig. 122, "50 west of south;" in Fig. 123, "20 east of south." 89. The direction of a line, when indicated by the angle it makes with the north-south line, is called the bearing of the line. Problems in Proportion and Similarity 117 Surveyors use the surveyors' compass, Fig. 128, to measure the bearings of lines. yw. -t. Si i FIG. 121 *] FIG. 122 ** FIG. 123 I I I t FIG. 126 FIG. 127 1. Read the bearings of the arrows in Figs. 124, 125, 126, 127. 2. With a ruler and protractor draw lines having the fol- lowing bearings: 65 east of south 47^ west of south 65 east of north 43 west of north. 90. The bearing of a point B from a point A is the bear- ing of the line A B with reference to the north-south line through A. n8 First- Year Mathematics i. In Fig. 129, read the bearings of A from O B from O C from O O from A O from B O from C. 2. The bearing of fort A from fort B, both on the sea- coast, is 65 west of north. An enemy's vessel at anchor off the coast is observed at A to bear northeast; at B, north- west. The forts are known -SJL.J FIG. 128 to be 7 mi. apart. Find by drawing a plan (scale: 2 cm. = i mi.) the distance from each fort to the vessel. In solving problems like 2, draw first a sketch of the distances and directions, and then make the scale drawing. 3. Find the distance P Q, if Q is 6.4 mi. east and 9.8 mi. north of P (scale: i in. = 2 mi.). What angle does P Q make with the north-south line through P ? 4. A hill in a battle-field obstructs the view from a battery at B, Fig. 130, to the enemy's fort at F. A point H is found at the bottom of the hill, from which F is observed to- bear FIG. 130 Problems in Proportion and Similarity 119 4 mi. northeast. If H is 6.25 mi. northwest of B, what is the distance F B, and the bearing of F from B ? The triangle in the drawing, problem 4, is similar to the triangle in the field, for the first triangle is the same as the second with all sides minified or reduced in a definite ratio, the angles remaining unchanged. 5. A tree at Q is 6. 5 rods north of R, and 9 rods west of S. What is the distance and bearing of S from R? Show that the triangle in the drawing is similar to the triangle in the field. 6. A man wishes to measure the width of a river without crossing it. The river flows due west. Standing at A, on FIG. 131 12C First- Year Mathematics the bank, he observes a tree on the other bank in the direction, 20 east of north. He walks 50 rods east along the bank to B, and there observes the tree in the direction, 60 west of north. Find the width of the river. 91. In the preceding problems angles have been measured by referring them to the north-south line. By means of the engineers' transit (Fig. 131) angles in any position may be measured. For rough measurements, an angle-measurer can be constructed by tacking a protractor on a board (Fig. 132). A ruler with a pin stuck in it at each end can be used for sighting. C, ,B FIG. 132 FIG. 133 1. Draw a plan of a garden plot from the data of Fig. 133. Find the length, in rods, of B C, and of the perpendicular from B to A C. (Scale: i in. =10 rods.) 2. Draw triangles having the following parts: 4 in., 4 in., and the included angle 60 3 in., 4 in., and the included angle 90 3 in., 3$ in., and the included angle 50. Measure the third sides of the triangles. Compare your results with those of other members of the class. What truth about tri- angles do you infer ? 3. A railroad surveyor wishes to measure across the swamp A B, Fig. 1 34. He measures the distance from a tree at A to a stone at C and finds Problems in Proportion and Similarity 121 it to be 165 feet. The distance from a tree at B to the stone is 150 feet. Find the distance in feet across the swamp, the angle at C being 80 (scale: i cm. = 15 ft.). Show that the tri- angle in the drawing is similar to the surveyed triangle. 4. To measure the width, A C, of a stream (Fig. 135), without crossing it, an engineer lays off a line, B C, on one side of the river, and measures (with a transit) the angles at B and C. Draw a triangle FlG - X 35 to scale from the data in the figure, and determine the width of the river. 5. Draw triangles from the following data: A 6 = 3^ in., angle A = 15, angle 6 = 130 A B=3^ in., angle 6=90, angle C= 60 A 6=3^ in., angle = 55, angle A = 20. In each triangle measure the sides not given and compare as in problem 2. What do you infer? 6. A boy wishes to determine the height, H K (Fig. 136), x . of a factory chimney. He places the angle-measurer first at B and then at A and measures the angles x and y. The x /7 x / ' / <' As* & # FIG. 136 angle-measurer lies on a box, or tripod, 3^ ft. from the ground. A and B are two points in line with the chimney and 50 ft. apart. What is the height of the chimney if the ground is level and if #=63 and ^=33^? 122 First-Year Mathematics 7. Classify triangle ACK, Fig. 136, as to its angles. Triangle BCK. 8. How many degrees are there in each angle at K, Fig. 136 ? In the angle adjacent to angle x ? 92. A telescope is pointed hori- zontally toward a tower (Fig. 137), and the farther end is then raised \H (elevated) until the telescope points to the top of the tower. The angle FlG - J 37 through which the telescope turned is the angle of elevation of the top of the tower, from the point of observation. 1. From point A, Fig. 136, what is the angle of elevation of the top of the chimney ? From point B ? 2. When the angle of elevation of the sun is 25, a building casts a shadow 90 ft. long, on level ground. Find the height of the building. 3. Find the angle of elevation of the sun when a tree 40 ft. high casts, on level ground, a shadow 60 ft. long. 4. On the top of a tower stands a flagstaff. At a point, A, on level ground, 50 ft. from the base of the tower, the angle of elevation of the top of the flagstaff is 35. At the same point A, the angle of elevation of the top of the tower is 20. Find the length of the flagstaff. 93. A telescope at T, on //_ Ari"-.. u l' l '"l-_ . the top of a cliff (Fig. 138), ,, \.^" *,** is pointed horizontally, and v^ then the farther end is lowered ^.^" (depressed) until the telescope " B points to the buoy at B. The FIG. 138 angle through which the telescope turned is the angle of depression of the buoy from the point T. Problems in Proportion and Similarity 123 1. If the height of the cliff, Fig. 138, is 100 ft., and the angle of depression of the buoy, as seen from T, is 40, what is the distance of the buoy from the bottom of the cliff ? 2. A boat passes a tower on which is a search-light 120 ft. above sea-level. Find the angle through which the beam of light must be depressed, from the horizontal, so that it may shine directly on the boat when it is 400 ft. from the base of the tower. 3. From the top of a cliff 150 ft. high, the angle of depres- sion of a boat is 25. How far is the boat from the top of the cliff? 4. From a lighthouse, situated on a rock, the angle of depression of a ship is 12, and from the top of the rock, it is 8. The height of the lighthouse above the rock is 45 feet. Find the distance of the ship from the rock. Proportion 94. Comparing the ratios of the areas of figures with the ratios of corresponding dimensions leads to proportion. i. Two rectangles, as ABCD and * f EFGH (Fig. 139), have equal altitudes, h. The bases are 7 in. and 4 in. respectively. What are the areas? Find the ratio of the areas. Find the ratio of the bases. How does the ratio of the areas compare FIG. 139 with the ratio of the bases ? 2. The altitude of a rectangle is 10 in. and the base is 4 feet. The altitude of another rectangle is 20 in. and the base is 4 feet. What is the ratio of the areas ? Of the alti- tudes ? Compare the ratios. 3. Two rectangles have bases 20 ft. and 25 ft., and an altitude of 15 feet. Express by an equation that the ratio of the areas equals the ratio of the bases. 124 First- Year Mathematics 4. The dimensions of one rectangle are a and b, and of another a and c. Compare the ratio of the areas with the ratio of the unequal dimensions, and express the result by an equation. 5. How does the ratio of the areas of rectangles having equal bases compare with the ratio of the altitudes? How does the ratio of areas of rectangles having equal altitudes compare with the ratio of the bases? 6. The area of rectangle ABCD (Fig. 140), is 80 sq. ft., and the base is 10 yards. What is the area x of rectangle EFGH, having the same altitude and a base equal to 24 yards ? so FIG. 140 FIG. 141 7. Express by an equation the relation between the areas and bases of the rectangles in Fig. 141. Find the base x. f *Y* 8. The equation - - expresses the relation between the 4 o ratio of the areas and the ratio of the altitudes of two rec- tangles. Find the altitude x. 9. Triangles ABC and ADC (Fig. 142), with the same base, have altitudes as shown. What is the ratio of the areas ? Of the altitudes ? How do the ratios compare ? 10. If two or more triangles have equal bases, how does the ratio of the areas of any two compare with the ratio of the altitudes? n. Two triangles (Fig. 143) have equal bases, b, and altitudes as shown. FIG. 143 The ratio of the areas is f . Express by an equation the FIG. 142 Problems in Proportion and Similarity 125 relation between the ratio of the areas and the ratio of the altitudes. Find the altitude, x, by solving the equation. 12. Compare the ratios of areas and of bases of triangles with equal altitudes (Fig. 144). Express the result by an equation. FIG. 144 FIG. 145 13. Compare the ratios of areas and of bases of the parallelo- grams in Fig. 145. Express the result by an equation. 95. An equation of two ratios is called a proportion. For 4 2 a ac ,ac example, -=-, T=r~ and 7=- are all called proportions, o 3 b be b a and are sometimes written thus: 4:6 = 2:3, a:b=ac:bc, and a:b=c:d. The last maybe read "a is to b as c is to d." Read the other two. Numbers that form a proportion are said to be proportional. Four lines are said to be proportional if their lengths are proportional. Ha FIG. 146 i. In Fig. 146, the letters a and b denote the same num- bers throughout. How do the areas of triangles I, II, and III compare ? I and IV ? II and VI ? Ill and V ? IV and VIII ? II and IX ? IX and X ? VII and X ? Ill and X ? 126 First- Year Mathematics 2. Show that (i) areas of rectangles are proportional to the bases, if the altitudes are equal, i. e., the ratio of the areas equals the ratio of the bases; (2) areas of rectangles are proportional to the altitudes, if the bases are equal; (3) areas of rectangles are proportional to the products of the bases and altitudes; (4) areas of triangles are proportional to the products of the bases and altitudes. 3. The altitude and base of a triangle are 4 ft. and 15 ft. re- spectively. What are the dimensions ot other triangles of the area ? Of J the area ? Of the area ? 4. The altitude and base of a rectangle are 6 in. and 8 in. respectively. What are the altitudes and bases of triangles whose areas are: (1) equal to the area of the rectangle ? (2) twice the area of the rectangle ? (3) one-half the area of the rectangle ? 5. Answer the same questions when the altitude and base of the rectangle are a and b inches respectively. 6. A triangle and a rectangle have equal bases and are equal in area. How do the altitudes compare ? 7. The dimensions of a rectangular block are 4 ft. X 15 ft. X 25 feet. What are the dimensions of other rectangular blocks having the volume ? the volume ? the volume ? the volume ? ^ the volume ? 8. Two rectangular flower beds have the same shape, but are different in size. One is 3 ft. wide and 5 ft. long; the other is 12 ft. wide. How long is it? What is the ratio of the corresponding sides? 9. Two books have the same shape. One is 5^ in. wide and 7$ in. long. The other is 15 in. long. How wide is it ? Problems in Proportion and Similarity 127 10. The top of a desk and a rectangular sheet of paper, 12 in. by 18 in., have the same shape. The desk is 2 ft. wide. How long is it ? n. A city block and a lot within the block have the same shape. The lot is 100 ft. by 150 ft. and the block is 300 ft. wide. How long is it ? 12. Is 2:5 = 5:15 a proportion? Is 2:7=8:25 a propor- tion ? Give reasons for your answers. 96. The first and last terms of a proportion are called the extremes ; the second and third, the means. 1. Compare the product of the extremes with the product of the means in the proportion 2:5=6:15; in 3:7=6:14; in 20:2 = 10:1; in 12:3=4:1. What do you find true of the products ? 2. For which of the following expressions does the product of the first and last number equal the product of the other two ? (1) 1:3= 4:12 (6) 2:3 = 20:30 (2) 3:4= 6:12 (7) 8:80 = 3:33 (3) 2: 3= 8:i1 ( 8 ) x:y=W-4y (4) 5:6 = 10:12 ' (9) 30:3^=60:6^ (5) 8:3 = 15:3 (10) *:3* = i:3. Is there any proportion in this list in which the product of the means does not equal the product of the extremes ? Is there in the list any expression that is not a proportion, for which the product of the first and fourth number is equal to the product of the second and third ? 97. In a proportion, the product of the means equals the product of the extremes. This is a convenient test of proportionality, i. By this test tell what expressions in problem 2, 96, are proportions. 128 First- Year Mathematics 2. If T = j, prove that ad=bc. b a 3. Divide 85 into two parts in the ratio 2:3. 4. Divide 84 into three parts proportional to 3:4:5- 5. What number added to 12 and subtracted from 30, gives results that are to each other as 5:10? 6. Two numbers are in the ratio of 5:6. If 12 is sub- tracted from each, the differences are in the ratio 3:4. What are the numbers ? 7. The ratio of two lines is 2^:3!. The longer line is 30 centimeters. Find the shorter line. FIG. 147 8. In Fig. 147 the ratio of the parts of AC equals the ratio of AB to BC. AD is 2 in. less than DC. Find the lengths of AD and DC. 9. On squared paper make an accurate drawing for problem 8. Measure the angles at B. How do they compare in size ? 10. If 6 be taken from one of two complementary angles and added to the other, the ratio of the two angles, thus formed, is 2:7. Find the angles. n. The ratio of 2 times one of two supplementary angles to 8 times the other is 1:2. Find the supplementary angles. 12. Three angles just covering the plane around a point are to each other as 2:3:4. Find them. 13. The angles of a triangle are as 1:2:3. Find them. Problems in Proportion and Similarity 129 14. The acute angles of a right triangle are as 2:5. Find them. 15. Find the value of x in the following proportions: (i) ?=M (7) 10 - 28 i-ifl (8) x # 13 x (l\ J 8 s . = -v-- 7 \jJ 810- ' J (9) 51 (10) x 2 : 64 =400:1 S~X_2X X 3 _ 3/ j_ i j_//> 7 192 . 2 ,_^ * + 5_ 16 1 V T >l *V 1 F* o 4 * v) 98. Proportions may be written from equations that express the equality of products. 1. The statements below are different arrangements of the four factors in the equation 8-7 = 14-4. (i) Apply the test of proportionality and point out which statements are proportions: (2) 8:14=4:7 (6) 8:7 = 14:4 (3) i=Y (7) rV=f (4) 4:7=8:14 (8) 8:4 = 7:14 (5) =1 (9) A-t- 2. From what place in the given equation (i) were the means taken in the proportions ? The extremes ? 3. Is the same thing true of any of the statements that are not proportions ? 4. Try to write a false expression of proportion by making both factors of either side of equation (i) the means of the expression. i jo First- Year Mathematics 5. Try to write a correct statement of proportion using any one of the four factors of the given equation for a mean and the other factor on the same side for an extreme. 6. Write four proportions from 3-28=4-21, and apply the test of proportionality. 7. Write four proportions from a i2b = T,a 46, and test. 99. // the product of two numbers equals the product of two other numbers, either pair may be made the means and the oilier pair the extremes of a proportion. i. From each of the following equations write at least four proportions and show that they satisfy the test of propor- tionality: (1) yiod=c-3od . 4 _2X (2) 6-2ia=i8. 7 a " 5" (3) IS' 7' = i5 '* (6) a-b=ab-i. (4) a bc=ab c 100. Proportions may be written from other proportions. 1. Using the numbers of the proportion 4:7=12:21, write another expression in proportion form: (1) without changing the positions of the 4 and the 21 (the extremes), (2) without changing the positions of the 7 and the 12 (the means). 2. Interchange the means only in 5:7=15:21, and test for proportionality. Interchange the extremes and test. 3. If a: b=c: d, is a: c=b: d? Reason for answer. 4. If a: b=c: d, is d: b=c: a ? Reason for answer. 101. If the means or the extremes of a proportion are interchanged the resulting expression is a proportion. When a second proportion is made from a given propor- tion by interchanging the means, or 'by interchanging the Problems in Proportion and Similarity extremes, the second proportion is said to be obtained from the given proportion by alternation. A AB AC AB AE Show that =- Z FIG. 148 2. In two equilateral polygons having the same number of sides, the corresponding sides are proportional. that Then A"R A "R PROOF: Since ^7^=1 (Fig. 149), and -^ = i (Why?), it follows B(_, BjCj ^4'- Why? AB BC ArBx In a similar way prove BC CD . Why? DE C,D, 3. If two parallel lines are intersected by three or more parallel lines, the ratio of any two parts of one of the first two parallels equals the ratio of the corre- sponding parts of the other. Suggestion for the proof: AB BC . Why? FIG. 150 I 3 2 First- Year Mathematics Variation i. Let the vertical side of a square (Fig. 151) represent i mile and the horizontal side i hour. Graph the distance passed over in i hour, 2 hours, 3 hours, 10 hours, by a man walking at the rate of 2 miles an hour. Draw the lines O A,, A I B I , BjCr What kind of a line is O E t ? How do the angles of triangles O A A,, O B BJ, O C C I} etc., compare in size ? What is the ratio of A A, to O A ? of B B, to O B ? of C C, to O C ? FIG. 151 2. Show that the triangles O A A., O B B,, , etc., are similar. What is the ratio of any distance -line to the corresponding time-line ? How does the distance change when the time is doubled, trebled, quadrupled, etc. ? How does the distance-line change (vary) when the time- line changes ? 3. Write the equation for the time I and the distance d passed over by a man walking at the rate of 2 miles an hour. What are the values of d as / takes the values 1,2, 10? What is the ratio of a value of d to the corresponding value of/? 4. On squared paper let the vertical side of a small square denote i cent and the horizontal side i pound. Graph the cost of flour at 3^ cents per pound, for i lb., 2 lb., 3 lb., 8 pounds. How does the graph show the cost to change (vary) as the weight is doubled, trebled, etc. ? How does the graph show the cost to vary as the weight is varied ? What is the ratio of the cost to the weight ? Problems in Proportion and Similarity 133 5. Write the equation for the weight iv in pounds, and the cost c of flour at 3^ cents a pound. Find the values of c as w takes the integral values from i to 10. What is the ratio of the values of w to the corresponding values of c ? 6. Graph the areas of rectangles with the altitude 5 and bases i, 2, 3, 6. How does the area vary as the base varies ? Write the equation for the area A and the base b of rectangles with altitude 5. Does a change in value of b change the value of A ? A Does it change the value of the ratio ? 1 02. The number x is said to vary directly as the number y, if the ratio, x:y, remains constant (i. e., does not change). 1. Graph the areas of triangles with altitude 6 and bases i, 2, 3, 7. Show that the area A of a triangle varies directly as the base b. 2. Write the equation for the area A of a triangle having the altitude h and the base b. Show from the equation that the area A varies directly as the base b, when the altitude is constant. 3. Write the equation for the area A of a rectangle having the altitude a and the base b. Show that the area A varies directly as the base b, when the altitude is constant. 4. Cut (from card-board) circles with diameters of various lengths. With a string or thin wire measure the circumferences of the circles. Find the ratio of the circumference of each circle to the diameter. How do the ratios compare ? 5. The area of a rectangle varies directly as the base if the altitude remains constant; and when the area is 27, the base is 3. What is the constant ratio of the area to the base ? 134 First-Year Mathematics What is the equation connecting the area A and the base b ? When the area is 54, what is the base ? 6. The circumference of a circle varies directly as the diameter. The constant ratio of the circumference to the diameter is 3.14, approximately. Write the equation for the circumference c and the diameter d. When the circumference is 157, what is the diameter? 7. The distance d through which a body falls from rest varies directly as the square of the time / hi which it falls; and a body is observed to fall 400 ft. hi 5 seconds. What is the constant ratio of d to t* ? Write the equation for d and /. How far does a body fall in i second ? In 2 seconds ? In 3 seconds? 8. x varies directly as y, and when x = 20, ^=4. Find the value of x when ^ = 17. 9. If z varies as x, and 2=48 when #=4; find z when #=11. 10. The turning-tendency caused by a weight moved along a bar, or lever, varies directly as the lever-arm. The turning- tendency is 20 when the arm is 5. Find the turning-tendency when the arm is 7. 11. Rectangles with area 6 sq. ft. have bases of i, 2, 3, 6. For each rectangle write the ratio of the base to the reciprocal of the altitude. How do the ratios compare in value ? 103. The number b is said to vary inversely as the num- ber a, if the ratio of b to the reciprocal i of a, is constant. i. Rectangles have an area equal to 10. The base varies inversely as the altitude. What is the constant ratio of the base to the reciprocal of the altitude? Write the equation for the base b and the altitude a. Problems in Proportion and Similarity 135 2. y varies inversely as x. y = 2 when x = i. Find the constant ratio of y to the reciprocal of x. Write the equation for x and y. Find y when x=S. 3. When gas in a cylinder is exposed lop essure, the volume is reduced as the pressure is increased. It is found in physics that the volume varies inversely as the pressure. The volume of a gas is 4 cubic cm., when the pressure is 3 pounds. What is the volume under a pressure of 6 pounds ? 4. The speed of a falling body varies directly as the time. Write the equation for the speed v and the time t. A body, falling from rest, moves at the rate of i6oft. a second 5 sec. after it began to fall. What will be the speed attained in 8 seconds ? 5. The distance passed over by a body, moving at a con- stant rate, varies directly as the time. Find the rate of a train which travels, at uniform rate, the distance of 225 mi. in 6 hours. 6. A stone fell from a building 560 ft. high. In how many seconds did it reach the ground? (See 7, 102.) 7. The number of men doing a piece of work varies inversely as the time. Twelve men can do a piece of work in 28 days. In how many days can 3 men do the same ? 8. The time t of oscillation of a pendulum varies directly as the square root of the length /. A pendulum 39 . 2 in. long makes one oscillation in one second. Find the .length of a pendulum which makes an oscillation in two seconds. !^6 First-Year Mathematics SUMMARY 1. The ratio of two numbers is their quotient. 2. Similar triangles are triangles that have the same shape. Similar triangles are not necessarily of the same size. 3. All similar triangles may be regarded as the same tri- angle drawn to different scales. They may be regarded as the same triangle magnified or minified to a definite scale. 4. Two triangles are similar when the corresponding angles are equal and when the ratios of tlte corresponding sides are equal. 5. When surveying problems are solved by drawing tri- angles to scale, the triangle in the drawing is similar to the surveyed triangle. 6. A proportion is an equality of ratios. 7. Areas of rectangles (and triangles) have the same ratio (1) as the bases, if the altitudes are equal, (2) as the altitudes, if the bases are equal. 8. Areas of rectangles (and triangles) have the same ratio as the products of the bases and altitudes. 9. The extremes of a proportion are the first and last terms; the means are the second and third terms. 10. In any proportion tJie product of the extremes is equal to tfte product of the means. This is a convenient test of proportionality. n. If the product of two numbers is equal to the product of two other numbers, either pair may be made the means and the other pair the extremes in a proportion. 12. A proportion is obtained from a given proportion by alternation, if the means are interclianged, or if the extremes are interchanged. 13. One of two numbers varies directly as the other if their ratio is constant. 14. One number varies inversely as another if the ratio of the first to the reciprocal of the second is constant. CHAPTER VII PROBLEMS ON PARALLEL LINES. GEOMETRIC CONSTRUCTIONS Parallel Lines 104. The equation may be used to solve exercises on the angles made by lines intersecting parallel lines. 1. Point out the parallel edges of a rectangular box or table-top. Give other examples of parallel lines, as telegraph wires, latitude lines, plumb-lines, etc. 2. Draw two parallel straight lines, using the parallel edges of a ruler; draw a straight line inter- secting them (Fig. 152). Measure and / compare angles a and b, c and d, e and */ /, g and h. These angle-pairs are called */ f pairs of corresponding angles. <( t . 3. Draw two figures like Fig. 152, / with the crossing-lines in different / directions, and compare the corre- FIG. 152 spending angles. 4. Draw two straight lines that are not parallel, and com- pare the corresponding angles thai they make with a crossing-line (transversal). 105. Parallel straight lines are lines that have the same direction or opposite directions. i. x and y (Fig. 153) are equal. Show by rotating a pencil, as in- dicated, that AB and CD have the same direction. '37" 138 First- Year Mathematics 2. Draw two intersecting lines, AB and RS (Fig. 154). Show how to draw with the aid of a protractor a line parallel to A B through M. Y A FIG. 154 FIG. 155 3. Draw a triangle on paper. Cut out the triangle and place one side of it along a line X Y (Fig. 155), first in position (i) drawing line A B, and then in position (2) drawing C D. Show that A B and C D are parallel. What angles are equal ? 4. Fold a piece of paper so as to form a right angle (r,, Fig. 156). Using the right angle as in the figure, draw A B and C D and show that they are parallel. 1 06. In the preceding prob- lems, the following properties of parallel lines were studied: (a) If two straight lines are parallel, the corresponding angles made with a crossing-line (transversal) are equal. (6) Two straight lines that make equal corresponding angles with a transversal have the same direction, and are parallel. From (b) it follows that (c) Two lines that are perpendicular to the same line are parallel. Problems on Parallel Lines 139 1. In Fig. 157 read several pairs of supplementary adjacent angles (see p. 42); of vertical angles (see p. 47). 2. Draw two parallel straight lines and a transversal. Letter the angles as in Fig. 157. Give reasons for the fol- lowing: x a=e (i) / therefore, e+d = iBo. (3) Give reasons: 3. If a=e, then c=e (Fig. 157). M 4. If c=e, then c=g. 5. If c=g, then a =g. 6. Show that b is the supple- ment of a; that /is the supplement of e. a=e. 7. Give reasons for the following: FIG. 157 Show that &=/, if therefore, Give reasons: FIG. 158 (i) (2) (3) 8. If b=f, then d=f . 9. If d=f, then d=h. 10. If d=h, then b=h. n. Make a list of .aK equal angles of Fig. 157, and of the pairs of non-adjacent angles that were proved to be supple- c mentary. 12. Make a list like n for Fig. 158, A B and C D being parallel. 13. One of the 16 angles made by the rails of two intersecting railroads is 35. Find each of the other 15. 140 First- Year Mathematics 107. When two lines (Fig. 159) are cut by a transversal, a and e \ are called the angles of b and/ ( corre- the angle-pairs d and h ( spending c and g J angles, angles c, d, e, f are called interior angles, angles a, b, g, h are called exterior angles, r / FIG. 159 the angles of the angle-pairs the angles of the angle-pairs the angles of j the angle-pairs d and e \ are called interior angles on the c and / } same side of the transversal, d and / ) on opposite sides of the transversal, c and e j are called alternate interior angles, on opposite sides of the transversal, are called alternate exterior angles. 1. State the results of problem n, p. 139, naming the angle-pairs as just defined. 2. Draw a pair of parallels crossing another pair. Letter the angles as in Fig. 160. Prove a=t, t = m; therefore a=m, H X b and h a and g 3. Prove that a +q = i 80. m + q=i8o. Why? m=a. Why? ' a + q = i8o. Why? 4. Prove that r is the supplement of d, and that e is the supplement of d] therefore, r^=e. Problems on Parallel Lines 141 5. Prove the following: Fig. 160. (1) d=q (5) g = t (2) w=f (6) (4) b=l (8) 6. From the top of a cliff (Fig. 161), 200 ft. high, the angle of depres- sion of a buoy is 60. Prove that "-"- -'*- 6 - angle B is 60. Find, by a scale ' drawing, the distance of the buoy $/ from the bottom of the cliff . &/ 1 08. A quadrilateral whose oppo- \ site sides are parallel is a parallelo- ~ i c gram. FlG - l61 1. In the parallelogram, Fig. 162, prove that consecutive angles, as x and y, y and 0, are supplementary. 2. Prove that the opposite angles, x and 0, y and w, of the parallelogram, Fig. 162, are equal. 3. Prove that the sum of the interior angles of a parallelo- / / gram is four right angles. ' / f f Af iB FIG. 162 4. With ruler and protractor draw a parallelogram having adjacent sides 3 in. and 5 in. (Fig. 163), and included angle 60. How many degrees are there in a ? In e ? In b ? In c? Inrf? 142 First- Year Mathematics 5. Draw parallelograms having the following parts. Find the number of degrees in each of the remaining interior angles: ADJACENT SIDES INCLUDED ANGLE (1) 3 in. and 5 in. and 120 (2) 3 in. and 4 in. and 90 (5) 3 in. and 3 in. and 40 (4) 3 in. and 3 in. and 90 109. The following theorems have been studied: THEOREM I. Two consecutive angles of a parallelogram are supplementary, and the opposite angles are equal. THEOREM II. The sum of the interior angles of a parallelo- gram is four right angles. no. A quadrilateral with one pair c! parallel sides is a ^ j trapezoid. ^^ ^o i. In the trapezoid, Fig. 164, prove that FIG. 164 x and y are supplementary. 2. Prove that the sum of the interior angles of a trapezoid equals four right angles. 3. Draw a trapezoid A B C D from the data of Fig. 165. Prolong B C and A D until they meet at O. Show that the angles of triangle O C D are equal to the corresponding angles of triangle O B A, and that the two triangles are similar. 4. In Fig. 165, show that O C:O B=O D:O A, and that CD:BA = OC:OB. / \ FIG. 165 FIG. 1 66 Problems on Parallel Lines '143 5. Draw a triangle ABC (Fig. 166) having an angle of 60. Draw lines B X and C X, making angles as indicated. Prove that A B X C is a parallelogram. 6. Draw a parallelogram, starting with a triangle having an angle of 135; an angle of 90. / 7. Using Fig. 167, in which / D E is parallel to A C, prove that the sum of the interior angles of a triangle is two right angles. 8. What is the sum of the ex- terior angles (formed by prolonging a side) of a triangle taking one at each vertex ? 9. From Fig. 168, prove that an T\_ exterior angle of a triangle (formed FIG. 1 68 by prolonging a side) equals the sum of the interior angles not adjacent to it. Algebraic Exercises on Geometric Figures i. Two parallels and a transversal make angles that may be designated as shown in Fig. 169. Find the values of x, and of all the unknown angles. 'SO' FIG. 169 FIG. 170 2. Two parallels and a transversal form angles that may "V be designated as shown in Fig. 170. Find x and all the 8 angles. >^ 3. With parallels, and angles as shown FIG. 171 in Fig. 171, find x and all the 8 angles. 144 First- Year Mathematics 4. Two parallels are cut by a transversal making a pair of alternate interior angles of 5^+3 and 59 3y. Find y and all the 8 angles. 5. With two parallels and a transversal, a pair of corre- sponding angles are x + 2y degrees and 2(x y) degrees, the angle adjacent to the latter being 120. Find x, y, and the unknown angles. 6. With two parallels and a transversal, a pair of alternate exterior angles are $y zx, and gx+y. The angle adjacent to the latter is 86. Find x, y, and the unknown angles. 7. With two parallels, the interior angles on the same side of a transversal are 6x+y degrees and 14^ y degrees and their difference is 14. Find x, y, and all the 8 angles. 8. With two parallels, the interior angles on the same side of the transversal are 4X y and $(2y+x). $(2y+x) and 125 are alternate interior angles. Find x, y, and the unknown angles. 9. With parallels, transversal, and angles as shown in Fig. 172, find x, y, and all the 8 angles. FIG. 172 FIG. 173 10. With parallels, transversal, and angles as shown in Fig. 173, find x, y, and all the 8 angles. n. With two parallels and a transversal, the alternate ex- terior angles are 7(^ + 1) degrees and iSi 2X degrees. Find x and all the 8 angles. Problems on Parallel Lines 145 Problems on Construction in. The following geometrical figures are to be constructed by means of compasses and unmarked straight edge only. 1. To construct a line perpendicular to a given line, AB, at a given point, C, on the line AB. CONSTRUCTION: With C (Fig. I 174) as center and a convenient I radius, draw arcs of a circle meeting ^W* A B in two points as D and E. With D and E as centers and a ( I t convenient radius, draw arcs of two x circles meeting as at F and G. Connect by a straight line one of the points as F with C. FlG ' x ? 4 The line F C is the required perpendicular to A B at point C. How long must the radius D F be ? 2. Construct a right triangle having given the two sides, 3 in. and 4 in., which meet at the vertex of the right angle. A rectangle is a parallelogram one of whose angles is a right angle. 3. To construct a rectangle, having given the length / and the width w. 1 CONSTRUCTION: On any convenient line, A B (Fig. 175), lay off C D equal in length to /. At C and D construct the perpendic- ulars C E and D F. On C E and D F lay off the lengths C G and D H each equal to w. Draw G H. a The quadrilateral C G H D is the FIG. 175 required rectangle. 4. Construct the complement of a given acute angle ABC. 5. Construct a square having given one side s. 6. Construct two lines perpendicular to a given line and prove the two lines parallel. !46 First- Year Mathematics 7. Draw a right triangle. Construct a rectangle having for two of its sides two sides of the triangle. 8. From a paint, C, outside of a line, AB, to construct a line perpendicular to AB. Construct CF (Fig. 176) perpendicular to A B, following the directions given in the A* 5 construction for problem i. FIG. 176 FIG. 177 9. Construct a perpendicular from C to A B when C and A B are as in Fig. 177. 10. Draw a triangle of which all angles are acute. From the vertex of each angle construct a line perpendicular to the opposite side. It is proved in geometry that the three altitudes of a triangle meet in a point. This gives a test of the accuracy of the constructions in problems 10, n, and 12. Crease the three altitudes of a paper tri- angle and find whether they all pass through the same point. 11. Draw a triangle having an obtuse angle and construct perpendiculars as in problem 10. 12. Draw a right triangle and construct perpendiculars as in problem 10. 13. Draw a parallelogram, as A B C D (Fig. 178). Con- struct the altitudes from B and C to the base A D ; from D to the base A B. -f f FIG. 179 14. Angle F (Fig. 179) is a right angle and h is the altitude from F to D E. Find the values of x, y, and z. Problems on Parallel Lines 147 15. To bisect a given line, A B. CONSTRUCTION: Using the end points A and B as the points D and E were used in problem i, draw a line, as F G, perpendicular to A B. The point of intersection H is the I required midpoint of A B. The line F G is called the perpendicular ** iH a bisector of A B. 1 6. Draw a triangle. Bisect the y*^ sides and join the midpoint of each FIG. 180 side to the vertex of the opposite angle. A straight line drawn from the midpoint of a side of a triangle to the vertex of the opposite angle is a median. Show, by creasing the 3 medians of a paper triangle, how the accuracy of the construction of the three medians of a triangle may be tested. 17. Draw a right triangle. Construct the medians and show, by drawing a circle on the hypotenuse SLZ a diameter, that the median to the hypotenuse equals one-half of the hypotenuse. 1 8. Draw a right triangle, a triangle having an obtuse angle, and a triangle of which all angles are acute. In each of the triangles construct the perpendicular bisectors of the three sides. What seems to be the test for the accuracy of the construction? Crease the perpendicular bisectors of the three sides of a paper triangle and see whether the test is met. 19. To bisect a given angle ABC. CONSTRUCTION: With B -as center and any radius, draw arcs intersecting B A and B C in two points as D and E. With D and E as centers and a con- venient radius, draw arcs meeting as at F FIG. 181 an d G. Connect by a straight line one of these points, as F, with B. The line B F is the required bisector of angle ABC. ! 48 First- Year Mathematics 20. Draw a triangle. Construct the bisectors of the three angles of the triangle. What seems to be a test for the accu- racy of the construction ? Find a test for the accuracy of the construction by creasing the bisect- ors of the angles of a paper tri- '** angle. 21. Construct the bisectors (LO A and M O) of two supplementary adjacent angles (Fig. 182). Prove that the bisectors are perpendicular to each other. 22. Construct the bisectors of a pair of corresponding angles made by a transversal cutting two parallels, Fig. 183. Prove that the bisectors are parallel. \ \ \ \ r- A * ' \ FIG. 184 FIG. 183 23. Construct the bisectors of a pair of interior angles on the same side of a transversal cutting two parallels (Fig. 184). Prove that the bisectors are perpendicular to each other. 24. At a given point, A, on a line, B C, to construct an angle equal to a given angle, D E F. CONSTRUCTION: With E as center (Fig. 185) and any radius, draw an arc meeting E F and E D in the points, G and H. With A as center and the same radius, draw the arc M N. Problems on Parallel Lines 149 With M as center and radius equal to the distance from H to G, draw an arc meeting M N at O. Draw A O. Then O A M is the required angle. 25. Draw a line parallel to a given line passing through a point outside of the given line. Construction is as in problem 2, p. 138. Instead of using the pro- tractor to construct the angle, ruler and compasses are to be used. 26. Draw a triangle. Con- struct an angle equal to the sum of the angles of the triangle, accuracy of the construction ? 27. Draw a triangle. Construct a parallelogram having for two of its sides two sides of the triangle. 28. To construct triangles having the same base and equal areas. CONSTRUCTION: Construct a line C D (Fig. 186) parallel to A B. FIG. 185 How can you test the FIG. 1 86 Connect, by straight lines, the points A and B with points on C D as E, F, G, etc. Show that the triangles thus formed have equal altitudes and equal areas. 29. Draw a triangle. Construct another triangle having two sides and the included angle equal to two sides and the included angle of the first. Compare the triangles as to size by cutting out one of them and fitting it on the other. 150 First-Year Mathematics 30. Draw a triangle. Construct another triangle having one side and the two angles adjacent to that side equal to a side and its adjacent angles of the first. Compare the triangles as to size. Algebraic Exercises on Geometric Figures i. The angles made by two pairs of parallels intersecting as in Fig. 187 are designated as shown. Find x, y, and all 4 angles about the crossing-point, K. FIG. 187 2. With two pairs of intersecting parallels and angles as shown in Fig. 188, find x, z, and all 4 angles around any crossing-point. FIG. 1 88 3. With the sides and angles as shown in Fig. 189, find y, z, a, and 6, and all the 4 angles and 4 sides of the parallelo- gram. Assume that the opposite sides of a parallelogram are equal. FIG. 190 FIG. 189 4- In the trapezoid of Fig. 190, find x, y, and all 4 angles. Problems on Parallel Lines 5. In the trapezoid, Fig. 191, find x, y, and all 4 interior angles. /tf*~ 6. The non-parallel sides of a / trapezoid are extended until they yffiy y. meet, Fig. 192. The lengths of FIG. 191 lines being designated as shown, find x and y. 7. The non-parallel sides of a trapezoid being prolonged to intersect and the lengths of lines being designated as shown in Fig. 193, find x and y and M FIG. 192 the lengths of all lines. 8. The non-parallel sides of a trape- FIG. 194 FIG. 193 zoid are extended to meet, Fig. 194. Find the lengths of lines, b, c, and the area of the trapezoid in terms of a. 9. Find all angles of Fig. 194, lengths of lines being as shown and the angle at the top, 53 8'. SUMMARY 1. Parallel straight lines are lines that have the same direction or opposite directions. 2. If two lines are cut by a transversal, making a pair of corresponding angles equal, the lines are parallel. 3. Two lines that are perpendicular to the same straight line are parallel. 4. If two parallel lines are cut by a transversal, (1) the corresponding angles are equal; (2) the alternate-interior angles are equal; (3) the sum of the interior angles on the same side of the transversal is two right angles. i 2 First- Year Mathematics 5. A parallelogram is a quadrilateral whose opposite sides are parallel. 6. Two consecutive angles of a parallelogram are supple- mentary, and the opposite angles are equal. 7. The sum of the interior angles of a parallelogram is four right angles. 8. An exterior angle of a triangle (formed by prolonging a side) is equal to the sum of the two interior angles not adjacent to it. 9. A trapezoid is a quadrilateral with one pair of parallel sides. 10. The sum of the interior angles of a trapezoid is four right angles. 11. To construct figures by the aid of compasses and unmarked straight edge only, the following fundamental con- structions are used: (1) To construct a perpendicular to a line at a point on the line; (2) To construct a perpendicular to a line from a point without; (3) To bisect a line; (4) To bisect an angle; (5) To construct an angle equal to a given angle. 12. A median of a triangle is a straight line drawn from the mid-point of a side to the vertex of the opposite angle. CHAPTER VIII THE FUNDAMENTAL OPERATIONS APPLIED TO INTEGRAL ALGEBRAIC EXPRESSIONS Addition of Monomials 112. The tickets for a football game are sold at 25^ by John, Henry, Kenneth, William, and James. They report sales as follows: John sold 56 tickets, Henry 75, Kenneth 27, William 83, James, 69. At the gate 123 tickets are sold. Find the total receipts. SOLUTION I: John, 56X25? = 75X25?: $14.00 Henry, 75X25?= 18.75 Kenneth, 27X25?= 6.75 William, 83X25?= 20.75 James, 69X25?= 17.25 Gate, 123X25?= 30.75 Total receipts $108.25 SOLUTION II: John, 56 tickets, 56X25? Henry, 75 tickets, 75X25? Kenneth, 27 tickets, 27X25? William, 83 tickets, 83X25? James, 69 tickets, 69X25? Gate, 123 tickets, 123X25? 433 tickets, 433X25? = $108. 25. In SOLUTION II the addition is simplified, because the dif- ferent terms to be added have a common factor and they were therefore added by adding the coefficients of this factor in the different terms. i. The tickets being sold at xa*b + gab* + sas 463 - S, we write: ( 4a 2 6 + 706* I2a3 25^3) (i8a2& + 9a6 +Sa3 4b3), which is equivalent to: ( 4a 3 b + jab* 1 203 2563) + ( + i8o 2 & 906* 503 + 4b3) or: 43 1203 25&3 + i8a 2 & 9oZ> 1703 + I4o 2 & 20& 2 2163. EXERCISE XIX Simplify: 2. ( $a 2 x+iobxy+24b 2 y iSaxy) ( 3. (-9m 2 pq Fundamental Operations and Algebraic Expressions 161 4. 5. (45x 2 y 2 -27x4 +8i;y4) - (73*4 +45y 6. (+4X 3 ~- 7- ( 3 --& 8. (3 . 4^354 5 . 7^53 -f 9 . 8l> 6 S 2 ) ( I . 7^ 4 S3 3 . 2V 6 S 2 2 -6gh 3 + 25/^4) 12. ( 2 13 14. ( 5wi> 2 $mvu -\-4mu 2 ) 1 20. Compare the signs of the terms of the subtrahend in the separate exercises of EXERCISE XIX, before and after the parentheses are removed. 1. State a rule as to the effect of a minus sign in front of a polynomial in parentheses. 2. State a similar rule as to the effect of a plus sign. EXERCISE XX Perform the following operations and simplify results: 3. 4. 2/-[6/-3g-4/-(2g-4/)l 5. 4 ; 2 _!/ 2 _ 6- 9*y- 7. - 1 62 First- Year Mathematics 8. 7-5/ >3 --[3-4/> 3 -4-2/' 3 + i-6/> 2 -3-4/' 2 -4. 9. 2 k 3 -\2p 2 +k 3 -p a +r\-[p 2 -r-(k 3 +r)} 10. 11. -r-3r'-3 2 -4r*-sr- 12. - Multiplication of Monomials 121. The area of a rectangle and the surface and volume of a rectangular block can be easily found when the dimensions are known. 1. Find the areas of rectangles, whose dimensions are: (1) 5 in. and 7 in. (6) m 2 yd. and m yd. (2) 13^ cm. and i6 cm. (7) a 2 b mi. and b 2 c mi. (3) 7 ft. and a ft. (8) 2y 3 km. and sy 2 km. (4) x m. and 9^ m. (9) $\x 2 y ft. and $%xy 2 ft. (5) p rd. and q rd. (10) 4 . $abc in. and 3 . 2abc in. Illustrate the problems by drawings on square-ruled paper when- ever possible, and check results by counting the number of squares con- tained in each rectangle. In the above and in all the following problems with literal numbers test the values obtained by assigning numerical values to the letters. For example, the area of a rectangle of dimensions $ab and zbc is 6ab 3 c. Assigning to a, b, and c, the values 2, 3, and 4 respectively, the dimensions and area are 18, 24, and 432 respectively, which is correct, since 18X24 = 432. 2. Find the area of a square, whose edge is: 15 ft.; 8^ m.: a 3 cm.; 7.56^ in.; x 6 in.; a 3 b 2 mi.; &p 4 q 5 m.; f jx 3 y''z ft.; 4.T,2a 2 b s c 4 dm.; qmn 2 p 3 q 4 cm. 3. Find the volume and area of a cube, whose edge is: 7^ cm.; 2x 2 ft.; $%ab in.; 2.4^3 m .; ^xyz cm.; 5 . 6x 2 y 3 z 2 mi. : . ; I2a 2 6 2 yd.; 4^s ft.; i$p 3 q 2 r m. Fundamental Operations and Algebraic Expressions 163 4. Find the area and volume of a rectangular block, whose dimensions are: (6) 3.5^3, 4 . 504, ! 60s cm. (7) x 2 yz, xy 2 z, xyz 3 ft. (8) 2amp, $bpm, ^cmp in. (9) 4* 8 , 2j*', ^ m. (10) $x 3 y 2 , 4^x 2 y 3 , %x 4 y* cm. 5. We have seen (p. 21), that in x$, 5 is called the exponent, x is called the base, and x$ is called the 5th power of x. Give other examples of different powers of the same base. 6. In what way are powers of the same base multiplied? Can powers of different bases be multiplied in the same way? (1) 4, 9, and 15 in. (2) 3 J. 5 j, and x ft. (3) *> 8 > y m - (4) a 2 , a 3 , a 4 rd. (5) X 3 , 2X 2 , 5*5 yd. FIG. 195 7. What is the volume of a rectangular block of dimensions a, b, and c ft. ? b, c, and a ft. ? b, a, and c ft. ? c, b, and a ft. ? c, a, and 6 ft. ? a, c, and b ft. ? What is the effect of the order in which the factors are multiplied upon the value of the product ? 8. What is the total volume of 7 rectangular blocks of dimensions x, y, and z ? What is the volume of one rectan- gular block of dimensions jx, y, and z ? x, "jy, and z ? x, y, and 7z ? 'jx, jy, and z ? 7*, y, jz ? #, 7^, 72 ? 7*, 7^, 7Z ? 164 First-Year Mathematics 9. Which of the following expressions are equal ? (1) 7(*-?-) (5) 5(4-3-6) (9)8(7-6-5) (2) ys-yyyz (6)5.3-5.6-4 (10) 8 7 -8 6 -8 5 (3) 7* y ' z (7) S ' 4 3 ' 6 (n) 8-7-6-5 (tijx-w-z (8) 5-3-S-4-5-6 (12)8.7-8-6-5. 10. Compare also the following: SaXbXcXd, s^XbXcXd), saXs&Xs^Xs^ 11. State the law of signs for multiplication and interpret the meaning of positive and negative multipliers and multi- plicands (see pp. 77-85). EXERCISE XXI Simplify the following products: 2. _ 3- (-a)*X(-a)3X(-a) 4 4. 7- 8. 9- 10. II. i2X 2 y 3 Z S T.ZX 5 V Z 4 X 3 V 5 Z 2 XOJ ./ xx ^ A 5 4 7 - 5 a m --fias 12. 3 XN 25 13- 24 14 as k IO5* 14- ~ ~~. X 15. Fundamental Operations and Algebraic Expressions 165 16. (r--s) n X(r-s)3 17. (/ 2 -/ 2 )*X(/ 2 -/ 2 )* 18. ( 19. 3 20. -a- T.$x 5 y' ! (x-}-y) s 2ix 4 y s (x+y) 6 21. --- X - H 3 22. (a5) 2 X(-a3)4X(-a 2 )s 23- (3/> 2 ) 4 X(-/> 3 )sX(-5/> 4 ) 2 2 4 . (2i*3)2X(-f#5)* X (3*7) 2 . Division of Monomials 122. We have found the area of a rectangle when both dimensions are known. We shall now find one dimension if the other dimension and the area are known. 1. Find the altitude of a rectangle of area 144 sq. ft., the base being 9 ft.; 16 ft.; 12 ft.; 8 ft.; 72 feet. 2. Denoting the area of a rectangle by A, the base by 6, the altitude by a, find the second dimension of the following rectangles: (1) A=$2t 2 sq. ft.; 6 = 26* ft. (2) -4=#5 S q. in.; b=x 2 in. (3) A=4%x 4 y 3 z* sq. m.; a=3# 2 ;y 2 z a m. (4) 4= (5) A= (6) A=< (7) A= (8) A= (9) A = i . 96z 3 # 5 /> 6 ; b = i.4Z 2 p 2 x 3 (10) A=&$xsy 6 zi; a = i$x3y4zs. 3. State a short way of dividing powers' of the same base. 4. State the law of signs for division (see p. 86). 1 66 First- Year Mathematics EXERCISE XXII Reduce the following fractions to lowest terms: 2 (x 2 y 2 ) 3 - 3- i2(-a)46s(- c )6 4 ' 3(- I4r ' -28 -i2X l8 y'^z 1 '' l ' 6a*b 10 cv '' ' Multiplication and Division by Means of Exponents 123. Make tables of the first twelve powers of 2, 3, and 5; thus: 2=2 X 3=3' 5=5 r 4=2 a 9=3 2 25=5* 8=2 3 27=33 I2 5 = S 3 i6= 2 4 81=34 625 = 54 32=25 243=35 3> I2 5=5 S Fundamental Operations and Algebraic Expressions 167 64 = 26 7 2 9 = 3 6 15,625 = 56 128 = 27 2,187=37 78,125 = 57 256 = 28 6,561=38 390,625 = 58 512 = 29 I9)6 83= 3 9 1,953,125 = 59 1,024 = 2" 59,o49=3 10 9>765,625 = 5 10 2,048 = 2" 177,147=3" 48,828,125 = 5" 4,096 = 2" 531,441=3" 244,140,625 = 5" 1 . Using the tables, write as powers of 2, 3, or 5 : 16 6,561 i77,U7 2,187 243 729 1,024 128 15,625 390,625 48,828,125 2. Using the above tables, many large multiplications and divisions may be simplified, as will be seen from the following examples: (1) 32X64 = 2 5 X2 6 = 2" =2,048 (2) 15,625X625 = 56x54 = 5^=9,765,625 (3) (7 2 9) 2 = (3 6 ) 2 =3 I2 = 53 I >44i (5) (625) 3 = (5 4 ) 3 =5 12 =244,140,625. EXERCISE XXIII Carry out the following multiplications and divisions, using the tables of powers of 2, 3, and 5: 1. 6,561X81 7. 390,6254-3,125 2. 78,125X625 8. 244,140,625^-78,125 3. 512^-16 9. (243) 2 4. 729X243 10. (15,625)' 5. 2,0484-128 ii. 4,0964-64 6. 177,1474-729 12. 390,625X625. 1 68 First- Year Mathematics For the following problems the above tables have to be extended to the 25th powers: 282,429,536,481 I3 ' 3,486,784,401 14. 15,625X78,125X9,765,625 15. 2563 847, 288,609,443X53^441 ' 31,381,059,609X14,348,907 30,517,578,125X3,814,697,265,625 10,670,928,955,078,125 g 16,777,216X1,024X262,144 2,097,152X8,192 Multiplication and Division of a Polynomial by a Monomial 124. What is the total area of four adjacent flower-beds, whose length is 10 ft. each, and whose widths are 7, 2, 5, and 3 ft. respectively? The total area may be expressed in either of the two following ways : 10(7 + 2 + 5 + 3) or 10X7 + 10X2 + 10X5 + 10X3 Thence: 10(7 + 2 + 5 + 3) = 10X7 + 10X2 + 10X5 + 10X3. The area may thus be expressed as the product of a polynomial by a monomial or as one polynomial. 1. Represent in two different ways the total area of 7 adjacent rectangles having the same base b and having lengths 5, a, x, i\, 4^, 9.5, and y 2 respectively. Express by an equa- tion the equality of these two representations. 2. Represent in two different ways the combined area of 4 rectangles of length a 2 and of bases a 3 , 306*, 3a 2 6, b 3 respec- tively. Express the equality of the two representations. 3. How may the product of a polynomial by a monomial be reduced to a polynomial ? What do you notice concerning the terms of the resulting polynomial ? Fundamental Operations and Algebraic Expressions 169 EXERCISE XXIV Reduce to one polynomial: 2. a 3- -5 . 4. a 5. 6. 7 . ^x 2 ($x 2y) 2xy(T > x 2y) + 6y 2 (T,X 2y) 8. 9. 5#J4a-2(3a-4&) +5(40-36) [ 10. 2a)5(4a-76-3c)-6(5a + 46- 11. -4*[2* 2 + 3x|4(*-i) -5(^-2) f] 12. Solve the following equations: 13. 14- 15. $x(4X + 2y+6) +$x(']x+6y 9) (x a +&xy *]x} = 100 16. 125. To factor a number is to find the numbers which, multiplied, give the number to be factored. 1. The total area of three adjacent lots is x 4 + $x 3 y+4X 2 y 2 sq. m., each term representing the area of a lot. The lots all have the same length. What may their dimensions be ? 2. Sketch a rectangle whose area is: (1) 4^-3^ (2) $x 3 - (3) i4a 2 b 3 (4) 3w s (5) 1 5* 4 170 First-Year Mathematics 3. How can some polynomials be reduced to the product of a polynomial by a monomial? Can every polynomial be factored in this way? EXERCISE XXV Write the following polynomials as the product of a poly- nomial by a monomial: 1. 5a io& 2. i7# 2 289*3 3. i6x 2 2abx 4. ax+ay az 5. i4# 2 7 2 z 2 'jx 3 y 3 z 2 6. 6om 2 n 3 r 2 7. 4 8. 4 9. a 10. x 3 pqx 2 +p 2 q 2 xpq 3 x 3 . 126. When one of the factors of a number is known, a second factor is found by dividing the known factor into the given number. 1. The area of a garden is T.$abc + 2"ja 2 c2ib 2 c. The length is $c. Find the width. 2. Find the missing dimension of the following rectangles: AREA BASE ALTITUDE (1) 54+57 5 3 (2) &x 2 y 2 ioxy 3 4# 4 2X (3) pk+lk-rk+tk k (4) mu 2 +mv 3 +mw 4 mz m (5) as -\-bs- cs+ds s 3. How may the quotient of a polynomial by a monomial frequently be reduced to a polynomial ? Is this always pos- sible ? Fundamental Operations and Algebraic Expressions 171 EXERCISE XXVI Reduce: i. 2. Sab iox 2 y i $x 3 y 2 + $x 3 y -6x 2 y liab 4- >jab 3 c 3 5- I -^- ""' 5a 2 b 2 c 2 'jab 2 c+ 6ab 4 c) 7 ' " ~ 22 ' A /v"2/\i2\ S M 2 ( r* mn 2 (iT > \n gm) 10. (3)- 2 -45*;y+6:x: 2 )(3;y 2 -5* 2 ) n. (a#+6_y cz)(ax+by+cz) 12. (a^+ft^ cz) 2 13. (a^c by cz) 2 14. (a^c 6v+cz) 2 15. (fyns \sl %tm)(6m 125 + 18^) 16. (i .4/ 2 2.5w/+.9m 2 )(.2/ 2 /w .iw 2 ) 17. (a# 6j+cz) 2 (ax+by+cz) 2 18. (a+&)3 19. (a -6)3 20. 2^3 21. 22. 23. 24. 2 5 . Division of Polynomials 131. In the division of polynomials, all the preceding operations find application. It is therefore a subject by means of which the foregoing work may be reviewed. i. Write the product of x+y by a+b+c in the form of a polynomial. What is the contribution of a and x+y to the product? Of b and x+y ? Of c and x+y ? i-jB First-Year Mathematics From what terms of the multiplier and the multiplicand is the term ax of the product obtained ? 2. What is to be found by dividing ax+ay+bx+by-\-cx + cy by x+y? How is the first term of the quotient found ? How is the total contribution of the divisor and the first term of the quotient obtained ? Write down the remainder of the dividend, which is con- tributed by the divisor and the remaining terms of the quotient, as yet unknown. 3. Determine the 2d term of the quotient from the re- mainder of the dividend by the same method as was used in problem 2 to determine the first term of the quotient. Then proceed as in problem 2. Repeat this process, until all the terms of the dividend have been used. The following arrangement of the work is convenient: a+b + c x + y/ax + ay + bx + by + cx + cy First partial product: a(x+y) ax + ay bx + by + cx+cy Second partial product: b(x+y) bx + by cx+cy Third partial product: c(x+y) cx+cy 4. Compare the partial products of problem i at the beginning of this section with the partial products in the division. 5. Divide a3+3a 2 &+3a6 2 +6 3 by a+6. What is the first term of the quotient? How is the ist partial product found? What is the 2d term of the quotient ? How is the 2d partial product found ? etc. Fundamental Operations and Algebraic Expressions 179 EXERCISE XXXII Divide, and test by multiplying:* 1. qt 2 + 24st+i6s 2 by 3/+4S 2. i .2X 2 + .$xy 2.&y 2 by ^x+jy X 3. 6&3_ 3 i*/+47& 2 -42/3 by 2&-7* *4- lox* $xy 2 543/3 ^y by x 2y 5. 27 2 n. 8# 3 , at r%, for t years, show by an equation how i, p, r, and / are related. i>rt 47. Divide both sides of the equation, i = , by rt, then X OO multiply both sides of the resulting equation by 100, and inter- pret. prt 48. Divide both sides of *=- by pr, then multiply both 100 sides by 100, and interpret. prt 40. Multiply both sides of i= by 100, then divide by 100 pi, and interpret. What is the product of a fraction by the denominator ? Illustrate. 134. Equations, such as d=m s, P=m-M, D=q*d-\-r, n t n 2 n I d 2 n 2 d I , n 2 n^, br = - - + = - > P = , etc., are called a l a 2 a.! a a a, a 2 a^n 2 100 formulas; because they express laws of number in brief form, that is, they formulate laws of number. 1 88 First- Year Mathematics Graphing Percentage and Interest 135. The laws of percentage may be more clearly shown graphically. PERCENTAGE 1. What is 2% of $100 ? Of $200 ? Of $300 ? 2. Let a vertical side of a small square (Fig. 198), denote $2, and mark upward from the horizontal O X, on the verticals r^ 2 2 2 / f / /> / ^<^ ^ / ^ pf / t .-^ ^ 2 ^ f [2 ^ T7H jMii !> AIM }>* FIG. 198 through $100, $200, $300, $400, etc., the percentages at 2% of $100, $200, $300, etc., as shown. Draw a line, as O P, through the tops of the lines. What kind of line does O P seem to be ? 3. Starting again on the $100- vertical, mark off similarly distances above O X to show 4% of $100, of $200, of $300, of $400, etc., to $500, and connect the upper ends with a con- tinuous line, as O M. 4. Draw a similar line for the percentages at 3% of the same amounts. 5. Draw a similar percentage-line for i% of the same amounts. 6. Through what point do all these percentage-lines pass ? What does this mean ? Practice in Algebraic Language 189 7- How does the percentage-line at 3% lie with respect to the lines at 2% and at 4% ? 8. Show that the line O P might be represented by percentage = .02 X base, or more briefly, by p= . O2X&, Or />=i^y&. 9. Write similar equations for the 4%-line; the 3%-line; the i%-line. 10. Write similar equations for a 5%-line; a io%-line: a 6%-line; and an r%-line. 11. In Fig. iy8 what does the line, AB, drawn midway between the $300- vertical and the $400- vertical from O X up to O P, represent ? What would A C represent ? E F ? EG? 12. How would you read from the figure the percentage at 4% for $150 ? For $250 ? For $450 ? 13. Draw a number of equally spaced verticals upward from the horizontal and mark the lines o%, i%, 2%, etc., as shown. Taking $500 as a base, measure upward from O X on the i %- vertical, i% of $500, or $5, letting a side of a small square division denote $5. This gives the point a. On the 2 %-line measure and mark off 2% of $500; on the 3%-line, 3% of $500, and so on. Connect a, b, c, d, etc., by a line. 14. How could one read from the drawing 2^%, 2 J%, 4$%, .... of $500 ? 15. Taking a base of $250 construct a $25o-percentage line. 1 6. Construct a $ioo-percentage line. 17. Show that the $5oo-percentage line might be expressed as an equation, thus: Percentage = $500 X rate, $5oor or p=-2 -, 100 or />=$5r. FIG. 199 First- Year Mathematics 1 8. Write a similar equation for a $ioo-percentage line; a $4oo-percentage line. 19. All three lines go through the o-point. What does this signify? 136. The laws of simple interest may be shown and interest- problems may be solved graphically. 1. Find the interest on $100 at 3% for i year; 2 years; 3 years; 4 years. 2. Call a side of a small square $3, mark a point, as a, to show $3 on the i-year vertical. Mark off $6 on the 2 -year vertical, as at 6; $9 on the 3-year vertical, etc. Through the points a, b, and c, draw a straight line and extend it. This is the 3%-line. 3. Tell from the drawing the interest on $100 for i years at 3%5 f or 2\ years at 3%; for 3$ years at 3%. 4. Calculate the interest on $100 at 5% for i year; 2 years; 3 years; 4 years; etc., and draw the 5%-line. 5. From the 5%-line give the interest at 5% on $100 for i \ years; for 2 \ years; for 3^ years. 6. Calculate three points and draw the 8%-line. 7. All these lines go through the zero-point, o. What does this mean ? 8. Explain how the lines show that the interest at a given rate on $100 increases proportionally to the time. Such work as that above is called plotting, or graphing. Percentage-lines and interest-lines are called graphs. 137. Graphing Statistics Expressed as Percentage. i. Taking the average price of food in the United States from 1899 to 1900 as 100, the relative prices from month to / / / / / c / / 1 4 a / / i 3 ( FIG. ytr line. 2OO Practice in Algebraic Language 191 month for 1905-6 and 1907 are given below. Graph the values for 1905-6-7 on a sheet of rectangle paper as shown for 1905. Plot the tenths by estimate. Month r 1905 i car Igo6 1907 January 115.6 II7.0 121 .6 February 115.8 116.2 121 .1 March 113.8 II5.0 II9.6 April 112. II3-9 Il8. 3 May IIO-7 II3.0 II7.6 June 109.8 II3.0 II7.8 July 109.7 II3-3 II8.4 August IIO. I II3.8 II9-3 September 110.8 II5.2 121 .1 October 112. 1 II7.I 123-4 November II3.8 II9.4 124.2 December II5.2 121 .2 125.0 116 IU at at as 06, JU 111 JJO m xe lot 101 iaa 2 / I ** ' x. 1 V f 2 3 J / T V- / \ _) 3 J \ I \ / \ / \ f \ f S 5 / * \ / \ Jt s >fc- IHttt * ~ff> .ii : , a 190S 7 i/aM. Apr July Vet , means is greater than. The length of the curve of a circle is about 3^ times the diameter. More accurately, the ratio is 3.14159. For con- venience, the symbol ?r (called pi) is often used to denote the correct value of the ratio of the circle to its diameter. The Triangle 144. To find the distance straight through a hill from A to B, a point, C, was chosen on level ground, so that the 202 First-Year Mathematics distances, A C and B C, could be measured. C D, in the prolongation of B C, was made equal to B C, and C E, in the prolongation of AC, was made equal to A C. The points, D and E, were marked with stakes, and DE was measured and found to be 68 . 5 rods. 1 . How long is A B ? 2. What is the relation of angles x and y ? 3. Recalling how D and E were located, what parts (angles and sides) of triangle ABC are known to be equal to certain parts of triangle C D E ? 4. What relation as to form and size do the whole triangles ABC and C D E seem to have ? 145. THEOREM: // two sides and the included angle of one triangle are equal, each to each, to two sides and the included angle of another triangle, the triangles are equal in all respects. Draw a triangle, as A B C, and make another triangle as A'B'C', having a'=a, c'=c t and angle B= angle B'. Trace triangle ABC through a piece of thin paper, or on tracing paper, and fit the trace over triangle A'B'C' by placing the trace of a along a', of B on B', and of c on c'. What seems to be true of the relations of b and V ? and ZC'? It is not necessary always actually to do all this fitting and measuring. It is quicker and better to reason out the equality of the triangles, thus: Imagine triangle ABC carried along and placed over tri- FlG. 2IO Of /.A and Z4'? Of ZC Practice in Algebraic Language 203 angle A'B'C', so that point B falls on point B', and a lies along a'. As angle B= angle B', c falls along c'. As a=a f , point C must fall on point C'. As c=c', point A must fall on point A'. b and b f must then be one and the same straight line (i. e., they must coincide), because only one straight line can be drawn connecting two points. Consequently, the two triangles, ABC and A'B'C', can be made to fit, or coincide throughout and hence they are equa) in all respects, or If a=a', then, b=b', c=c', ^A = ZA f , andZ=Z', andZC=ZC'. 1. If a=a', b=b', and ZC=ZC', compare c and c', Z.A and Z.A', /_B and Z'. 2. If b=b', c=c', and /_A = Z.A', compare a and a', Z.B and Z_B', ZC and ZC'. 3. With the triangles of Fig. 211 prove that (1) if a=a', c=c', and y=y'\ then b=b', x=x', and z=z' (2) if c=c', b=b', and x=sc f ; then a=a', z=z', and y=y (3) if a=a', b=b f , and z=z f ; then c=c', x=x f , and ^=y. A triangle that has two equal sides is an isosceles triangle. FIG. 211 4. Draw the bisector of the angle A of the isosceles tri- angle, Fig. 212 (c=b), and prove that angle 5= angle C. 5. Prove that the angle-bisector of angle A bisects a, at D. 2O4 First- Year Mathematics 6. Prove that the angle-bisector of A is perpendicular to side a. 7. Prove that if the sides including the right angle of a right triangle are equal, each to each, to the sides including the right angle of another right triangle, the triangles are equal in all respects. A' FIG. 213 8. The span of a roof is 48 feet, and c and b are 32-foot rafters. What is the rise AD? (A D is the angle-bisector of A.) 9. The span of a roof is 25, the length of the rafters, r. Find the rise h. 146. A triangle that has all sides equal is an equilateral triangle. 1. Prove that an equilateral triangle is equiangular (all angles equal). Apply problem 4, p. 203. 2. The length of a side of an equilateral triangle is a. Find the length of the angle-bisectors, shown in Fig. 215. 3. How many degrees are there in one of the angles of an equilateral triangle ? Let x denote the value of one of the angles, then 3* = 180. Why ? 4. How many tiles of the shape of equilateral triangles will just cover the plane if placed as shown in Fig. 216 around a point as O ? Practice in Algebraic Language 205 5. Draw 6 equal equilateral triangles in positions as shown in Fig. 217. What kind of figure is formed ? Compare the lengths of the sides of the large figure. Compare the angles of the large figure. Compare the distances from O to each of the corners A, B, C, etc. A plane figure that is bounded by 6 equal sides making equal angles is a regular hexagon. 6. If O were used as a center and a circle were drawn with O A as a radius, through what points would the circle pass? 7. How many chords equal in length to the radius of a circle, if placed end to end, will reach around the circle ? 8. How may a regular hexagon be readily constructed by the aid of a circle ? 9. Six lights are placed equal distances apart around a circle of radius r. Find the distances from any one light to each of the others. L B and A C bisect each other perpendicu- larly. The distance L A, Fig. 218, is the radius of the regular hexagon. The distance LN perpendicular to A B, Fig. 218, is the apothem. 10. Find the area of an equilateral triangle whose side is 1 2. 11. Find the area of an equilateral tri angle whose side is a. Fig. 219. 12. Find the area of a regular hexagon, one of whose sides is 10. FIG. 219 FIG. 218 206 First-Year Mathematics 13. Find the area of a regular hexagon (Fig. 220), one /of whose sides is s. 14. A wheel of 85 cogs works into a wheel of 27 cogs. In how many revolu- tions, x, of the larger does the smaller gain g revolutions ? 15. A wheel of a cogs works into a f wheel of b (a>b) cogs. In how many FIG. 220 revolutions, x, of the larger, does the smaller gain g revolutions? 16. Draw two concentric circles of radii 6 and 10. Find the length, /, of the chord of the outer circle that is tangent to the inner circle. A radius, drawn to the contact-point of a tangent, is perpendicular to the tangent. See p. 201. 17. Draw two concentric circles of radii r and R (rI d-2 a^dy n l n, Wj d, Dividing one fraction by another is shown by -4- = ~ X . 208 First-Year Mathematics br Percentage is shown by p= . 100 prt Interest is shown by %** . 100 Percentage and interest problems may be solved graphically. Statistical laws may be shown graphically. Problems in motion and mensuration may be solved by substituting in the formulas that express the laws of motion and mensuration. Problems on tlte circle and the spJtere may be solved by substituting in the formulas that express the properties of the circle and the sphere. In a right triangle the square of the hypotenuse is equal to the sum of the squares on the other sides. A circle is a plane curve all of whose points are at the same distance from a fixed point, called the center. The circumference is the length of the circle. A cJwrd is a straight line connecting two points of the circle. A radius (plural ra/di-i) of a circle is a straight line con- necting the center with a point on the circle. A diameter is a chord through the center. A secant is a straight line cutting the circle in two points. A tangent is a line that touches but does not cut the circle, however far produced. A circumscribed square is a square whose sides are tangent to a circle. An inscribed square is a square whose sides are chords of a circle. The symbol < means is less than, and > means is greater than. If two sides and the included angle of one triangle are equal, each to each, to two sides and the included angle of another triangle, the triangles are equal in all respects. Practice in Algebraic Language 209 A regular hexagon is a plane figure that is bounded by 6 equal sides, making equal angles. The apothem of a regular hexagon is the perpendicular distance from the center to the side of the hexagon. The side of a regular hexagon is equal to the radius of the circumscribed circle. The radius of the hexagon is the radius of the circum- scribed circle. The symbol of continuation is , meaning, and so on according to the same law. CHAPTER X THE SIMPLE EQUATION IN ONE UNKNOWN 147. Algebraic expressions may be pictured or graphed, and equations may be regarded as having been obtained by giving to the expressions the particular value o. i. A man walks along a straight road, AB, in the direc- tion of the arrows (-) at the rate of 3 mi. an hour. How far from the house, H, is he 4 hr. before reaching it ? 3 hr. before reaching it ? 2 hr. before reaching it ? i hr. before reaching it ? 1 hr. after reaching it ? 2 hr. after reaching it ? 3 hr. after reaching it ? 4 hr. after reaching it ? FIG. 223 2. Show the times to a scale (i large space = i hour) on a horizontal line, and at each time draw, or measure, a vertical downward to represent to scale (i small space = i mile) his distances below H, and draw verticals upward to represent his distances above H. The Simple Equation in One Unknown 211 The row of points at the ends of the verticals pictures the successive positions of the man with respect to the house, H. 3. Show on the graph how far the man was from the house at/=+Jhr. AU=-hr. At t=o. 4. Show on the graph when the man was 7 miles below H; 4 miles above H. 148. By putting in dots more thickly more distances from H would be represented and the line through all the points represents all distances of the man from H between 4 hr. and +4hr. Since the distances from H for all the times from t hours before to t hours after arriving at H are also represented by FIG. 224 3^, the line drawn through the row of points also pictures the expression 3*. The line through the points is called the graph of 3/. If the distances of the traveler were expressed from a point, P, 3 miles below H, the row of points and the line through them would be as in Fig. 224. The algebraic expression of these same distances is 3^+3. The line through the dots of Fig. 224 is the graph of 3^+3. 2i2 First- Year Mathematics 149. Of course, any other letter, as x, might be used to stand for the times. If x had been used instead of t, the line of dots of Fig. 224 would have been the graph of 3^ + 3. The distances that are represented by the verticals might have been found by substituting 4, 3, 2, i, o, +i, + 2, +3> +4 i 11 turn f r x > calculating the corresponding values -9, -6, -3, o, +3, +6, +9, +12, +15, of 3^+3, and then representing these numbers to scale on the verticals. Graphs may be drawn for any expressions in the same way. 1. Draw a graph of 3/ 3; of 2/+3; of 2^3; of of 2t 2. 2. Draw graphs of the following expressions: (i) 2X+i (4) 3*-4 (7) (a) 2x-i (5) 4*-3 (8) (3) 3* +4 (6) 4* +3 (9) 2* -4- 150. The graphs show: (1) that there are a great many values of any one of these expressions, one, indeed, for every value of t, or of x, we care to substitute; (2) that there is but one place, or point, where the value of the expression, as 3^+3, is o; (3) that the value of x that makes 3^+3, or other like expression, o, is the distance from O in the graph to the crossing-point of the line of points and the horizontal line. This same value of x that makes 3^+3 equal to o can be more quickly found by solving the equation, 3^+3=0, for x, thus: 3*= -3 x= i CHECK: 3 . ( 151 When any equation as 3^6=0 is written down, it may be regarded as merely symbolizing the question: The Simple Equation in One Unknown 213 What value, or what values, if any, of x are there, which substituted in 3^6 will make 3^6 equal to o ? Two kinds of equation are used in algebra: I. The identical equation, or the identity, is an equation that is true for any, or all, values of the letters. EXAMPLES: i. (x + s)(x 3)=# 2 9. Test whether example i is true for #=4; for # = 7; for # = io; for x = i. Test whether example 2 is true for x=3, a=i; for x = $, a=z; for x = i2, a = 10. Identical equations are sometimes written with the sign =, when it is desired to distinguish them from other kinds of equations. II. The conditional equation is an equation that is true only for one or for a definite number of values of its letters. EXAMPLES: i. 5^15 = 25. 2. a;* 16 = 0. 3. *- Test whether example i is correct for x=8; for any other value of x. Test whether example 2 is correct for x=4; for #=4; for any other values of x. Test whether example 3 is correct for x = i; for x = 2; for x=3', for any other values of x. When nothing is said to the contrary, the word "equation" means conditional equation. The left side of an equation is called the left member, or the first member, and the right side, the right member, or the second member. Point out which of the following equations are (i) identical equations, or identities, (2) conditional equations, and give reasons for your answers: 214 First- Year Mathematics 1. x 1=0 10. a(b 5)=o6 50 2. * 2 -4=o ii. 3. (#-2) 2 = 4. * 2 -i6 = 5. a(x-$)=ax-i > a = 6. a(#-3)=a 4 ~ 4 7. x(x i)=x 2 x <; 7 8. *(*-i)=o I5 ' 5 ' 9. #3-27=0 Z 6. w4-w4 = (w 2 +n 2 )(w 2 -w 2 ). While any letter may be used to denote the primary unknown (or primary variable), it is customary to use x, or y, or z, or some one of the later letters of the alphabet for this purpose. The earlier letters of the alphabet, as a, b, m, I, k, etc., are quite commonly used to denote fixed, or constant, numbers, like 6, 15, 75, etc. The letters that abbreviate words, as i for interest, b for base, are also much used to denote numbers. Stating Algebraic Problems 152. The problems of the foregoing chapters of this book, as well as those of modern arithmetic, show that letters may be used to advantage to express numbers, and to simplify stating and solving problems. Algebraic skill means nearly the same thing as skill in stating and solving problems. This skill can be acquired only through much practice, and not by committing rules and directions. Still, it may aid to know that in algebra stating a problem usually means expressing in the language of algebraic symbols certain number-relations that are given in verbal language. The statement of a problem in most cases takes the form of an equation. To obtain this equation, the following may serve to guide the beginner: I. Denote the unknown number by a letter, then translate the verbal statement of the number-relations into a symbolic statement in equation form. The Simple Equation in One Unknown 215 The first letters of words are convenient letters to use to denote numbers, while they are yet unknown; as n for num- ber, / for time, a for age, etc. Give statements of the following problems, then solve and check them. 1. Two-thirds of a number diminished by 12 equals 4. Find the number. STATEMENT: Two-thirds of a number diminished by 12 equals 4. X n 12 = 4. Solve the equation and check by substitution. 2. Twice a number decreased by 12 is the same as the number increased by f of itself. Find the number. 3. Six times a number increased by ^ of itself equals n. Find the number. 4. Four times a number increased by of itself is the same as twice the number increased by n. Find the number. 5. The acute angles of a right triangle are 4# and 5*. Find the acute angles. 6. One of two supplementary adjacent angles is 3 times the other. Find the two angles. 7. The angles for a triangle are x, 2X, and 33; degrees. Find the numerical values. Another guide that is often useful in stating a verbal problem in equation form is the following: II. By the aid of literal and of arithmetical numbers express some number of the problem in two different forms, and write the two expressions equal to one another. Thus an equation is a statement, in symbols, that two different number-expressions stand for the same number. 8. A rectangle is 3 times as long as wide, and the perim- eter is 48 inches. Find the width and length. Equate two different expressions of the perimeter. 2 1 6 First- Year Mathematics g. Double the number of years in a boy's age is 16 more than his age 2 years ago. How old is the boy ? Equate two different expressions for the difference of double the boy's present age and his age two years ago. 10. The difference of double a boy's age and 3 times his age 10 years ago is his present age. How old is the boy ? STATEMENT IN SYMBOLS: 2x-3(x-io) = x (i) 3(3: 10) = 3^ + 30, hence 2X 3^ + 30= x (2) 2X 3# = x, hence #+30 = x (3) Adding x to both sides 30 = 23; (4) Dividing by 2 15= x. (5) Check by testing in the conditions of the problem. The boy's age is 15 years. 153. The solution of an equation consists in getting another equation in which the unknown stands alone in one member, with the numbers in terms of which the unknown is to be expressed, in the other member. The required changes are made by means of the axioms stated on p. 26, 21, or by means of the fifth axiom now added below. The five axioms that are to be the reasons for changes in equations, are now stated in final form. I. // the same number, or equal numbers, be added to equals the sums are equal. (Addition-axiom.) 11. If the same number, or equal numbers, be subtracted from equal numbers, the remainders are equal. (Subtraction- axiom.) III. // equal numbers be multiplied by the same number, or by equal numbers, the products are equal. (Multiplication- axiom.) IV. If equal numbers be divided by the same number, or by equal numbers, the quotients are equal. (Division-axiom.) Division by zero, or by any expression in which the letters have values that make the divisor zero, is not permitted. The Simple Equation in One Unknown 217 V. Any change may be made in the form of an expression, that does not change its value. One of the most important form-changes is the removal and introduction of parentheses. When a reason for a change is asked, the pupil should be able to quote, or to cite, the axiom that justifies the change. 154. Give the reason for the truth of the conclusion in each of the following: 1. a =9 and &=3, thena+6 = i2 2. m=iT, and n = 2, then mn = 26 3. a =x and b=y, then a-f &=# +;y 4. c =16 and ^=4, then c d = i2 m 5. w = 28 and w = 7, then =4 ft c . d c+d 6. = < and = 1, then =8. 12 12 12 Give reasons for the following: 7. Ifa = 7, then 50=35 u. If #5=3, then #=8 8. If 4=3, then/=48 12. If = 7, then r = 70 16 10 o. If mn=6n. then m=6 b 13. If ax = b, then#=-- 10. If 2y = iS, then y = g a I 55- To solve an equation means to find the number, or the numbers, which, substituted for the unknown in the equation, reduces both members to the same number. Any number that fulfills these conditions is a root of the equation. The solution of an equation means two different things in mathematics, viz.: (1) The solving of the equation. (2) The value, or values, found by solving the equation. 2 1 8 First- Year Mathematics In this book the word "solution," from now on, usually has the second meaning. To check tlie process of solving an equation means to show that the work of finding the unknown is correct. This is often done by checking the "solution," i.e., the result, though it is generally desirable to examine the reasoning, or to use an independent process. Distinguish between root of a number (pp. 24 and 28) and root of an equation. To check the solution of an equation means to show that the result is a root of the equation. This is commonly done by substituting. To check the solution of a problem means to show that the result answers the conditions stated in the problem. To do this the result must be tested by the stated conditions, and not merely by substituting in the equation solved. This equation may itself be wrong. Exercises for Practice 156. It should be borne in mind that finding the root of an equation includes checking the result. Find the roots of the following equations: FULL SOLUTION: Subtract 3* Subtract 9 Divide by 2 3* -3* (by Axiom II) 9 = i? 9 (Ax. H) 2X X = 8 4 (Ax. IV) CHECK: 5*+ 9 = 5.4+ 9 = 29 3*+i?=3- 4 + 17 = 29 Hence, 4 is a root of 5^+9 = The Simple Equation in One Unknown 219 SHORTENED SOLUTION: 5#+9 = 3je+i7 Subtract 3^ + 9 from both sides (Ax. II) getting CHECK: 5 -4 + 9 = 29 = 3 Hence, 4 is a root of 5^+9 = 3^ + 17. NOTE: The reasons need not be written, but they should be thought. In practice only this is needed: CHECK: 5.4 + 9 = 29=3 4. 6x 7 = 5- 6. <;. 2*-* = ve-7 i8.'*-+s=*+3 o o / 4 19. 3# 2^ + 5) =o# 20 7. 9^ 8 = 25 2X If J ,,-. o/v ' Q ' ' * SlW o. 9- 20-4^=8-10^ 2I- 3^ +5=91 _ I0 ^ 10. 18 #=4* + '? 4 ByAx,omV ( 8x-22=^-i 8(3-2^-2(5-^=28 12- 6*- 7 -8* + il 5 =0 W 13. 8^- 22 = -2^ + io8 24 ' -3*-24 = 33(2-*) I4> 7^-9-10^ = 3^-15 2 5- 5(i-i3^)=35-io5^ * i , * i , 2X 15. + 2^C 8 = 3^ 5 20. S + 2X-\ = lf-l 4 43 27. 3* + i4-5(:x;-3)=4(*+3) 28. 2(x \ 29. 30- X 2_7 ^~~5 22O First-Year Mathematics 33. 35. 7 Problems Leading to Equations in One Unknown 157. State and solve the following problems and interpret the results: 1. A rectangle is 2 units longer than three times the width, and the perimeter is 60. Find the length and the width. 2. The area of a triangle, whose altitude is 5 units less than the base, is equal to 10 square units less than half the area of a square on the base of the triangle. Find the base and the altitude of the triangle. 3. One-fourth of the difference of 3 times a number and 8 is 10. Find the number. 4. A is 1 60 yd. east and B 112 yd. west of a gate. Both start at the same time to walk toward the gate, A going 3 yd. and B 2 yd. a second. When will they be at equal distances from the gate ? 5. Baking powder is made up of as much starch as soda and twice as much cream of tartar as soda. How much of each ingredient is there in 2 Ib. of baking powder ? 6. An express train whose rate is 40 mi. per hour, starts i hr. 4 min. after a freight train, and overtakes it in i hr. 36 min. How many miles per hour does the freight train run ? Let some letter, as x, be the rate of the freight train and notice that the trains go the same distance. 7. Two trains start at the same time from S, one going east at the rate of 35 miles per hour, and the other going west The Simple Equation in One Unknown 221 at a rate \ faster. How long after starting will they be exactly 100 miles apart? 8. Sound travels 1,080 ft. per second. If the sound of a stroke of lightning is heard 3^ seconds after the flash, how far away is the stroke ? 9. If sound travels / ft. per second, how far away is a lightning stroke if the sound is heard 5 seconds after the flash ? 10. A tree half a mile distant was struck by lightning. It took 2\ seconds for the sound to reach the ear. Find the rate in feet per second, at which the sound traveled. 11. A tree m feet distant was struck by lightning. It took t seconds for the sound to arrive. What was the rate in feet per second at which the sound traveled ? 12. A man walks beside a railway at the rate of 4 mi. per hour; a train 208 yd. long, running 30 mi. per hour, overtakes him. How long will it take the train to pass the man? 13. A man walks beside a railway at the rate of m mi. per hour. If a train I yd. long and traveling n mi. per hour overtakes him, how long will it take the train to pass the man ? 14. Solve the following exercises, as many as possible mentally : (1) ^+6=8 (8) 3 *-I^=_7 . 2T 4X T, (2) --3 = i (9) i2-~^=x + 3 O / (3) 1058=35 (10) ax bx=6a 6b (4) 3* + i3-5*=* (") 3*+* = I2a +4 (12) 8-(s-2*) = -* 2 x-a (13) ax --- =a 2 (6) 10 x= 20 c . . 3* 2 , , . a(x c) $ac (7) - 1-6 = 2* (14) ax - - -=- 222 First- Year Mathematics 15. A train moves at a uniform rate. If the rate were 6 mi. per hour faster the distance it would go in 8 hr. is 50 mi. greater than the distance it would go in n hr. at a rate 7 mi. per hour less than the actual rate. Find the actual rate of the train. 1 6. Two trains go from P and Q on different routes, one of which is 15 mi. longer than the other. The train on the shorter route takes 6 hr., and the train on the longer, running 10 mi. less per hour, takes 2\ hr. Find the length of each route. 17. The distance from A to B is 100 mi. A train leaving A at a certain rate, meets with an accident 20 mi. from B, reducing the speed one-half and causing it to reach B i hr. late. What was the rate per hour before the accident ? 1 8. An express train whose rate is r miles per hour, starts h hours after a freight train and overtakes it in t hours. Find the rate per hour of the freight train. 19. Two trains start from the same place at the same time, one going east at the rate of m miles per hour, the other going west at the rate of n miles per hour. How long after starting will they be c miles apart ? 20. A man rows downstream at the rate of 6 miles per hour and returns at the rate of 3 miles per hour. How far downstream can he go and return in 9 hours ? 21. At what time between 3 and 4 o'clock are the hands of the clock together ? Let x denote the number of minute-spaces over which the minute hand passes from 3 o'clock until it first overtakes the hour hand, then show that 1-15=*, whence, x = i6&. Hence, the hands are together at i6j* r minutes past 3 o'clock. 22. At what time between 2 and 3 o'clock are the hands of the clock together ? The Simple Equation in One Unknown 223 23. At what time between 3 and 4 o'clock are the hands of the clock at right angles (two results) ? 24. At what time between 7 and 8 o'clock are the hands of the clock pointing in opposite directions ? 25. Solve the following for x: ** T 'Y* ^* ^* / \ /O\ a 20 346 (3) - (4) a 2 xb !l x=ab (5) b 2 xa a x=ab xv (10) #= c T. 2 (n) 5# - IS 4 (7) *-~-f =7f (12) * - = (13) - or m about 26. Venus makes its orbit in 224.7 months ; the earth starting as shown in the Fig. 225. In how many days will Venus next be in line between the earth and the sun? The rate per month of Venus is T \ of the orbit, that of the earth ^. Let x be the required number of months. Express by an equation that Venus must make one more revolution than the earth. 27. Calling the revolution of Venus about the sun 7$ months, and that of Mercury 3 months, how many months after Mercury is in the line between Venus and the sun will it next be in the same relative position ? 28. Seen from the earth, the moon completes the circuit of the heavens in about 27 days, 8 hours, and the sun in 365 FIG. 225 224 First-Year Mathematics days, 6 hours, in the same direction. Required the time, to .001 day, from one full moon to the next, the motion supposed uniform. 29. If a revolution of one planet is a days and of another b days (a x-ax=a 2 +9-6a 13. 3ra; 2cm = 3 4. m 2 x n 2 x=n+m 2Xl_x2s 5. s 2 x+r 2 x-2rsx=r 2 -s 2 I = a; jc_& 2 a 3 2# a_x a_a a b ab I ^ < c d ~c x x 6aSb cxd dxc cd n ^= ..... - - A __ L _ . - 2 J ___ r __ 4& 3& i2ab dx ex cdx , x . i 2 2.T+a #+a o 8. 3e = i+ 17. - a a 2 a c c c 9. (2X 1)(3 x) = 2(x 2 9) x 2 dd+x 2X d 18. 1 10. x 2 c 2 = y 2 2cx+x 2 ex c c x 19. r s st rt 20. A = x t xr x r-s s-t (s-t)(t-r) 21 23. y '' xt xr (xf)(xr) s-r t-s (s-t)(t-r) ------ ^:r x+t r+x (x+t)(x+r) ex , dx H -- x+d x+c (2X-$r) 2 = x-$r (2X-$S) 2 X-SS i i 2Xpq ' xp xq x(xpq) 26. 5.8^+3.69=3.96 + 2.8^ 27. .374^-. 53 + 1. 2^ + .o6=. 8 + 1. 28. .i.*-.8) = .6(5.i + .2#) 232 First-Year Mathematics 29. .05(20* 3.2)= .8(4*+. 12) ii .256 .2IX+ .OI2 30. 1.4* 1.61 -- = 1.3* . o 1-2* 23C-.5 311 " 32. .25 12.5 5 3 .2X .66 .08*4-. 38 - - 2 SUMMARY Algebraic expressions may be pictured, or graphed. An equation symbolizes the question: What value, or what -values, if any, of the unknown, are there -which, substituted for the unknown, makes the first mem- ber identical with the second member ? The identical equation, or the identity, is an equation that is true for any, or all, values of its letters. The distinguishing sign of identity is =. The conditional equation is an equation that is true for one value, or for a definite number of values, of its letters. The left side of an equation is called the left member, or the first member, and the right side the right member, or the second member. An algebraic problem is stated I. By denoting the unknown by a letter and then translating the -verbal statement of the number relations into a symbolic statement in equation form. II. By expressing some number in the problem in two different forms and writing the two expressions equal to one another. To solve an equation in one unknown means to find the number, or the numbers, which, substituted for the unknown (the variable), reduces both members to the same number. The Simple Equation in One Unknown 233 Any number that fulfills this condition is a root of the equation. To check the solution of an equation means to show that the result is the root of the equation. To check the solution of a problem means to show that the result answers the conditions stated in the problem. CHAPTER XI LINEAR EQUATIONS CONTAINING TWO OR MORE UNKNOWN NUMBERS. GRAPHIC SOLUTION OF EQUA- TIONS AND PROBLEMS Indeterminate Linear Equations 1 60. The meaning of the solution of equations and of problems is made clearer by means of graphic methods. i. How many 8 and 10 candle-power bulbs are necessary to obtain an 82 candle-power light? (More than one solu- tion.) Let x and y denote the number of 8's and ID'S respectively Then 8s + icy = 82, or 4# + 5v = 41 4i-4* and y=> . From which, if x = i and if x = 14 then v =,Q then, y 3. These solutions are the co-ordinates of two points on the graph; viz., A (-1, 9) and B (14, -3). (Fig. 226.) s j "X, A -1 1 u u t f 1 ^ X, c M 15 L 4, i> B s 4 | \ ft \ il) X a F i fs, | /> d c y H \ s, B (1 4j ft \ \ \ FIG. 226 Since x and y must be positive integers to satisfy the prob- lem and since there are on the graph only two points whose 234 Linear Equations Containing Unknown Numbers 235 co-ordinates are positive integers, namely, C (4, 5) and D (9, i), then x=4, y = 5', and #=9, y = i are possible solutions. What are the co-ordinates of point E? Do the co-ordi- nates of point E check in the equation ? In the problem ? What good reason is there for always checking in the problem ? What is the origin? What are the co-ordinates of the origin ? Find the co-ordinates of the point P. (Find L P by pro- portion.) Where in respect to the x- and ;y-axes do you look for points having positive integral co-ordinates ? What important relation is there between the equation and the co-ordinates of all points on the graph of the equation ? Why may 8# + io;y= 82 be called an indeterminate equa- tion ? A simple equation ? A linear equation ? Why were only two points necessary to construct the graph ? Why should the two points which determine the graph be chosen some distance apart and have integral co-ordinates ? 2. A contractor has cornice stones 3 and 4 feet in length. How many of each may he use, without cutting, to lay the cornice of a wall 46 ft. long? Give all the positive integral solutions of the equation. 3. In how many ways can a merchant make an even ex- change of hats at $4.00 apiece for gloves at $3.00 a pair? Give the least positive integral solution of the equation. 4. At how many feet from the fulcrum will a 75-pound weight on one side balance a 45-pound weight on the other? Give the least positive integral solution. 5. Weights of 12 and 8 pounds are placed at opposite ends of a lever resting on a fulcrum. By moving the i2-lb. weight 3 ft. toward the fulcrum, the weights balance. How far is 236 First- Year Mathematics the fulcrum from each end of the lever and how long is the lever ? (Several solutions.) Let x and y denote the required distances. Then x 3 will denote the number of feet the ra-lb. weight is from the fulcrum. Interpret each solution. What is the least positive solution of the equation ? What form have the equations of problems 3 and 4 ? Which term is missing ? Their graphs have what point in common ? What are the co-ordinates of this point ? Do the graphs of all the equations of this form, ax = by, pass through the origin ? Do all of them have the solution x=o and y=o ? Test your answers with the following equations: 4^ = 7^, T ) x+2y=o, and 4^5^=0. A System of Two Linear Equations 1 6 1. Some problems lead to two linear equations in two unknowns. i. Weights of 12 Ib. and 8 Ib. are placed at opposite ends of a lever resting on a fulcrum. (a) If the i2-lb. weight is moved 3 ft. toward the fulcrum they balance. (6) Changing the weights from end to end, and moving the i2-lb. weight i ft. toward the fulcrum, they balance again. How far is the fulcrum from each end of the lever ? (See problem 5, p. 235.) Letting x and y denote the required distances, then from Condition (a): i2(x 3)=&y; or 3* 2^ = 9 (i) Condition (&): Sx = i2(y ij), and zx sy = 4. (2) Construct the graphs of (i) and (2) on the same axes. Find the co-ordinates of several points on the graph of equation (i). Do they satisfy equation (i) ? Condition (a) ? Linear Equations Containing Unknown Numbers 237 Consider similarly co-ordinates of points on the graph of equation (2). ZS&?A FIG. 227 What must be true of the co-ordinates of a point common to both graphs ? What is evidently the solution of the problem ? Check in both equations and in the problem. 2. Find two integral numbers such that, (a) 4 times one diminished by 3 times the other is equal to 22, and (b) the sum of the numbers is equal to 9. 3. Find two integral numbers such that, (a) their sum is equal to 9, and (6) 6 times the first diminished by 5 times the other is equal to 45. 4. Find two integral numbers such that 6 times the first diminished by 5 times the other is equal to 45, and the sum of 3 times the first and 2 times the second is equal to 9. 5. Find two integral numbers such that the sum of 3 times the first and 2 times the other is equal to 9, and 4 times the first diminished by 3 times the second is equal to 22. 162. It was seen that each of the preceding problems 238 First-Year Mathematics produced a pair of linear equations having one and only one common solution. Such equations are commonly called a system of two linear equations. EXERCISE XXXV 163. Solve the following systems by the graphic method, and check: (i) x + 2y = iy (4) 2X 3^=4 -36 (5) (6) A Pair of Contradictory Linear Equations 164. Find two numbers such that, (a) their sum is 10, and (b) 3 times the first increased by 3 times the second is equal to 15. Letting x and y denote the required numbers, then from Condition (a) x+ y = io Condition (b) 3*+3y = I 5 or *'+>' = 5 That is, x+ y = io And, at the same time, x+ y = $. (2) (3) o*- 6 :v = 36 2X = ^ N i n< \ \ \ a ltj s jj "U B \ * ^ ; \ MO ^ ,v * / \, ' \ 9- s^ X, :S It \ >, X \ ! \ \ * \ \ fe > D 5, iii < 1 ' h K; J 1 's J* -t X IS " \ \ S \ FIG. 228 Construct the graphs of the last two equations on the same axes. Are OB and OA divided proportionally by D C ? Why ? Is CD parallel to AB? Why? What does this tell about the problem? The equa- tions ? Linear Equations Containing Unknown Numbers 239 What two contradictory statements are expressed in the equations themselves ? The equations of such a pair are called inconsistent, or contradictory equations. A Pair of Equivalent Linear Equations 165. Find two numbers such that, (a) their sum is 8, and (b) % of the first increased by of the other is equal to z\. Letting x and y denote the required numbers, then from Condition (a) x+y = 8 Condition (b) $x + $y = 2% , or x + y = 8. The graphs evidently coincide. Hence the problem and its equations have an unlimited number of solutions. Give several of these solutions. The equations of such a pair are called equivalent, or dependent equations. EXERCISE XXXVI 1 66. Are the following pairs of equations contradictory, or equivalent, and are their graphs parallel, or coincident ? Give a reason for each answer. , 3- 7*~ 8 =4y~2X 4- 3*-2>' = i4 gx 6^=36 5. 3X + 2y 7-x = 2^+5^ = 6. The Solution of a System of Two Linear Equations by Algebraic Methods 167. (a) John sets out on a walking trip and travels at a uniform rate for 5 hours when he meets with an accident. He continues, however, at a slower pace, and 3 hours later reaches 240 First- Year Mathematics a point 26 miles from home. (6) If he had turned back at the time of the accident he would have reached in 3 hours a point 14 miles from home. What was his rate of speed both before and after the accident ? Letting x and y denote the rate before and after the accident respec- tively, Condition (a) 5^+3^ = 26 (i) Condition (b) 5* 3)1 = 14 (2) (Add. Ax.) iox=4o (Sub. Ax.) 6y = i2 (Div. Ax.) x = 4 (Div. Ax.) y = 2 Check: 5(4) +3(2) = 26, and 5(4) -3(2) = 14. Check in the problem also. The equation 10^=40 was obtained from equations (i) and (2) by what operation ? What terms were canceled (eliminated) from equations (i) and (2) by this operation? What is true of the coefficients of the y-terms which makes possible the elimination of these terms by addition ? The equation 6y = i2 was obtained from equations (i) and (2) by what operation? What terms were eliminated from equations (i) and (2) by this operatibn ? What is true of the coefficients of the j-terms which makes possible the elimina- tion of these terms by subtraction ? 1 68. What must be done when the coefficients of x and y are not alike, as in the following example ? Solve: 5*+3}' = z6 (i) -iy = 2 (2) J ( If equation (i) is multiplied by the coefficient of x in equation (2), and equation (2) by the coefficient of x in equa- tion (i), what is the coefficient of x in each of the resulting equations ? What operation will then eliminate the #-terms from the resulting equations ? How may the ^-terms be eliminated from equations (i) and (2) ? Linear Equations Containing Unknown Numbers 241 Show from the following whether your answers are correct : ,,, 4X(i), 20*+ 127 = 104 )> x 2iy = ) v ' 5X(2), 20^357 = 10 5 47* =188 47V = 94 * =4 y = 2. Having found the value of x as shown above, the value of y might have been found by substituting the value of x in one of the given equations, thus Check in (i) 5 4 + 3 2 = 26 3y= 6 Check in (2) 4 47 2= 2. V= 3 Do the values x=\ and ;y = 2 satisfy system (a) ? System (ft) ? System (c) ? (a), (6), and (c) are called equivalent systems. How were systems (&) and (c) obtained from system (a) ? From your answers to the questions above, make a rule for solving a system of two linear equations. 169. This method of obtaining from two equations a single equation with one unknown is called elimination by addition and subtraction. EXERCISE XXXVII i. Tell how to obtain from each of the following an equiva- lent system having the coefficients of the first unknown the same; of the second unknown the same. (i) (2) 5#+4;y = 22 (6) 6a 46 = 2 3 *_ 7:V =_ I5 50 + 76=43 (3) 3^-5> l== 5i (7) -nx+gy=i6 2^ + 7^=3 4^+8^ = 28 (4) 2^-75 = 58 (8) 3- 2*> = 4 (9) mx- 9^ = 242 First-Year Mathematics 2. Solve the systems of problem i and check. 170. Solve '-5 + ^=^ (0 Removing 2oX(i), 60 + 556 = 128 (3) denominators 6X(2), 340 + 156 = 132. (4) When the coefficients of an unknown have a common factor, the work of eliminating that unknown can be made easier by multiplying each equation by the remaining factor of the coefficient of that unknown in the other equation after removing the common factor. To eh'minate the a-terms from equations (3) and (4), above, what are the smallest integral multipliers ? To eliminate the 6-terms, what are the smallest integral multipliers ? Show whether your answers are correct by comparing them with the following: i7X(3), 1020 + 9356 = 2,176 3X(3), 180 + 1656=384 3X(4), 1020+ 456=396 nX(4), 3740 + 1656 = 1,452 8906 = 1,780 3560 = 1,068 6 = 2 0=3. Supplement the preceding rule, 168, so as to include systems like the foregoing. EXERCISE XXXVIII 171. Tell how to obtain from each of the following in the first column an equivalent system having the coefficients of the first unknown the same; of the second unknown the same. 1. i2x+i$y=66 3. 33^-28.6=38 i6x2$y=2 22^+35^ = 79 2. 8& 2r= Linear Equations Containing Unknown Numbers 243 5. 2i+69^ = iii 7. (2u+$) : 5 = ( 3 ^+s) : 7 i4u z6v = 2 JH : 47; =77 : 40 6. 51^+76^ = 203 g + ?*z2 + 34^-38^= -42 4 10. r- u = w i 10 ii. ti^:_^_z=^ 2 3 = 3 4 12. a i 3 . Solve each of the 12 systems above. 172. Elimination by Substitution and by Comparison ELIMINATION BY SUBSTITUTION Solve: 5# 7;y = i (i) 3 *+4;y = i7 (2) From(i) ^= i2 ' (3) Substitute (3) in (2) 3 + 4 >' = i7 (4) (Mult. Ax.) 3+2i;y+2o;y=85 (5) (Sub. Ax.) 41^=82 (6) (Div. Ax.) y=2 (7) Substituting y = 2 in ( 3 ) #=3- (8) 244 First- Year Mathematics ELIMINATION BY COMPARISON Solve: 5* 7; y = i (i) 3* +4; y = i7 (2) From (i) i+7? (3) .v 5 From (2) 17 4y (4) 3 (Comp. Ax.*) 1+7? 17 4y (5) 5 3 (Mult. Ax.) 3 + 2i;y = 85-2o;y (6) (Add. and Sub. Ax.) 41^=82 (7) (Div. Ax.) y = 2 (8) Substituting x=3- (9) EXERCISE XXXIX 173. Solve each of the following by the method of substi- tution and check by the method of comparison: 1. 4*+53' = I 4 5- 9 R ~ 2r =44 3X2y=i 6R r=$i 2. \x+= 6. i 6v = 22 3. 7^ 27=8 7. 7^22=46 3^+4^ = 18 w+ 2 = 13 4. 2W+ 11^ = 50 8. 174. If the time required to do a piece of work is: (1) 10 days, what part of it is done in i day ? In 3 days ? In 10 days ? (2) x days, what part of it is done in i day ? In 3 days ? In x days ? * Comparison Axiom: Numbers that are equal to the same number are equal numbers. Linear Equations Containing Unknown Numbers 245 (3) y days, what part of it is done in i day ? In 3 days ? In y days ? If A worked 3 days on a piece of work and B 2 days, they can do -ff of it. But if A works 2 days and B 3 days, they can do -fo of it. In how many days can each one do it, working alone ? Letting x and y denote the number of days required by A and B, respectively, then - and - will denote the parts A and B, respectively, x y can do in one day. Whence, +=H (i) These equations are not linear in x and y, but are linear in - and - and should be solved for - and - . They should x y x y not be cleared of fractions. From the values of - and - , x y x and y can easily be found. What operation would eliminate the ^-terms if their numera- tors, were the same ? Solve the equations. EXERCISE XL 175. Solve the following without clearing of fractions, and check: IO 9 _JL_ 5_U^ 24 II 2 4,3, 13 " i i y a >> t " i o r 4- ~~H = T05 ~+~ i ;* y X y X y 56_ 15 2i_ 4 9_i_ 20 _ jc^""^ icy~^ T ^cy~ 246 First-Year Mathematics Systems of Linear Equations in Two Unknowns Having Literal Coefficients 176. Solve for x and y: a 2 x+b 2 y=a+b (i) abxaby = ba (2) aX(i) aix+ab*y=a'+ab (3) &X(2) ab*x-ab*y=b 2 -ab (4) (Add. Ax.) (a*+ab*)x=a 2 +b 2 (5) a a +b 2 a 2 +b 2 i x a(a 2 +b*) a Find the value of y, first by eliminating the #-terms from lations (i) and (2); and thi tion (2). Check your results. equations (i) and (2); and then by substituting x=- in equa- EXERCISE XLI i. Tell how to solve each of the following; then solve and check: (1) x+y=i ~+~=- axby=o x y n (2) cx+nyi. I_I = ax by=o x y . k (3) ax+by=h (g) ^ + ^ =c bx+ay=k x y (4) cx+dy = 2cd -+-=/ bxcy=dc x y (5) ax+by = 2 ab fo) +":=k 2bx+ $ay =2b 2 + $a a n m (6) a I x+b l y=c 1 --\ a 2 x+b a y=c 2 Linear Equations Containing Unknown Numbers 247 . 2a 3& , . x + i h + i+k (IO) 7" (I3) j+i-*+7=* 2& 3a_ :v ;y = 2& * y ^-ft_&-5 #+a_a+c y 6 b+c ' ~ = x+bb-c * a+b v ' a+b a-b a-b (12) -= r y ab x y i x+c a+b+c a+b ab a+b _ _ yc abc Systems of Three Linear Equations 177. The sum of 3 times the first, $ times the second, and 3 times the third of three numbers is equal to 22. The sum of ^ times the first, and 3 times the second, diminished by 4 times the third, is equal to i. If from the sum of 4 times the first and twice the second, $ times the third is subtracted, the remainder is 7. What are the numbers ? Letting x, y, and z denote the first, second, and third num- bers respectively, the equations are: 22 (i) -i (2) -5z= -7. (3) Eliminating by combining (i) and (2), and again by combining (i) and (3), or (2) and (3), what two unknowns will the two resulting equations contain ? Can these equations be solved for x and y ? Having found x and y, how is z found ? If we eliminate first z and then y, will the resulting equations contain the same two unknowns? What two unknowns will they con- 248 First- Year Mathematics tain? Can such a pair of equations have one and only one solution ? Eliminate z from (i) and (2) thus: 4X(i) i2# + 2o;y + i2z = 88 (4) 3X(2) 15^+9^-122= -3 (5) (Add. Ax.) 2 7* + 29? = 85 (6) Next, eliminate z from (i) and (3) thus: 5X(i) 15^ + 25^ + 152 = 110 (7) 3X(3) i2X + 6y-isz=-2i (8) 27^+31^=89 (9) Solving (6) and (9), x = i and y = 2 Substituting in (2), 5 i +3 2 42= i 2=3. Check in (i): 3-1+5- 2+3 -3 = 22; i.e., 22 = 22. Check in (2) : 5-i+3-2-4-3 = -i; i.e., -i = -i. Check in (3) : 4-i+2-2-5-3 = -7; i.e., -7 = -7- Write out a rule for solving a system of three linear equa- tions. Compare your rule with the following: Make two different pairs of equations out of the three equa- tions, and eliminate the same unknown from both of these pairs. This gives two equations in two unknowns. Solve the two resulting equations as a system of two linear equations. EXERCISE XLII 178. Tell how to obtain from each system a single equa- tion containing only the first unknown ; the second unknown ; the third unknown. Linear Equations Containing Unknown Numbers 249 2Z = i 4. 4^ + 2^+32=9 3a 26 4^ = 5. 7722 = 27 32 43;= 12 6. 4^-^ = II. - M J+I 7 4_i j+ 2 -- 5 U IV 2+1 z +^. W _ i 12. Solve the above systems and check. Problems Involving Two Unknown Numbers 179. The student should make a practice of re-reading each problem until the conditions imposed upon the unknown numbers are fully understood. i. A street railway company receives a certain sum for each cash fare and a different sum for each transfer. On one trip the conductor of a car collects 13 cash fares and 18 trans- fers amounting to $1.10 for the company. On the return 250 First-Year Mathematics trip there were 7 cash fares and 24 transfers and the amount for the company was $0.95. What does the company receive for a cash fare ? For a transfer ? 2. Three tons of hard coal and two tons of soft coal cost $32. The price remaining the same, 2 tons of hard coal and 6 tons of soft coal cost $43 . 50. What were the prices per ton of the two kinds of coal ? 3. One of the base angles, x, of an isosceles triangle is equal to twice the vertex-angle, y. Find all the angles of the triangle. 4. The difference of the acute angles of a right triangle is 36. Find the number of degrees in each acute angle. 5. The difference of the adjacent angles of a parallelogram is 20. Find the values of all the angles of the parallelogram. 6. The consecutive angles a and b of a parallelogram are so related that $a 6=30. Find the values of both of the angles. 7. Three times one of two adjacent sides of a parallelo- gram exceeds twice the other by 45, and the perimeter of the parallelogram is 80. Find the length of the sides. 8. One dimension of a rectangle is 5 and one dimension of another is 3. The sum of the areas is 65 and the difference 35. Find the dimensions of the rectangles. 9. Two trains pass each other going in the same direction, with a relative speed of 10 miles an hour. Going in opposite directions, they would pass with a relative speed of 70 miles an hour. Find the speeds of the trains. 10. The sum of the areas of two rectangles of dimensions 5 and x, and 3 and y, is 49, and the area of a rectangle of dimensions 3 and x exceeds the area of the rectangle of dimen- sions 5 and y by 9. Find the dimensions x and y. Linear Equations Containing Unknown Numbers 251 11. The following advertisement was published by an electric company: The price of electricity has dropped since 1902 by a certain num- ber of cents. The number of units of electricity this amount of money will buy has increased by a certain number. The sum of these two numbers is 59 and the difference is 19. Find the numbers. Solve the problem. 12. A man invests part of $3,200 at 6 per cent, and the rest at 5 per cent. If his annual income is $180, how much did he invest at each rate ? 13. A man "gained 8 per cent, on one investment and lost 3 per cent, on another. If the money invested amounted to $22,000 and the net gain was $440, what was the amount of each investment ? 14. Two investments, one at 3^ per cent, and the other at 5^ per cent., yield annually $150. If the first had been at 8f per cent, and the other at 3^ per cent., the annual income would have been $175. What was the amount of each invest- ment? 15. The areas of two triangles, having equal bases, are 72 sq. in. and 60 sq. in. Twice the altitude of the first plus 3 times the altitude of the second is equal to 54 inches. Find the altitudes. 1 6. The areas of two circles are to each other as 4 is to 36. One-half the radius of the first, plus \ of the radius of the second, is equal to 6J. Find the radii of the circles and their areas. The formula for the area of a circle is: A =vr*. Find the ratio of the radii. . 17. The ratio of the circumferences of two circles is 2, and 6 times the radius of the first minus 4 times the radius 252 First- Year Mathematics of the second is equal to 14. Find the radii of the circles and their circumferences. The formula for the circumference of a circle is: c = 2irr. 1 8. The altitude of a trapezoid is 8 and the area is 56. If the lower base is increased by a length equal to the upper base, the area is 72. Find the bases of the trapezoid. 19. The lower base of a trapezoid is 24 and the area is 150. If a length equal to of the lower base is added to the upper base the area is 170. Find the altitude and upper base of the trapezoid. EXERCISE XLIII 20. Solve (he following: (3) (4) (5) (6) 1 i-ft r = 7 ( mxky=mR \kx+my=kR (for x and y) ( 6-H R + $ = 12 i-H 20-2K $K = H 3 6 "6 5 Linear Equations Containing Unknown Numbers 253 ( (7) 3^+4^+6^ = ( ( (8) < K + R = 2 a (for H, K, and R) (R+H = 2 b (9) Construct the graphs of (2) and (6) . 21. In the right triangle A B C, C D is perpendicular to the hypotenuse AB. Find m and n. The segmsnts of the hypotenuse are to each other as the squares of the two sides adjacent to the segments. FIG. 229 FIG. 230 22. In the right triangle E F G, h is perpendicular to the hypotenuse E F. Find m and h. Apply the principles of similar triangles. 23. In Fig. 231 x has the same value throughout. The same is true of y. Find x and y from the relations indicated FIG. 231 (i) in triangles i and 2; (2) in triangles i and 3; (3) in triangles 2 and 3. 254 First- Year Matfomatics 24. The perimeter of the first triangle, Fig. 232, is 288. Find x and y from the relations indicated in the triangles. 25. The corresponding altitudes of two similar triangles are represented by the expressions 3^+4^7 and 2X y FIG. 232 respectively, and the corresponding bases by the expressions 5# 2y+i and 4* zy respectively. The perimeter of the first triangle is 4 and of the second is 6. Find the values of x and y, The perimeters of two similar triangles are to each other as the corresponding sides or altitudes. 26. In Fig. 233 D E is parallel to B C. The sides A B and AC are represented by the expressions 4#+3;y + i and 6x +4y+2 respectively. Find x and y. 27. A number of two digits is equal to 9 less than 7 times the sum of the digits. If the digits are reversed the number is 18 less than the original number. What is the original number? 28. Three times the reciprocal of the first of two numbers and 4 times the reciprocal of the second are together equal to 5. Seven times the reciprocal of the first less 6 times the reciprocal of the second is equal to 4. What are the numbers ? Linear Equations Containing Unknown Numbers 255 29. If the numerator of a certain fraction is increased by 2a and the denominator by 36, the resulting fraction is equal to r- If the numerator and denominator are decreased by 1 2a and 6b respectively, the resulting fraction is equal to -r . What is the original fraction ? 30. A and B, working together, build a fence in 4 days. They can also build it if A works 3 days and B 6 days. In how many days can each alone build the fence ? 31. A boat crew rows 4 miles down stream in 20 minutes and the same distance up stream in 35 minutes. Find the rate of the boat crew in still water and the rate of the current. EXERCISE XLIV 1 80. Solve the following exercises: 5^ + 14^=385 "1 m n "64 m n o~i~, ~2 3 6 r2 v+4 r+i = z>+3 r+2 v4 (for w and k) ah ok , (bc)h 2(0 b)k , ,, , , , x h * =bc (for h and k) 256 First-Year Mathematics 8. 12. 13- 14. is- a b a ABC I ---- =i 2 4 5 ABC ---- 1 -- =12 3 2 3 ABC ----- = 4 4 2 (w n) c 27 = 16 2d = 17 2C 5a { 3 2 2C d 4 5 2. 2g+. 8 -5/i-.4 -3 2 2.5 6-4^ 3^-8 3 2 3 - 4 =^ -+-+ =4 w v 2r 12 l8 10 | ---- =4 V 2V W II- (for C x and C 2 ) n 2 (for R! and 16. i Given s = |a/, and T = Ra. (i) Solve for / and a; (2) Solve for R and a. (3) When in the equations R = 2, t=6, and 5=81, what do T and a equal ? (4) Check by substituting in the formula obtained by solving (i). Linear Equations Containing Unknown Numbers 257 IV Given l=a + (n i)d and s=-(a+l). (i) Solve for a and I; (2) Solve for a and d Find the values of the remaining letters when (i), n=S, d=2, and 5 = 104 when (2), = i6, a = 5, and 5=40 when (3), a = 7, 1=49, an d 5=812. Given l=ar n ~' i a(r n -i) and j ri (1) Find a and / when 5 = 765, (2) Show that lr=ar n (3) Show that r= C " = 2, and w= 5 (4) Find r and n when a =3, = 384, and 5 = 765. Problems Involving Three Unknown Numbers 181. Some problems lead to systems of three equations in three unknowns. FIG. 234 1 . In the triangles ABC andA / B'C / ,B=B / ,AB=A / B / , and B C^B'C', and the sides may be designated as shown in Fig. 234. Find the values of x, y, and z. 2. The triangles of Fig. 235 are equal in all respects. a FIG. 235 258 First-Year Mattiematics Find the values of x, y, and z, the sides being designated as folbws: (i) (2) (3) (4) z, anda' = =3# y+2Z, , and a' = i4 # , and b / = io y , and c'= =2( b=x+z, , and c'=z , and a' = 2(4^ 2 =#+5z, and b /= 4^+8 =3^-z, and c' = 5(2^-4). 6 FIG. 236 3. The triangles of Fig. 236 are similar and the ratio of their perimeters is 3. Find the values of x, y, and z under the following conditions: Linear Equations Containing Unknown Numbers 259 (i) a = , and a' = 7 z, and b' = 5 c= #+3;y + 5z, and c'=6 3, and a'=io+- 3 i=#+z, and b'=4+- 3 (2) 4. The sum of the three digits of a number is 1 6. If the order of the digits is reversed, the new number is 396 less than the original number. If the middle digit be placed first, the resulting number is 90 less than the original number. What is the number ? 5. A and B can do a piece of work in 35 days. B and C can do it in 17^ days. C and A can do it in 21 days. How long will it take each to do it alone ? 6. In the triangle ABC, Fig. 237, find the values of x, y, and z. (1) Ifa=6andb=8, c=8# jy (2) If a = 5 and b = 12, (3) If a=9 and b = i2, c=/ 7. The angles of a triangle are A, B, and C; and \A + ^B = \C + 30. Find the values of A, B, and C. 8. $18,000 is invested as follows: One part at 3^ per cent., a second part at 5 per cent., and the rest at 4 per cent., and the total annual in- terest is $730. If the first part had been invested at 4 per cent., the second at 3 per cent., and the third at 6 per cent., 260 First-Year Mathematics the total annual interest would have been $840. How much was each part ? SUMMARY Problems involving two unknown numbers may be solved graphically. The graph of a linear equation is a straight line. The solutions of a linear equation are the coordinates of points on the graph of the equation. The coordinates of two points are sufficient to locate the graph. The coordinates of every point on the graph of an equation are solutions of the equation. Positive integral solutions and least positive integral solu- tions of linear equations may be found from the graph of the equation. The solution of a problem involving two linear equations is shown to be the coordinates of the point of intersection of the graphs of the equations. The graphs of a pair of dependent equations are two parallel lines. The conditions of a problem having an infinite number of solutions are represented graphically by two coincident lines, the graph of two equivalent equations. Two equations having all solutions in common are called equivalent equations. Their graphs are coincident lines. The method of elimination by addition and subtraction ex- plained. The method of elimination by substitution explained. The method of elimination by comparison explained. The solution of literal equations and equations involving the reciprocals of the unknown numbers explained and applied. A system of two equations, that are linear in the reciprocals First- Year Mathematics 261 of the unknowns, of the type: --\ =- , may be solved as x y d linear equations by regarding - and - as the unknowns. The solution of systems involving three unknown numbers explained and applied. Problems involving two and three unknown numbers. CHAPTER XII FRACTIONS 182. The operations of addition, subtraction, multiplica- tion, and division are extended to apply to fractions. Reduction of Fractions 1. How many fourths equal J? How many sixths? How many eighths ? 2. How many halves equal ^? How many fourths? How many sixths ? 3. How many sixths equal J? How many ninths? How many twelfths ? 4. How many fourths equal -? How many eighths? How many twelfths ? 5. Show that ff=^f = . 6. Show that is equal to - . ac c 7. Show that - is equal to . c ac 8. Mike a rule for reducing fractions to lower terms; to higher terms. 183. Multiplying numerator and denominator of a frac- tion by the same number does not alter the value of the fraction. By this principle we may reduce a fraction to higher terms. 184. Dividing numerator and denominator of a frac- ion by the same number does not alter the value of the fraction. By this principle we may reduce a fraction to lower terms. 262 Fractions 263 EXERCISE XLV 185. Reduce the following fractions to lowest terms, doing as many as you can mentally: I. 2. 3- 4- 5- 6. 7- 8. 9- 1 It u 185 ab IO. II. 12. 13- 14. 15- akx 2 l6 a(x+y) aky 3 a 2 b 3 m m 3 (xy~) a a 3 b 2 m 2 u 3 ii^y m s (xy) ig cx+cy rx+ry amnbmn ^2^6,y7 x n+ 3 X+ a ^ (a+6) 2 (6-c)3 bx abc x n+4 (u+v) 3 (m w)7 acy a 3 & ] oo a* O-J f/jt _1_ <7 1 | 5 I -*M * ^ 7 I i* |^ U j \fiv ^^ fir J 186. Adding fractions that have the same denominator. EXERCISE XL VI Add the following fractions mentally: i. f +f g a_b 2- i+f ' C C 3- Kf-f !^ + ^ + * 4- iV+A-A+A w *' * 5- *-t-t+* r_ ^ v 6- 1-l-f-V- ^ ;> y 7 -+- ii -2 ^-- 7 - ^ " 12. Make a rule for adding fractions having the same denominator. 264 First-Year Mathematics EXERCISE XL VII Combine each of the following expressions into a single fraction: i. +&+ *-y , ?y t -_2 22 c ax + 30* 3- tt+A-tt 45 45 * 3<* 30 -8& *__ 10 I0 * | I2 * 5* 5 ' ' Ja !<* x x ,a-6 5^- 2a- 3 6 -- u . ii. ~ 2 2 *-y x-y 187. Adding fractions that have different denominators. 1. Reduce and f to fifteenths. 2. Reduce -| and f to fractions having the same denomi- nator. (See 183.) 3. Reduce - and - to fractions having the same denomi- x y nator. (See 183.) 4. What is the sum of and ? and ^? and ? and 5. What is the sum of i and 4? 2-7 2 7 7-2 Give reasons. 6. In the same way give the sum of \ and \. and \. iV and TV Fractions having i for numerator are called unit frac- tions. 7. Make a rule for writing at once the sum of two unit fractions. Fractions 265 EXERCISE XL VIII 1 88. Using the rule find these sums mentally: 8 I+L 2. + ' n m 3- Hi j_ i 4. '' b " 20+36 30 2& ii i i 7. h- 12. x y zx 3 zy 13. What is the difference of and $ ? J an d i ? i and ^-? ^ and i ? ^ and ^ ? 14. Write out the work of the parts of problem 13 as is 7^7 *> done in this explanation. 1lwm-i --- ^- =- - . Give rea- 5-7 5-7 5-7 sons. 15. Make a rule for writing at once the difference of two unit fractions. EXERCISE XLIX 189. Apply the rule to find these differences: 1. i-i i_I 2. - '' b x i e. 7 4- -- O. -- T 3- x c b 266 First- Year Malftematics b a+b i i 11. V^-J--\f2 3"2__^y2 I I 9 ' a+b a-b i i a+6 a+b+c 2 5 a+6' 14. Show that the addition and subtraction of unit fractions may be indicated thus: - - 190. Addition and Subtraction of Fractions .Add* and J. |+l - + - 2. Subtract -f- from f . 3. Add -1- and -^ where w t (read "n one") and d^ (read a x a 2 "d one") stand for first numerator and first denominator; n 3 (n two) and d, (d two) stand for second numerator and second denominator. d 2 n 2 d. n^dy+n-^d^ T + 1TT = TJ -- 5 give reasons. 5. Subtract - from -^ . a 2 di 6. From 3 and 4 obtain a rule for writing at once the sum or difference of two fractions. 7. By this rule write the sum and difference of T and - . b d EXERCISE L By this rule give the values of the following indicated sums and differences, giving as many as you can mentally: Fractions 267 I. i + yr 71 T 2 0+30+5 II. '" - + * 7 i a a 5 7 3- A+A 8 8 + 7 or b x 2 +a 2 x 2 -a 2 5 10 15. AddfV-t JL4 , 5-8+3" 12 JL. 8-12 8-12 This method is especially useful in adding two fractions whose denominators have no common factors. In case the denominators of the fractions to be added have a common factor, the fractions should be reduced to fractions having for their denominators the least common multiple of the given denominators, and then added, thus: 1 6. This method is used in adding three or more fractions, thus: x a xy y 2 ' a b c _ ay 2 bxy ex 2 _ay 2 +bxycx 2 x 3 xy y 2 x 2 y 2 x 2 y 2 x 2 y 2 x 2 y 2 EXERCISE LI 191. Simplify the following: 2 5 10 268 First- Year Mathematics 3- 12 2O 3 4- I_Z 18. &-* a ax 4 a b a b 5- -+- ex cy 19. 1 x 2 y xy 2 6. i b a+b t ab 20. 1 c ca * $x x z a 2b 40 56 i 2ab 6b 3* 5* ab bc ca T T T 8. 1 ab be ca 22. T a 3 a 2 a 9- 2 $y 2 xys+y3 a 6 c 2-?. 7-H 1 r bc ac ab xy xy3 x 2 y 6 EO a ab i i a i a(a i) 24. *-? y i i . . / \ ^ M 5 4y 1 3* i ' -5y 4* ' 5 - * 3^ 12. T x+y 2X 3y 4# 2 6xy _ la+^b 4-C 20 + 40 3 C 20. , a b NOTE. 4x*-6xy = 2x(2x-3y) 13- b ab 27 1 - " /- 5C 2 )/ x'y 3 ' *^ 2 . 7 1 7 . s-1 J a + 20 saa + ooa 0+6 a 14. -+b a a o NOTE. Put &=-. i (a+6) 2 29. i 400 15- a x x+s x$ X 2 I X I X+I 1 6. i 2<* 30 a 2ac 5 a+ 3 c+d cd c 2 d 2 ' Fractions 269 192. Multiplication of Fractions 1. Multiply | by 8; by 12; by 5; by 25; by a; by xy. 2. Multiply \ by \\ $by; by; -by; -by; -by- (i of |, iXl, and \ \ are all equivalent). 3. Multiply | by 4; by |; by ; by ; by ^; by-; by ^ . 4. Make a rule for multiplying two fractions together. Fractions are multiplied by multiplying their numerators for the numerator of the product, and multiplying their denomina- tors for the denominator of the product. 5. Multiply f -J-f . Since 12-15 will be the numerator of the product (why?), and 35 16 will be the denominator of the product (why?), any factor of 12 or 15 which is also a factor of 35 or 16 will divide both the numerator and the denominator of the prod- uct. It is simpler to divide out, or cancel, such factors in advance, thus: 3 3 yf ^f 3 3 9 ~. jjg- _jg=7 4=28 ' VTO reasons. 7 4 EXERCISE LII 193. Simplify the following: a. .c 10. -X- x a ? 4. V-S. O' 5 ^ 4- & XI 8. -X^ ii. ^X^-' 5- 57 9 ' * q I2 ' 6, b, ' 270 First- Year Mathematics EXERCISE LIII 194. Find the values of these products, using cancellation when possible: i. $ f f ,..* 8 ^!.^!.*! 2 - i * f * I b d y ' b 3 a 3 a 3. A. * fi ^i.^?.^ _ A.^.l 3 ' * 7 6 d, d 2 - 30+36 ' (*-;y) 2 ' a a ab x 2 +xy x 2 xy a'+ab 21. Show that 7= H = H -- r . o o o Division of Fractions EXERCISE LV 196. Multiply the following fractions: !.$- . x y a i 6. - -. 10. - - 2. f y x i 7 i . A w <2 # s -y 3- t 7 7. . _ . _ . JL , 3 d n y xs 4 ' 3 'x 8- f i a+& | C+jy , . xy ioa 2 2. ao-j - 4. 2$a 6 -. Fractions 273 5 . I8 ^ 24 ^ 6 (-06) * ab ab 7. SX3V3Z3 -. -- ~ x r x r ~ 2 II. x n + i x n + 3 2ox 2 y 3 4axy a k ~ 2 a k+3 10. 24(x i) 2 _ 30(^ " I4X 2 TX 2X I 14. SUMMARY Multiplying both the numerator and the denominator of a fraction by the same number does not alter the value of the fraction. Dividing both the numerator and the denominator of a fraction by the same number does not alter the value of the fraction. Fractions having the same denominator are added or sub- tracted by dividing the sum or the difference of the numerators by the common denominator. Fractions having different denominators are added or sub- tracted by first reducing them to equivalent fractions having a common denominator (usually the least common denomina- tor) and then adding or subtracting the resulting fractions. Fractions are multiplied by multiplying their numerators for the numerator of the product, and their denominators for the denominator of the product. 274 First-Year Mathematics Fractions are divided by multiplying the dividend by the inverted divisor. Cancelling, or removing, the same factor from both the numerator and the denominator of a fraction is equivalent to dividing both the numerator and the denominator by the same number. The numerator and the denominator, when spoken of together, are called the terms of the fraction. CHAPTER XIII FACTORING. QUADRATICS. RADICALS 198. The factors of a number are its exact divisors. The process of factoring is therefore inverse to the process of multi- plication. In this chapter we consider only factors which are non-fractional (integral) and which do not involve radical signs (i. e., which are rational numbers). We, therefore, admit only those factors of a number which divide it exactly with a rational integral quotient. A prime number is a number which has no factors except itself and unity. Give examples. A prime factor is & factor that is a prime number. A number has only one set of prime factors. Monomial Factors 1. a(b+c)=ab+ac, therefore the factors of ab+ac are a and b+c. The monomial a appears as a common factor of the terms of ab+ac. 2. X 2 y(k+l+m)=kx 2 y+lx 2 y+mx 2 y, therefore the factors of kx 2 y+lx 2 y-\-mx 2 y are x 2 y and k+l+m. The monomial x 2 y appears as a common factor of the terms of the given expression kx 2 y+lx 2 y-\-mx 2 y. 3. Multiply $x 2 y 3 and (2xy $x 3 $y 2 ). What are the factors of 6x 3 y 4 gx 5 y 3 i$x*y 5 ? How does the monomial $x 2 y 3 appear in the expression 6x 3 y 4 gx$y 3 i$x 2 y 5 ? 4. Factor T > a 2 b i2ab 2 . Since ^ab is a factor of each term, it is a factor of the whole expres- sion (each term in succession). We obtain the quotient a 46. Then 306 and a 46 are the factors of $0,^1 zab*, or 30*6 i2ab* = T > ab(a ^b). Test by multiplication: 46 275 276 First- Year Mathematics 199. An expression is equal to the product of all of its prime factors for all -values of its letters. It will be recalled that such equality is called identity. (See p. 213.) By this principle, we can test, or verify the correctness of the factors. If the factors of T ) a 2 b i2ab 2 are T,ab and a 46, then 3 a 2 b i2ab 2 and $ab (a 46) being 15 for a = 5 and b = i, the expression is correctly factored. Verify by substituting other values for a and b. EXERCISE LVII 200. Factor the following expressions and test results both by multiplication and by substitution, doing as many as you can mentally: 2. am 2 +bm 2 3. abcabd 4. T,nx+6ny 5. 2cdl+4cdm 6. ax 3 y-\-bx 2 y 7. 8. 9. 10. ii. 12. Factoring, Quadratics, Radicals 277 13. i $a 3 x i oa 3 y + $a 3 z 14. 15. 1 6. 8# 2 ;y 2 -f 1 6xyz 17. 1 8. 19. 20. 21. 22. 23. 24. 25. 26. x"+x 2n 27. 28. 29. 30. 5f 2 1 "^ 3 2^c 3m ^ 2n + zx 2m y n 31 32 EXERCISE LVIII 20 1. Reduce to lowest terms: 36 + y ab+ac $a 2 b ioab 2 bx+cx 2km 2 6bm 2 *" *j ___i__l. /* 3 + 2'3 + i-4 + 2'4 = 3+6+4+8 = 21 and, (c+d)(a+6) = (3+4)(i +2) = 7X3 = 21. Therefore, ac + be + ad + bd = (c + d) (a+b). 280 First- Year Mathematics 4. Factor 14^ 6x 2 2ix + g. 6x 2 -2ix+g = 2X*(jx-3) -3(7^-3) = (7*- Test by substitution and by multiplication. -3). EXERCISE LX 205. Resolve into factors the following expressions and test results, doing as many as you can mentally: 1. ax+bx+am+bm 2. ar+br+as+bs 3. ad+bd+at+bt 4. sa+^b+ay+by 5. akbk+albl 6. ax 3 bx*+ay 3 by* 7. abc+abx+nc+nx 8. a 2 ^+a 2 /+6 2 yfe+fe 2 / 9. 5aw 5az/+wu mv 10. m 2 a+ma 2 +m 3 11. a 2 ad+abbd 12. 13. 6x 2 ()X 14. 2w 3 +w 2 29. 30 15. ^ac+^ax 5^ $x 16. 9 i^r + 2'jr 2 45^ 17. Sgh + i2ah + iobg + 18. 152 6 2ozw +Sw 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. \\mn- 2ikn 2 + 2bx+b+x 2(a+w) 3^ 2X r y-\-T ) y s+l EXERCISE LXI 206. Reduce to lowest terms: ax + bx+am+bm ar+br+as + bs ^uyv+auav ax 2 bx 2 +ay 2 by 2 mx 2 + my 2 +nx 2 +ny 2 X 4 2X 3 + "JX 1 4 2X 3 AX 2 +6x 12 Factoring, Quadratics, Radicals 281 Perform the indicated operations in the following: 3036 ex dx+cy dy x 2 xy 6x + 6y bx ay+by xy6y i i ab+acbk ck bm+cm + bk+ck n ^ $ax$bx 2cm $cn + 2dm $dn ' 6cy + 6dy EXERCISE LXII 207. Solve the following equations : 1. ax+bx=ad+bd+ac+bc. Solve for x 2. 3a*bx + 1 2bcd=4ab 2 x +qacd. Solve for x 3. axbcac=ab+bx. Solve for x 4. am + bm = T > am + 3bm+ak+bkanbn. Solve for m 5. av+bv yv = $a+sb i$c + 2aw + 2bw dew. Solve for v. Solve for w ad ac be 6. ab= | ---- . Solve for y y y y b d bd c be 7. i-| ---- = -- 1 I -- . Solve for x a x ax x ax bm 2 am 2 8. -. -A 7= ,H --- . Solve for w 6cd 50 2cd i$a 10 T, Solve for u 2U+I 4M+2 4 2_ x*+x*+x + i x 3 $x 2 +x 3 =o. Solve for x j 2 7 ii. - I -H - =4. Solve for y. 2 y- 2 3? 3 4^-4 208. Solve the following problems: i. In one pile of brick there are r times as many bricks as in a second pile. If placed in three piles of a, r, and ar 282 First- Year Mathematics bricks respectively, there is one brick left over. How many bricks in each of the two original piles ? 2. In an alloy of two metals, the ratio of the weights of the metals is a: b. A certain mass of the alloy is a +b pounds more than another mass, and the weight of the two masses together is 3*1 +3&- How many pounds of each metal are there in each of the masses of alloy ? The Trinomial Square EXERCISE LXIII 209. Multiply: 1. (a+b) 2 8. (m-n} 2 15. (yn? -5^)' 2. (r+s) 2 9. (x + s) 2 16- (a& + 3* 2 ) 2 3. (c+d) 2 10. 0>-3) 2 17- 2 w~5* 3 ) 2 4. (a-b) 2 ii. (7-a) 2 18. (abc*+$x 2 yy 5- (g~~h) 2 I2 - (2u T,v) 2 19. (x 2 y 'jxy 3 ) 3 6. (rs) 2 13. (4& + 7;y) 2 20. (y 2 s^^rs 2 y 7. (c-d} 2 14. ($x 2 + 2y) 2 21. (x + i/2) 2 . 22. From these exercises make a rule for squaring a binomial. When a binomial is multiplied by itself the result is a tri- nomial which consists of the square of the first term of the binomial, plus or minus (according as the binomial is a sum or a difference) twice the product of the first and second terms of the binomial, plus the square of the second term of the binominal. Then any trinomial, in which two terms are squares (and positive) and the other term is plus or minus twice the product of the square roots of those two terms, is the square of the sum or difference of those two square roots according as the third term is plus or minus. 23. Factor k 2 +6kl+gl 2 . k 3 and gl 2 are squares, and 6kl is twice the product of their square roots, therefore, Factoring, Quadratics, Radicals - 283 EXERCISE LXIV 210. Factor the following expressions, and test each re- sult, doing as many as you can mentally: 1. x 2 -\-2xy-\-y 2 13. 49 2. a 2 2ab + b 2 14. 3. m 2 +4tnn+4n 2 15. 4. 4 2 645 is obtained. The square ^04 of this number is 6.996025, which is less than 7 by not quite .004. While the square of 2.646 is larger than 7 by more than .0003. Then 1/7 lies between 2.645 an< ^ 2-646. Continuing the process of extracting the square root of 7, we obtain 2.6457. The square of this number is less than 7 by only .0003, while the square of 2.6458 is larger than 7. So 1/7 lies between 2 .6457 and 2 .6458. Expressed in symbols: 2.64 < 1/7 < 2. 65 2.645 (i) = 9-61/7 + 7. (v/7) 2 = 7> because the square root of a number when multiplied by itself (or squared) produces the number. Thus 1/3X1/3 = 3; i/#X i/#=*. The way to express the product of a rational number, as 6, and an irrational number, as 1/7, is to write the rational and irrational numbers side by side thus: 61/7. The product, 61/7, is read: "6 times the square root of 7." Equation (i) may now be written: 961/7 + 7 18+61/7 + 2=0 9 + 7-18 + 2+61/7-61/7=0 (2) 0=0. The equation then is satisfied for the value of m= 3 + 1/7. Show by substituting that the second value of m, 3 1/7, also satisfies the equation, w 2 +6m + 2=o. Thus, (-3-v/7) 2 + 6(-3-l/7) + 2 = o. 230. Simplify: 1. (3 + 1/2)' 4. 7(3 + 1/2) 2. (5 + 1/6) 2 5- 4(5 -J'a) 3- (7-1/3) 2 6. 5(7-1/3)- 7. Find the sum of the results of i and 4. Note that 6j/2 added to 6j/2 is i2|/2. the terms being similar with respect to the factor 1/2. 8. Find the sum of the results of 2 and 5; of 3 and 6. See the note to problem 7. Factoring, Quadratics, Radicals 297 9. Simplify: (1) (-2-V/5)* (4) 4(-2-i/2) (2) (- 7 + l/8) 2 (5) - 9 (-7+l/8) (3) (-5-1/) 2 (6) -3(-5-|/^>). 10. Find the sum (1) Of (i) and (4) of problem 9 (2) Of (2) and (5) of problem 9 (3) Of (3) and (6) of problem 9. 1 1 . Solve and check the results by substitution : (1) r 2 +r 7=0 (5) m 2 iom= 6 (2) d* + jd + 3=o (6) (3) x*+4X-i=o (7) (4) -y*=6;y + 2 (8) s 2 + 205= +28. EXERCISE LXVII 231. Reduce to lowest terms: 4- 232. Perform the indicated operations: 5 7 x a xy xy x 2 2xy+y 2 x 2 +xy - 6 -+ ' 15 4JC 6 4a 2 6ab 298 First-Year Mathematics The Difference of Two Squares EXERCISE LXVIII 233. Multiply: 1. (a+b)(a-b) 9. (2X+y)(2X-y) 2. ( a +y)(x-y) 10. (a + s&)(a-5&) 3. (m n)(m+ri) u. (3/w 2ri)(T > m + 2n) 4. (r s)(r+s) 12. (a6+c)(a6 c) 5. (M+V)(W i>) 13. (m 2 n+x*)(m 2 n x*) 6. (je 2 +^) (a; 2 y) 14. (2u 2 vw)(2u 2 v+w) 7. (^+^3)(^ ^3) I5 . 8. (a 2 -> 2 )(a 2 +^ 2 ) 16. 17. From these problems make a rule for multiplying the sum of two numbers by their difference. 1 8. Make a rule for factoring the difference of two squares. EXERCISE LXIX Factor the following expressions and test the results, doing all you can mentally: i. x 2 y a 14. 4a 2 c' 3 ) 2 ~ I 6z 4 ii. ax 4 looa 24. (x 2 -y) 2 -x ( > 12. p 2 q 2 rr 25. & 2 -( 3 a + 2c) 2 13. 225&'-/ 4 6 /i 14 26. i6a 2 (2W 3) 2 Factoring, Quadratics, Radicals 299 27. gc 2 2a y 34. 28. (30 2&) 2 (2C 3& 3 + 'Zf/=M/ 3 + 2i;/ + it;'ZfA 3 . Solve for t U Factoring, Quadratics, Radicals 301 n 2 m 2 = bolve for x x x 8 . * 4* = _l_4 + 4 solve for* b 2 a 2 b 2 ab a 2 m 6 . g. = i H . Solve for m 8 m u* u v* v _ , , , 10. = -H --- h- Solve for g; for /. Trinomials of the Form x 2 -}-ax+b 237. Multiply: 1. (* + 3) by (*+4) 5. (x + s) by (x + u) 2. (# + 7) by (*-2) 6. (* + 9) by (ae-8) 3. (ae-s) by (# + 3) 7. (#-io) by (^ + 7) 4. (*-3) by (*-8) 8. (x-2) by (^-13). 9. From these cases make a rule for multiplications of this type. 10. The general law thus obtained is proved as follows: x+a x+b x 2 +ax +bx+ab Therefore x 2 + (a+b)x+ab = (x+a)(x+b). The sum of +a and +6=a+6, the coefficient of x in the trinomial. The product of +a and +& = +a&, the term of the trinomial not containing x (i. e., the absolute term). This gives us a method of factoring by inspection. ii. Factor x 2 + 12^ + 35 by inspection. Find two factors of +35 whose sum is +12. The factors are +5 and +7. Then the factors of x 2 + 12* + 35 are (# + 5) and 302 First-Year Mathematics 12. Factor x 2 $x 40 by inspection. The factors of 40 whose sum is 3, are 8 and +5. Therefore x 2 x o = x 8 EXERCISE LXXII 238. Factor by inspection the following expressions: 1. x 2 +$x+2 15. y 2 12^85 2. m* + $m+6 16. d 2 7^30 3. k 2 + i2kb + 3sb 3 17. p 2 pgo 4. >- 2 + i4^+45 18. r 2 -i3r- 4 8 5. A 2 13^4-40 19. k 2 +4okb + mb 2 6. a 2 +8a 20 20. x'y 2 gxyz ii2Z 2 7- ^ r2 + 35 r ~ I 8o 21. r 2 " 1 19^+48 8. 6 2 + i9k+48c 2 22. ^'7^-78 9. 9 6 +^3_ 4 2 23. J Ior + 23^/3/ + I0 2/ 6/ 10. i 2 14/ 51 24. k 2m 2ok m L r ,6<)L 2r ' n. r 2 2r5 3235 2 25. 12. v 2 6vw()iw 2 26. 13. a 2 6 2 4ak 2 165^ 27. (a+&) 2 7(0+6) + 10 14. z 4 102 24 28- a 2 + 2a6 + 6 2 + 3a+36 10 29. a 2 6a6+9& 2 + 7ac 2i6c 44C a 30. m 4 x+m 2 x 650^. Quadratic Equations Solved by Factoring 239. Solve the equation x 2 8#+i2=o. x 2 8# + i2=o Factoring (x 2)(x 6)=o (i) This equation is satisfied if x 2=0 (2) because The product of two or more numbers is zero if, and only if, at least one of the numbers is zero. Factoring, Quadratics, Radicals 303 From equation (2) x = 2. CHECK: 2 2 8 2 + 12=4 10 + 12 = 0. Likewise from (i) ar 6=0, Whence, x = 6. Check. Consequently, both 2 and 6 are roots of the equation. 1. Solve the same problem by completing the square. (See pp. 287, 288.) 2. Solve the equation x 2 = x 3 x 4.2 = (*-7)(*+6)-o x = j, or 6. State the principle used in the last step. Test both results. Solve the same problem by the method of completing the square. EXERCISE LXXIII 240. Solve the following quadratic equations by the method of factoring, and test results: 1. x 2 3^+2=0 ^!_4^_4 2. ' 3. 4. 5. c 6. 2 7. *-+* = S 6 16. 8. r 2 + i=2or 9. 10. ^ -2*| ^+ 15 5 gy-! 3 ;y-i^io-y 4 ^!_ = 13^ 9 5 " 9 9? I2 _, = 10 10 a+7 i-a , 4 20. - - = - ; -- 1 ---- . 9 4 am + T ) m=bn 2 +bn 2b. Solve for a; for b 2. i$kx 2 +kxy6ky=gx 2 -{-3xy 2y. Solve for k 3. tx 2 lx = 2ot-\-2X 2 7# 15. Solve for t 4. 2a 2 x+i'jax+2ix3a 2 y3$y=o. Solve for x; for y 5. mx 2 + i$m-\-i3mx=8mx + 2nx 2 + 2in. Solve for w; for n. 253. Solve the quadratic equation, 2X 3 13^+6=0. Factoring, (sx i)( 6) =o. Then, by the principle stated in 239, the equation is satisfied if 2x 1 = 0; whence x = %. The equation is also satisfied if x 6 = 0; whence x = 6. The two solutions of the quadratic equation, 2^ 13^+6=0, are = i and ^ = 6. Check both results. 3io First- Year Mathematics 254. Solve the following quadratic equations: 1. 2X a 2. 6 3. 6x*=mx+3$m. Solve for m; for x 4. 36& 2 = i9&/+6/. Solve for k; for/ 5- 5:v 2 + i3:y= 6. 24w 2 + i5n 2 =38wm. Solve for m; for 7. 2C 2 +6<2 2 = i3cd. Solve for c; ford 8. 8a 2 + i4o& = i5& 2 . Solve for a; f or & 9. i4^ 2 + 2ij 2 = 55^. Solve for x; for ^ 10. 8k 2 l 2 = 22klm + 2im a . Solve for /, for , and II. = - 7 2 14 I 2 12. X 2 X+2 _ y+4 y+2 y x i x 2 4^+4 2X 2 6x+4~ 2 5S- Solve the following problems: 1. The length of a rectangle is 5 greater than the width, and the area is i8f. Find the dimensions. 2. B can do a piece of work in 3 less days than A, and both can do the work in 5^ days. How long will it take each alone ? 3. A walks faster than B by a quarter of a mile in an hour. It takes A a quarter of an hour less than it takes B to walk 15 miles. Find the rate at which each walks. 4. A freight train takes i hours longer to travel 100 miles than a passenger train takes. The passenger train runs 10 Factoring, Quadratics, Radicals 311 miles per hour less than twice the rate of the freight train. Find the rate of each train. 5. In three hours a boatman rowed 10 miles up a stream and 4 miles back. If the velocity of the current was 2 miles an hour what was his rate of rowing ? The Sum or the Difference of Two Cubes 256. Multiply: 1. (x+y)(x 2 -xy+y 2 ) 4. (a-b)(a 2 +ab + b 2 ) 2. (x y)(x 2 +xy+y*) 5. (a+2b)(a 2 2a&+4& 2 ) 3. (a + b) (a 2 -ab + b 2 } 6. (3a-b)(ga 2 +3ab+b 2 ) 7. ( 8. 9. 10. (2a n b n - y n ) (4a 2n b 2n + 6a n b n c n 11. Make a rule for multiplying the sum of two numbers by the square of the first number minus the product of the first and second numbers plus the square of the second number. 12. Make a rule for multiplying the difference of two num- bers by the square of the first number plus the product of the first and second numbers plus the square of the second number. 13. Make a rule for factoring the sum of two cubes. 14. Make a rule for factoring the difference of two cubes. 15. Factor 6403 + 27 & 3 . 640' + 27 b 3 = (40 + 36) ( i6a 2 1 2ab + gb 3 ) . The expression is the sum of two cubes (40)3 and (36)3. Therefore one factor is the sum of the cubes of the numbers (40 and 36). The other factor is the square of the first number of the first factor, (40) * = i6a", minus the product of the first and second numbers, (40X36) = i2ab, plus the square of the second number (^b) 3 =gb 3 . Check the result. 1 6. Factor &x 3 -125^. Sx 3 I2$y 3 = (2x $y)(4x* + ioxy + 2$y 3 ). Check the result. 312 First- Year Mathematics 257. Factor the following expressions, doing as many a< you can mentally: 1. a 3 +b 3 ii. 2. 8x 3 y 3 12. 3. m 3 + 2'jn 3 13. 4. Sc 3 -d 3 14. 5. 343 -x 3 15. 6. P+64 16. (a+b) 3 +c 3 7. 8# l8 + 27z ia 17. (a+w) 3 n 3 8. 279 7 ;6-)- I !8. a 3 + (&+ c )3 9. 8iJ l8 + 27w l8 19. (w+3) 3 a 3 10. 2a 6 + 2i6c 6 20. The Remainder Theorem 1. What is the remainder after dividing 3^ + 7 by x i ? What is the value of 3^ + 7 if i is substituted for x ? 2. What is the remainder after dividing 4^ + 5 by x 2? What is the value of 4^+5 if 2 is substituted for x ? 3. What is the remainder after dividing 3^6 by x 2 ? What is the value of 3^6, if x = 2 ? 258. In the following problems give the remainders after divid ng and evaluate as required: 1. Divide 4^ 3 by #3. Evaluate the dividend for # = 3. 2. Divide n# 13 by x 10. Evaluate the dividend for # = io. 3. Divide 2^+3 by x + i. Evaluate the dividend for x= i. 4. Divide 11^ + 25 by x + 2. Evaluate the dividend for X= 2. 5. Divide 20^37 by # + 5. Evaluate the dividend for *--S- 6. Divide ax+b by x r. Evaluate the dividend for x=r. Factoring, Quadratics, Radicals 313 7. Divide myn by y c. Evaluate the dividend for y=c. 8. Divide ax+b by x-\-r. Evaluate the dividend for x=r. 9. Divide mv+n by v+t. Evaluate the dividend for v= t. 10. Divide x 3 + $x 2 6x n by x 2. Evaluate the divi- dend for x = 2. 11. When an expression in x is to be divided by x plus a number, or x minus a number, how may the remainder be obtained without dividing ? In the following problems give the remainders without dividing and check the results by division: 1. (3*-7)-3-0*-i) 4- 2. (7* + 3)-K* + i) 5- 3 . (3* + Il)-5-(# + 2) 6. 7. Divide ax 2 -\-bx-\-c by x r. x r)ax 2 + bx + c(ax + b + ar ax 2 axr bx + axr bxbr axr + br + c axr ar 2 Remainder, ar*+br + c. Obtain the remainder by the rule given in problem 1 1 . 8. Without dividing, give the remainder after dividing kx 3 +lx 2 +mx+n by x i. Check by division. 9. Without dividing, give the remainder after dividing x 2 +px-\-q by x+k. Check by division. 259. If in a division problem the remainder is zero, the division is exact, and the divisor is a factor of the dividend. 260. Determine, without dividing, whether the first ex- pression is a factor of the second in the following, and factor the factorable expressions: i. x 2 ...... 2. x 314 First- Year Mathematics .3. x i ...... x 2 i2X 13 4. x 2 ...... x 3 x*+$x + 2 5. #+7 ...... x 3 + sx 2 2gx 105. 261. The proof will now be given of the theorem we have been using, i. e., that, if in a given expression containing x (call it for brevity an expression in x), r is put for x, the ex- pression in x reduces to the remainder R, that is obtained by dividing the expression in x by x r. In division there is always a divisor d, a dividend D, a quotient Q, and a remainder R (sometimes zero), and the fol- lowing relation connects them: D=QXd+R. Then, if the expression in x is divided by x r, by this principle we have the expression in x=Q(x r)+R. Substitute on both sides, for x, the number r, then the expression in r=Q(r r)+R = QXo+R=R- Therefore, R, the remainder found by dividing the expression in x by xr, equals the expression in r, i. e., the given expres- sion with r put in place of x. This is called the remainder theorem. 262. Multiply the following expressions and make a rule for finding the constant term in the product from the constant terms of the factors: 1. (*+ 4 )(* + 5) 6. (x-s)(2x'+x-8) 2. G*-3)G* + 7) 7- 3. (*- 2 )(*-6) 8. 4. (* + 2)(* 2 +5) 9. 5. (*+3)(*'+*-4) 10. (x-3)(x-4)(x-s). ii. From problems i-io it is clear that if an expression with integral coefficients has a factor x a, then a is a factor of the constant or absolute term of the expression. Factoring, Quadratics, Radicals 315 12. Factor by the remainder theorem, x 3 jx 3 -\-i6x 12. The exact divisors of 12 are i, i, 2, 2, 3, 3, 4, 4, 6, 6 and 12, 12. For as = i, the value of xs jx 3 + i6x 12 is 2. For #= i, the value is 36 For x = 2, the value is o. Therefore, x 2 is a factor of x3 jx 2 + i6x 12. By division, the other factor is #' 5* + 6. Then, X3 jx* + i6xi2 = (x 2)(x 2)(x 3). 263. Factor by the remainder theorem: 1. x 3 i 6. ^+64 2. #3 7# 2 + 7# + i5 7. r 3 +r 2 22r 40 3. # 3 5# 2 2^ + 24 8. 2W 3 +gm 2 + i2w+4 4. # 3 +8 9. # 4 3# 3 2i# 2 +43^+60 5. # 3 + 2a3; 2 + 50 2 :x:+4a 3 10. x 4 i5# 2 + i 0^ + 24. 264. Solve ^ 3 + 3^ 2 13^ 15=0. x=> i, reduces the expression on the left to zero, then * ( i), or x+ 1 is a factor. By dividing we get: 3- 13*- 15 = (*+ l)(#2 + 2 *- 15) Then, Whence, x = - i, 3, - 5. Why ? Check. 265. Solve the following equations: 1. m 3 6m 2 + iim6 = o 4. x 3 +$x 2 45=93: 2. # 3 + 2;x; 2 = ii# + i2 5. ;y3 + i2o=7;y 2 + i4y 3. 4x+i6=x*+4x 2 6. k 3 +zk 2 4k 8=0 7. x* = $x 2 8. 9. a 4 na 3 +44a 2 760+48=0 io. 12 3i 6 First- Year Mathematics 266. Show by the remainder theorem that the following have a & as a factor and find the other factor. 1. a 2 b 2 5. a 6 b 6 2. a3-&3 6. a7_&7 3. a 4 b 4 7. a 8 b & 4. a5-&s 8. a9-&9. 9. Make a rule for forming the other factor in 1-8. 267. Show by the remainder theorem that the following have a -\-b as a factor, and find the other factor. 1. a3 + &3 6. a 2 -b 2 2. as+fcs 7. a 4 -b 4 3. a?+&7 8. a 6 -b 6 4. a 9 +6 9 9. a 8 b B 5. a^+fc 11 10. o IO -6 10 . 11. Make a rule for forming the quotient of the sum of the same odd powers of two numbers by the sum of those numbers. 12. Make a rule for forming the quotient of the difference of the same even powers of two numbers by the sum of those numbers. Miscellaneous Exercises 268. Factor, and check the results: 1. m 2 n 2 8. 450 2a 2 2. 4a 2 gb 2 9. g 6 +h 6 3. i r 4 10. a 9 + & 9 (3 factors) 4. x 2 y y* ii. x I2 +y 12 5. U3+V* 12. m 13 n 12 6. c 6 d 6 13. w 4 +4 7. a s +b 5 14. a 4 b 2 6a 3 b+ga 2 Factoring, Quadratics, Radicals T 5- x y +2X' e y'z-' +yz 4 29. 16. I 3 mn 2 I 2 ntn 2 xI 2 m 2 n 2 30. T > oc 2 cd 2od 2 17. y 2 ay 42& 2 31. 18. 5& 2 + io& 15 32. 19. 3a 2 gab 2iob 2 33. 20. gm 2 24#w + i6tt 2 34. a 2 +6 2 c 2 206 21. 4# 2 ~i~ $2xy -{- 3<)y 2 35. a 2 6 2 + 1/7? etc., that can be expressed only approximately without the use of the radical sign are called irrational numbers. When expressed with the aid of the radical sign, irrational numbers are called radicals. The product of two or more numbers is zero, if, and only if, at least one of the numbers is zero. In division the dividend, D, the quotient, Q, the divisor, d, and the remainder, R, are connected by the relation D = Q d+R. The remainder, R, found by dividing an expression in x by x r, equals the expression in r, i. e., the given expression with r put in place of x. This is called the remainder theorem. CHAPTER XIV POLYGONS. CONGRUENT TRIANGLES. RADICALS Interior Angles of Polygons 275. A plane figure bounded by straight lines is a polygon. 1 . Find the sum of the interior angles x, y, z, t, w, of a five- sided polygon (Fig. 242). From a point P, inside the figure, draw lines to the vertices (Fig. 243). How many triangles are thus formed ? What is the sum of the angles of each triangle ? Of FIG. 242 FIG. 243 all the triangles? What is the sum of the angles around P ? Prove that <+w+#+y + z = 5 180360. 2. Prove that the sum of the interior angles (1) of a 6-sided polygon is 6 180360 (2) of a y-sided polygon is 7 180360 (3) of a lo-sided polygon is 10 180360 (4) of an ^-sided polygon is s 180360 (5) of an n-sided polygon is n 180360. 276. A diagonal of a polygon is a straight line joining two vertices that are not consecutive. Convex Polygon Concave Polygon A polygon of three sides is a triangle. A polygon of four sides is a quadrilateral. 323 324 First- Year Mathematics A polygon of five sides is a pentagon. A polygon of six sides is a hexagon. A polygon of seven sides is a heptagon. A polygon of eight sides is an octagon. A polygon of ten sides is a decagon. A polygon of fifteen sides is a pentedecagon. A polygon of n sides is an w-gon. 277. If all sides are equal, the polygon is equilateral. If Equilateral Quadrilateral Equilateral Octagon all interior angles are equal, the polygon is equiangular. A Equiangular Hexigon Equiangular Quadrilateral polygon that is both equilateral and equiangular is a regular polygon. Regular Hexagon Regular Quadrilateral 278. THEOREM: If d represents the number of degrees in the sum of the interior angles of a polygon, and n the number of sides, then d=n - 180-360 (see 275). (i) Polygons, Congruent Triangles, Radicals - 325 1. Show that equation (i) may be written d = (n 2)180. (2) 2. Letting r stand for the number of right angles in the sum of the interior angles of a polygon of n sides, show that equation (2) may be written r = (n-2)2. (3) 3. Find from equations (2) and (3) the ratio, d: r. 4. Using equation (2) as a formula, find the sum of the interior angles of a quadrilateral; of a pentagon; of a hexa- gon; of a triangle. 5. Using equation (3) as a formula, find the sum of the interior angles of a heptagon; of an octagon; of a decagon; of a pentedecagon ; of a polygon of 18 sides. 6. In the equation d = (n 2)180: (1) What is the value of d, if n is 3 ? 9 ? 16 ? s ? (2) What is the value of n, if d is goo ? 7200 ? 7. Solving for n the equation d = (n 2)180, Show that =. (4) 1 80 8. By means of formula (4), find the number of sides of a polygon, the sum of whose interior angles is 3600; 17100; 1 80. 9. Solving for n the equation r = (n 2)2, Show that n = r -^. (5) 10. By means of formula (5), find the number of sides of a polygon, the sum of whose interior angles is 80 right angles; 192 right angles; 2 right angles. 326 First- Year Mathmatics 11. Each interior angle of a regular pentagon is x. Find x in degrees. In formula (2), p. 325, let w = 5; then d = 53: (why?), and $x = (5 2) 1 80. Solve for x. 12. Find the number of degrees in each interior angle of a regular hexagon; of a regular octagon; of a regular i7-gon; of a regular w-gon. 13. Using the equation obtained in the last part of problem 12 as a formula, find the number of degrees in each interior angle of a regular heptagon; of a regular decagon; of a regu- lar pentedecagon. 14. An interior angle of a regular polygon is 120. How many sides has the polygon ? In formula (2), p. 325, d = i2on. Why? Then i2ow = ( 2)180. Solve for n. 15. How many sides has a regular polygon, one of whose interior angles is 135? 140? 144? 170? 1 6. Can a tile floor be laid with tiles all having the shape of regular polygons of 3 sides? Of 4 sides? Of 5 sides? Of 6 sides ? Of 8 sides ? Of 1 5 sides ? 17. In Fig. 244, show that x=x r . FIG. 244 FIG. 245 18. In Fig. 245, a=a f , b=b'. Show that c=c'. 19. In Fig. 246, a=a', b=b'. Show that c=c'. Polygons, Congruent Triangles, Radicals 3 2 7 20. In triangle ABC, Fig. 247, A O is perpendicular to B C, and ZB = ZC. Show that y=y f . B FIG. 246 o FIG. 247 B 5 O FIG. 248 21. In Fig. 248, w=w', x=x'. Show that A O is perpen- dicular to B C. If two angles of one triangle are equal to two angles of another, the third angle of the first triangle is equal to the third angle of the second. Exterior Angles of Polygons 279- If ne side of an -gon is prolonged at each vertex, n exterior angles are formed, taking one at each vertex; for example, angles a, b, c, d, e, f (Fig. 249). i. Find the sum of the exterior angles of an w-gon. The sum of the interior angle and the exterior angle at each vertex is 180. In Fig. 249, FIG. 249 i8o, etc. Since there are n vertices, the sum of all the angles, interior and exterior, is n times 180 degrees, or i8ow degrees. i8on is the sum of all interior and exterior angles 180/1 360 is the sum of all interior angles 278. Subtracting, 360 is the sum of all exterior angles. 328 First- Year Mathematics 2. Rotate a pencil through all the. exterior angles of a polygon, taking one at each vertex. Through what part of a complete turn has the pencil rotated ? 3. The sum of the interior angles of a polygon is 7 times the sum of the exterior angles. Find the number of sides. Show that (n 2)180 = 7 " 3^- Solve for n. 4. How many sides has a polygon in which the sum of the interior angles is 9 times the sum of the exterior angles ? 5. How many sides has a polygon in which the sum of the interior angles is a times the sum of the exterior angles ? 6. Each exterior angle of a regular polygon is 15. Find the number of sides. 7. Each exterior angle of a regular 8-gon is x. Find x in degrees. 8. Each exterior angle of a regular w-gon is y. Find y in degrees. 9. An exterior angle of a regular polygon is of the adja- cent angle. Find the number of sides. 280. Some of the following definitions have been stated in preceding chapters; they are here given again for convenience. TRIANGLES 281. An isosceles triangle is a triangle that has two equal sides. Isosceles Triangle Scalene Triangle 282. A scalene triangle is a triangle that has no two equal sides. Polygons, Congruent Triangles, Radicals 3 2 9 283. A right triangle is a triangle that has a right angle. The side opposite the right angle is the hypotenuse. The symbol for triangle is A ! for tri- angles is A- QUADRILATERALS 284. A trapezoid is a quadrilateral having Right Triangle one pair of parallel sides. Parallelogram Trapezoid Rhombus Rectangle Square 285. A parallelogram is a quadrilateral having two pairs of parallel sides. 286. A rhombus is an equilateral quadrilateral. 287. A rectangle is an equiangular quadrilateral. 288. A square is a regular quadrilateral. 1. Show that a parallelogram is a trapezoid. 2. Show that a square is a rectangle. 3. Show that a square is a rhombus. 4. Is every trapezoid a parallelogram ? 5. Is every rectangle a square ? 6. Is every rhombus a square ? Congruent Triangles 289. Inaccessible distances frequently may be determined by triangles that are both equal and similar. i. To find the distance from a point A to a point B on the opposite side of a river (Fig. 250) a surveyor lays off line C D making angle x equal to the angle x', and the line A D, making angle y equal to the angle y*. 33 First-Year Mathematics What corresponding parts of triangles ABC and ADC are equal ? FIG. 250 How do triangles ABC and ADC seem to compare as to size and shape ? A D is measured and found to be 310 feet long. What seems to be the length of A B ? 2. Draw on paper a triangle as A B C (Fig. 251). Con- struct another triangle A'B'C' as follows: make A'B'=AB ZA'=ZA ZB'=ZB. Use the protractor, or the construction given in problem 24, pp. 148, 149. Cut triangle A'B'C' from the paper and see if it can be made to fit upon triangle ABC. How do the triangles ABC and A'B'C' compare as to size and shape ? Triangles which can be made to fit one over the other, or to coincide, are called congruent triangles. Polygons, Congruent Triangles, Radicals 331 If A'B'C' can be made to fit on A B C, what must be true of the relations of the parts B C and B'C' ? Of the parts C A and C'A' ? 3. Draw the triangles of problem 2 on the blackboard. On thin paper make a trace of A'B'C' with colored crayon, and fit the trace over ABC. In congruent triangles the parts (the angles and the sides) which coincide, when the figures are made to coincide, are called corresponding parts. Hence, it may be inferred that any part (angle or side) of one of two congruent triangles is equal to the corresponding part of the other. In problem 2, triangle A'B'C' was constructed so that one side and the two adjacent angles were equal to one side and the two adjacent angles of triangle ABC. It was seen that the two triangles can be made to coincide (are congruent). 4. It can be proved, without actually placing one triangle over the other, that any two triangles are congruent if a side and the two adjacent angles of one triangle are equal to a side and the two adjacent angles of the other. The proof of this theorem may be reasoned out thus: Let ABC and A'B'C' (Fig. 252) be any two triangles having AB=A'B' Imagine triangle ABC placed on A'B'C', the side A B coin- FIG. 252 ciding with side A'B'. (Why can A B be made to coincide with A'B' ?) 332 First- Year Mathematics Since ZA=ZA', the side AC falls along A'C' and the point C somewhere on A'C'. Since ZB = ZB', the side B C falls along B'C' and the point C somewhere on B'C'. Since C must be on both lines A'C' and B'C', it must fall on the point of intersection of A'C' and B'C'. (Why ?) Therefore, C falls on C'. Then the triangles ABC and A'B'C' coincide throughout and are congruent. In the same way in which ABC and A'B'C' were made to coincide, any two triangles can be made to coincide if a side and two adjacent angles of one triangle are equal to a side and the two adjacent angles of the other. 290. THEOREM: If in two triangles a side and the two adjacent angles of the one are equal respectively to a side and the two adjacent angles of the other, the triangles are congruent. i. PROVE: In two right triangles if the hypotenuse and a side of the one are equal respectively to the hypotenuse and a side of the other, the triangles are congruent. Let the hypotenuse, c, and the side, a, of one triangle (Fig. 253) be equal respectively to the hypotenuse, c, and the side, a, of the other triangle. FIG. 253 To prove the triangles congruent. Proaf. Place the triangle at the left so that the equal sides, a, coincide, as shown. Polygons, Congruent Triangles, Radicals 333 The line V +b is straight. Why? The figure formed of the two triangles is an isosceles tri- angle. Why ? Complete the proof. 2. Give the proof, using Fig. 254. 6' FIG. 254 3. In triangles RST and XYZ (Fig. 255), the parts marked in the same way are equal. Show, as in problem 4, p. 331, that triangles RST and XYZ are congruent. How do the sides T R and X Z compare in length ? T S and Z Y ? How do angles T and Z compare in size ? 4. In triangle D E F (Fig. 256), EG bisects angle E and makes equal angles with D F (i. e, is perpendicular to D F). What parts of the two triangles are known to be equal ? Show that triangles D E G and E F G are equal in all respects. FlG - 2 S 6 Hence, side DE=side E F; side D G=side G F; and angle D= angle F. 334 First-Year Mathematics 5. In triangles ABC and A'B'C' (Fig. 257) the parts marked in the same way are equal. Prove that'C A = C'A'. What other parts are equal? Give reason. FIG. 257 6. In Fig. 258, A B is equal to B C and the angles at A and C are equal as indicated. What other angles of the tri- angles A E B and BCD are equal ? Prove the triangles con- gruent, and angle A E B equal to angle B D C. A E < x >py FIG. 258 FIG. 259 7. In Fig. 259, a =b and c=d. Prove, by means of con- gruent triangles, that R S=U T, and R U = S T. 8. In the parallelogram A B C D, Fig. 260, x=x', y=y, AB = DC. Prove that A O = O C, B O = O D. r r' SR FIG. 261 FIG. 260 9. Triangles R S T and R'S'T' (Fig. 261) are congruent. Find x and the corresponding sides R T and R'T'. Polygons, Congruent Triangles, Radicals 335 10. The triangles in Fig. 262 are congruent. Find 5 and t. How long is B C ? A B ? 5S 3 A 2t+19 FIG. 262 n. Draw a triangle, as A B C (Fig. 263). Construct a triangle having the sides equal respectively to the sides of triangle ABC. FIG. 263 On a line, as D E, lay off A'B' equal in length to A B. With A' as center and a radius equal to A C, draw an arc F G above D E. With B' as center and radius equal to B C, draw an arc meeting arc F G at C'. Draw lines A'C' and B'C'. Triangle A'B'C' is the required triangle. 12. Prove that triangle A'B'C', Fig. 263, is congruent to triangle ABC. Proof. In triangle ABC (Fig. 263), with A as center and A C as radius, draw a semicircle x (see Fig. 264). In /^A B C (Fig. 263), with B as center and B C as radius, draw a semicircle y (see Fig. 264). 336 First- Year MatJiematics In AA'B'C' (Fig. 263), extend arcs through C', com- pleting semicircles x f and y' (see Fig. 264). Imagine triangle ABC placed on A'B'C'. Then A B can be made to coincide with A'B'. (Why ?) Since A C is equal to A'C' (why ?), semicircle x coincides with semicircle x', for circles drawn with the same center and radius coincide. Therefore point C must fall somewhere on the semicircle *'. (Why ?) Since B C is equal to B'C', semicircle y coincides with semicircle y', and point C lies somewhere on semicircle y. Therefore point C must be on the point of intersection of x' and y. Then A C coincides with A'C', and B C with B'C'. The triangles coincide throughout and are congruent. 13. Two triangles as in Fig. 265 or Fig. 266 have three sides of the one equal to the corresponding sides of the other. Prove as in problem 12 that A A B C and D E F are con- gruent. 291. Problems n and 12 establish the truth of the follow- ing theorem: If two triangles have three sides of one equal to the three corresponding sides of the other, the triangles are congrtient. Polygons, Congruent Triangles, Radicals 337 292. We have thus far proved three theorems on con- gruent triangles, viz. : I. If two sides and the included angle of one triangle are equal, each to each, to two sides and the included angle of another triangle, the triangles are congruent. See 145, p. 202. II. If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of another, the triangles are congruent. See 290, p. 332. III. If three sides of one triangle are equal, respectively, to the three sides of another, the triangles are congruent. See 291, P- 33 6 - These three theorems on congruent triangles are used in proving other theorems in this chapter. The statements of the theorems should therefore be carefully memorized by the student. 293. Circles and Triangles i. Prove the theorem: If a point is on the perpendicular bisector of a line, it is equally distant from the extremities of the line* Let L M be perpendicular to A B at the middle point O, and P be any point on L M. To prove P equally distant from A and B. Proof. Draw P A and P B and A prove triangles P O A and P O B congruent. (145.) ThenPA = PB. Why? B A.' M But P is any point on A B. IG 26 Therefore, any point on the per- pendicular bisector of a line is equally distant from the extremities of the line. * In the statement of this theorem, and of some others following, it is implied that there is only one perpendicular bisector of a line. 338 First- Year Mathematics FIG. 268 2. Prove the theorem in i, using Fig. 268. 3. Prove the theorem: If a point is equally distant from the extremities of a line, it is on the perpen- dicular bisector of the line. Let P be any point equally distant from the extremities of the line A B ; that is, PA=PB. (Fig. 269.) To prove that P is on the perpendicular bisector of A B. Proof. From O, the mid- point of A B, draw a line to P. Prove triangles P O A and P O B congruent. (291.) Then x=y and P O is perpendicular to A B. Why ? But P O also bisects A B. Why? Therefore P is on the perpendicular bisector of A B. But P is any point equally distant from the extremities of AB. Therefore, any point that is equally distant from the ex- tremities of a line is on the perpendicular bisector of the line. 4. Prove the theorem in 3, using Fig. 270; Fig. 271. FIG. 269 p FIG. 270 FIG. 271 5. Assuming that the perpendicular bisectors X M and Y R of two sides of triangle ABC meet at O, Fig. 272, prove (i) OB=OC, OC = OA, andOB-OA, Polygons, Congruent Triangles, Radicals 339 (2) O is on the perpendicular bisector of A B. Draw a circle with O as center, and O B as radius, and notice that the circle passes through C and A. FIG. 272 A circle passing through the three vertices of a triangle is circumscribed about the triangle. 6. Draw a triangle having an obtuse angle. Circumscribe a circle about the triangle. Construct the perpendicular bisectors of two sides (see problem 15, p. 147). Prove that the point where the perpendicular bisectors meet is equally distant from the three vertices. Notice the position of the center of the circle with respect to the triangle. 7. Circumscribe a circle about a right triangle. Notice the position of the center of the circle with respect to the triangle. In problem 5, where does the center of the circle lie with respect to the triangle ? 294. The distance from a point to a line is the length of the perpendicular from the point to the line. The symbol _1_ stands for "perpendicular," "perpendicular to," or "is perpendicular to." The symbol JJ stands for "perpendiculars." i. Prove the theorem: 340 First- Year Mathematics If a point is on t/ie bisector of an angle it is equidistant from the sides of the angle. Let M O be the bisector of A O B and P be any point on M O. To prove P equidistant from A O and B O. Proof. Draw PC _L A O and PD x BO. Prove triangles P O C and POD congruent. (292.) ThenPC = PD. Why? 2. Construct the bisector of an obtuse angle A O B. From any point P on the bisector, construct perpendiculars, P C and P D, to the sides. Draw a circle with P as center, that passes through C and D. The sides A O and B O (Fig. 274) are tangent to the circle. 3. Prove the theorem: If a point is equidistant from the sides of an angle it is on the bisector of the angle. FIG. 274 Let P be any point equidistant from the sides of angle A O B; that is, P C _L O A, and P D _i_ OB,andPC = PD. To prove that P is on the bisector of angle A O B. Proof. Draw a straight line through P and O. Prove triangles P O C and POD congruent (problem i, p. 332). Then x=y and PO bisects A O B. Why? Therefore P is on the bisector of A O B. 4. The bisectors C M and A L of two angles of triangle FIG. 275 Polygons, Congruent Triangles, Radicals 34i ABC meet at O. Perpendiculars are drawn from O to the sides. A Prove (i) O X = O Y, O Y = O Z, and O Z = O X. (2) O is on the bisector of angle B. Draw a circle with O as center, and O X as radius, and notice that the circle passes through Y and Z. Polygons is a general name for triangles, quadrilaterals, pentagons, hexagons, etc. Polygon is another name for an w-gon. A circle entirely within a polygon, all sides of which are tangent to the circle, is inscribed in the polygon. 5. Draw a triangle having an obtuse angle. Draw the inscribed circle. Construct the bisectors of two angles (see problem 20, p. 148). Prove that the point where the bisectors meet is equidistant from the three sides of the triangle. 6. Inscribe a circle in a right triangle. 7. In Fig. 277, show that Area of B O C = $r a Areaof AOC = ir b Area of A O B = \r c. Therefore, area of A B C = \ra-\-\r\) -\-\rc r.--U FIG. 277 342 First-Year Mathematics 8. Letting A stand for the area of triangle ABC, Fig. 277, and s stand for $(a+b+c), show that A=s r A and r . s THEOREM : The radius of a circle inscribed in a triangle is equal to the area of the triangle divided by one-half of the per- imeter. Application of the Theorems on Congruent Triangles 295. The theorems on congruent triangles 292 may be used in proving the correctness of the fundamental construc- tions in chap, vii, 111. VJJT- i. Prove that CF (Fig. 174, /'!\ p. 145) is perpendicular to A B. / \ Proof. Draw the lines D F '""" ~~^ * ^ * and E F as in Fig 278. FlG - 2 ? 8 Then in triangles D C F and E C F the following relations are known. DC = CE Why? EF = DF Why? CF = CF. Therefore triangles D C F and E D F are congruent. Why ? x/DCF=ZECF. Why? CF_LDE. Why? Hence the line C F, if constructed as in m,is perpendic- ular to A B at the point C. The symbol .'. means "therefore," or "hence." 2. Prove that the construction in problem 8 (p. 146) makes C F perpendicular to A B. Proof. In Fig. 176 draw D C and C E as in Fig. 279. C is equally distant from D and E. Why ? Polygons, Congruent Triangles, Radicals 343 FIG. 279 .'. C lies on the perpendicular bisector of D E (see problem 3, p. 340). Since F is equally distant from C D and E (why?), F lies on the per- pendicular bisector of D E. (Why ?) .'. the perpendicular bisector of x~~ D E contains C and F. .*. line C F must be the perpen- dicular bisector of D E, for through two points only one straight line can be drawn. .'. C F is perpendicular to A B. Why ? 3. Give proof for problem 15 (p. 147). In Fig. 1 80, draw the lines A F, B F, A G, GB (see W Fig. 280). Show as in problem 2 that F G is the perpendicular bisector of A B. Then H is the midpoint of A B. Why? 4. Give proof for problem 19 (p. 147)- In Fig. 181, draw lines E F and D F. Prove that triangles B E F and B D F are congruent. Then angle C B F equals angle A B F. Why ? 5. Prove problem 24 (p. 148). In Fig. 185, draw G H and O M. 6. In Fig. 281, A C = C B and x=y. Prove ZA=ZB. \\ A~. X. H FIG. 280 Prove triangles A C D and BCD congruent (145)- FIG. 281 FIG. 282 7. In triangle ABC, Fig. 282, AC=BC; prove ZA=ZB. Bisect Z C (see 1 1 1) and use problem 6. 344 First- Year Mathematics THEOREM : // a triangle has two equal sides, the angles oj tlie triangle opposite the equal sides are equal. 8. In Fig. 283, A C = C B. C D is drawn from the vertex to the midpoint of A B. Prove xy z=u CDxAB. THEOREM: In an isosceles triangle tlie line joining the vertex to the midpoint of the base FIG. 283 bisects the vertex-angle, and is perpendicular to tlie base. 9. If two angles of a triangle are equal, the sides opposite them are equal. In this statement, it is assumed that two angles of a tri- angle are equal. From this it is inferred that the sides oppo- site them are equal. It is to be proved that this is true. Let A B C be a triangle having ZA= ZB. To prove AC =B C. Proof. Bisect ZC. J ZA=ZB Why? x=y Why? r=s Why? Prove triangles ADC and B D C con- gruent. FIG. 284 Whence A C=B C. Why? 10. Prove that the line bisecting the angle at the vertex of an isosceles triangle bisects the base, and is perpendicular to the base. What is assumed in this statement ? What inference is drawn from this assumption ? 11. Prove that the altitude of an isosceles triangle bisects the base and also the vertex-angle. What is assumed here ? What is the inference from the assumption ? Polygons, Congruent Triangles, Radicals 345 12. PROVE: A diagonal of a parallelogram divides it into two congruent triangles. What is the assumption ? The inference from it ? Given a parallelogram A B C D, and B D one of the diagonals. To prove that triangles A B D and C B D are congruent. Proof. Since A B C D is a parallelo- o c gram A B is parallel to D C. (285.) /*"^ v .'. x=x' (see Summary, p. 151) / A D is parallel to B C. Why ? J- and;y=/. Why? FIG. 285 .'. Triangles A B D and D B C are congruent. Why ? 296. In every theorem one or more assumptions are made. From these assumptions, inferences (conclusions) are drawn. The assumptions in a theorem are called the hypothesis and the inferences the conclusion. The hypothesis and conclusion together contain everything that is stated in the theorem. Nothing can be in the hypothesis or in the conclusion unless it is in the theorem. 1. State the hypothesis and conclusion in the following theorems : (1) page 332, 290 (4) page 336, 291 (2) page 335, problem 16 (5) page 338, problem 3 (3) page 332, problem 5 (6) page 340, problem 3. 2. PROVE: The opposite sides of a parallelogram are equal. Observe that the hypothesis in this theorem is: A figure is a parallelo- gram; the conclusion: the opposite sides of this figure are equal. The hypothesis and conclusion must be stated always with reference to the particular figure used in the demonstration. Hypothesis. A B C D is a parallelogram. (Fig. 286.) Conclusion. A B = D C and A D =B C. Proof. Draw A C. 346 First- Year Mathematics Then triangle A B C is congruent to A D C. Why ? /. AB=DC andAD = BC. Why? The symbol -^ may be used for "is congruent to" or "coincides with." The statement, triangle JT IG 2 86 ABC is congruent to triangle D E F, may be written thus: B C^AD E F. Prove the following theorems: 3. Parallel lines intercepted between parallel lines are equal (Fig. 286). 4. The diagonals of a parallelogram bisect each other. The proof for this theorem is suggested in problem 12, p. 334. 5. A parallelogram having two adjacent sides equal is a rhombus. 6. The diagonals of a rhombus are perpendicular to each other. Hypothesis. A given figure is a rhombus, in particular A B C D, and A C and B D are the diagonals of A B C D. Conclusion. The diagonals of the figure are perpendicular to each other, i. e., A C is perpendicular to BD. Proof. DC = CB Why? DO = OB Why? Prove ZDOC=ZBOC. Prove OC.LDB. Then, ACJ.DB. Polygons, Congruent Triangles, Radicals 347 7. The diagonals of a rectangle are equal. In Fig. 288 prove A B C^A B D (287). 8. If two adjacent sides and the in- cluded angle of one parallelogram are equal respectively to two adjacent sides and the included angle of another, the parallelograms are congruent. In Fig. 289, AI^AI' FIG. 288 Show that parallelogram A B C D can be made to coin- cide with parallelogram A'B'C'D'. FIG. 289 Prove the following theorems: 9. If two adjacent sides of one rectangle are equal respec- tively to two adjacent sides of another, the rectangles are con- gruent. 10. Two squares having a side of one equal to a side of the other are congruent. In 105 and 106 the following theorem was studied: Two straight lines that make equal corresponding angles with a transversal are parallel. Thus in Fig. 160, p. 140, if a=e, then A B || C D.* 11. Prove that if ce (Fig. 160), then ae and A B || C D. 12. Prove that if c=e, then h=d and A B |j C D. *The symbol || means "is parallel to," or "are parallel." 348 First- Year Mathematics t 13. Prove that if two lines make equal alternate interior angles with a transversal, the corresponding angles are equal. 14. Prove that if two lines are cut by a transversal making the sum of the interior angles on the same side equal to two right angles, the corresponding angles are equal. In Fig. 1 60 let d+e = iSo. Then d+a = iSo. There- fore a=e. 15. Prove that two lines making equal alternate interior angles with a transversal are parallel. 1 6. Prove that if two lines are cut by a transversal making the sum of the interior angles on the same side equal to two right angles, the lines are parallel. 17. State three theorems which furnish tests for parallel lines. 1 8. If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram. A FIG. 290 Hypothesis. B C=A D. B C || A D. Conclusion. A B C D is a parallelogram. Proof. Draw A C. Then AA B C^AA C D. Why ? ZBAC=ZACD. ABliCD. .'. A B C D is a parallelogram. Why ? 19. A rhombus is a parallelogram. (285 and 286.) 20. A rectangle is a parallelogram. (285 and 287.) 21. A square is a parallelogram. (285 and 288.) Polygons, Congruent Triangles, Radicals 349 The Equilateral Triangle 297. The altitude -lines of a triangle are perpendiculars from the vertices to the opposite sides. 1. Prove that an altitude-line of an equilateral triangle bisects the base. Show by 292 that triangles AOB and A O C (Fig. 291) are congruent. 2. Prove that a line drawn from the vertex of an equilateral triangle to the mid- FIG. 291 FIG. 292 point of the opposite side, is perpendicular to that side. Use 33, page 41. The medians of an equilateral triangle are also altitude- lines. 3. Prove that the median to one side of an equilateral triangle bisects the angle opposite that side. 4. Find the altitude and the area of the equilateral triangle (Fig. 292). 5. Find the altitude and the area of an equilateral triangle whose side is 6; 8; 12; 3; 5. Extracting Square Roots 298. The process of extracting square roots of numbers is used in finding approximate values of irrational numbers such as i/75> 1/27 l/^> l/^Aj met with m the preceding problems 4 and 5. 1. Give the squares of the following numbers: i, 2, 3, 4, 5, 6, 7. 8, 9 10, 20, 30, 40, 50, 60, 70, 80, 90 100, 2OO, 3OO, 400, 500, 600, 700, 80O, 900. 2. Show that the square (i) of any one-digit number has one or two digits 35 First- Year Mathematics (2) of any two-digit number has three or four digits. (3) of any three-digit number has five or six digits. 3. Illustrate by squaring 456 that if a number, like 207,936, be marked off into two-digit groups, thus 20' 7 9' 36', the num- ber of groups is the same as the number of digits in the square root. 299. The numerals of arithmetic express numbers as so many tens and so many units, or so many hundreds, so many tens, and so many units, and so on. To find the square root of a number means to find first the tens, then the units; or first the hundreds, then the tens, then the units, according as the root is a two-digit, or a three-digit number. By problem 3 (298) one can tell at the outset how many digits there are in the square root of a given number. i. How many digits are there in the square roots of the following numbers ? 1,681, 2,116, 961, 81, 121,104, 19,881, 964,324. 300. To understand the way to find square roots, it is necessary only to see how the tens and the units enter into the squares of numbers, and then, to learn how to reverse the order of steps. ARITHMETICALLY ALGEBRAICALLY 57=50 + 7 n = 500 + 2. 50. 7 + 7 2 = 5 2 + ( 2 - 50 + 7)7 = 2,500 + 700 + 49 = 3,249 REVERSING THE STEPS 3,249 50 + 7 = 57, root 2,500 = trial divisor, 2X50=100 second root figure, 7 = 7 50 + 7 =107 749 = remainder Polygons, Congruent Triangles, Radicals t + u, root, Trial divisor, Second root figure, = sq. of tens 2tu + u 2 = remainder i. Extract the square root of 183,184, omitting all unneces- sary figures. i8'3i'84/| 428, square root 16 So 231 82 164 840 6,784 848 6,784 CHECK: 428* = 183,184. 2. Extract the square roots of the following: (1) 784 (5) 42,436 (9) 120,409 (2) 1,521 (6) 52,441 (10) 281,961 (3) 2,209 (7) 99,225 (n) 389,376 (4) 2,401 (8) 112,896 (12) 515,524. 301. Proceed with decimals precisely as with whole num- bers, carrying the two-digit grouping toward the right of the decimal point, and bringing down always the next two figures of the given number. i. Extract the square roots of the following: 19.36 1.5876 -0361 2.89 2.1316 51.2656 5.29 9.6721 .5329 114.49 11.0889 78.4996 210.64 .1225 .984064. 352 First- Year .Mathematics 2. Find approximate values, to four decimal places, of the following: l/; ; 1/3; 1/5; 1/6; 1/7. Annex zeros and proceed as with decimals. 302. The square root of algebraic expressions is extracted in the same way as the square root of arithmetical numbers. i. Extract the square root of gx 2 $oxz 1 2Xy+4y 2 + 2z 2 + 2oyz. For convenience arrange the polynomial in the order of powers of x, y, and z thus: 30*2+ 2oyz + 2$z 3 square root $x zy 52 ist trial ) divisor ) 2d root ( 1 2xy number ( 5^ ist complete ) divisor \ fi 1 2xy + 4V 2 = 2y (t>x zy) 030*2+ 2oyz-\- 2$z 3 2d trial divisor 2(3^ zy) 6x 4y 3d root ( 30*2 = 53 number ( 6x 2d complete divisor 6^4^52 30*3 + 2072 + 252* = 52 ' (6x 4y 52 CHECK: (3* zy 52)* = qx* 1 2xy + 4y 2 30*2 + 20^2 +2522. 2. Extract the square roots of the following: (i) (2) (3) x 2 (4) ()a 2 - (5) (6) (7) gz 2 I2az+4a 2 +c 2 +6cz Polygons, Congruent Triangles, Radicals 353 (8) r 2 + 25+7-4- 2 >-3 + ior 2 -ior (9) s 6 (10) 4 303. Frequently the work of approximating the roots of numbers that are not squares is much shortened by simplifying the irrational numbers before extracting the roots. i. In an equilateral triangle with side s and altitude h show that if s = 4, then h = i/l2 if 5 = 10, then & = i/7~5 if s= 8, then ^ = 1/48 if 5 = 14, then h = 1/147. Instead of approximating directly the values of j/ilz, 1/75, 1/48, 1/147, we may approximate directly only 1/3, and multiply the result by 2, 5, 4, 7, respectively, thus: 1/12 =21/3 = 2X1.732 + = 3.464 + 1/75 =5* / 3 = 5Xi-732 + = 8.660 + 1/48 =41/3 = 4X1.732 + = 6.928 + V / i47 = 7l // 3 = 7Xi. 732 + =12.124 + . 304. In reducing the irrational numbers i/f2, 1/7^5, etc., in 303, the following principle is used: The square root of a product is equal to the product of the square roots of the factors. 1. Prove, by multiplying, that V / 3 2> 5=3 V$. 3- /I' 3 ' VS=3 -3 ' 1/5 ' V / 5 = 3 2 5- Why? 2. Prove the following by multiplying: (1) 1/2*^ = 2 5 (4) Vr ' 5 = 7 ' ^5 (2) Vx*y*=xy (5) 1/4 25 = 2 -j (3) l/o6=a6 (6) l/T^~7 = 2l/7 354 First- Year Mathematics (7) li6- 3=4 -_l3 (10) __ (8) 1/12 = 2 1/3 (n) 1/108=61/3 (9) 1/27=31/3 (12) l/a 2 6 2 c=a&l/c. 3. Reduce l/i8. 4. Reduce l/32a 3 6 s a; 3 . K 320365*3 = ^ i6a a b*x* 2abx = V i6a 3 b*x* V zabx = ^abxV 2abx. 5. Reduce the following irrational numbers, using the principle in 304: (1) 1/Js (8) 1/128*33, (2) 1/50 (9) 31/75^3,10 (3) 1/28 (10) (4) l/48 (n) _ (5) 1/125 (12) !/27a 2 -i86 2 (6) 1/200 (13) (7) 1/288 t \ I .* 305. It should be kept in mind that the square root of a sum is not equal to the sum of the square roots of the separate terms. 1. Prove, by multiplying, that I/a 2 +b 2 is not equal to a+b. Show that (a + b) (a + b) is not equal to a* + b*. 2. Prove the following, by multiplying: (1) Vx 2 y* is not equal to x y (2) 1/9 + 16 is not equal to 3+4 (3) I/a 2 b is not equal to a 1/6. 3. Find an approximate value of the side s, and the area, of the equilateral triangle (Fig. 293). Polygons, Congruent Triangles, Radicals In triangle A O B s 3 = + 36. Why? 4 Show that 5 = 6.928 + and area A B = 20.784 + . 355 FIG. 293 FIG. 294 4. Find an approximate value of a side, and the area, of an equilateral triangle having an altitude 3; 9; 12; 1551 8. 5. Find an approximate value of a side, and the area, of the equilateral triangle (Fig. 294). Show that 5 = 1/18^ It is shown in the following problems 6 and 7, that Then 5 = 10(1 -M. 732 + ) = $. 773 + and area A B C =14.430 + . 6. Prove the following by multiplication: ( 2 \ (4\2 = 1L6 (fa \i \SJ Us W/ (3) () 2 =H (7) (f) 2 = (4) (A) 2 =T-U (8) (f) 2 =- 7. Give the values of the following square roots: (2) l/Jj[ (6) Vflh M (3) I/If (7) ^SV (11) 1/fil (4) 1/dhf (8) ^ll (12) 356 First-Year Mathematics 306. It is clear from 6 and 7, p. 355, that the square root of a common fraction is the square root of the numerator divided by the square root of the denominator. i. Give the following square roots. Im 2 + 2mn+n* \ xoo(a + 6) 2 , a 2 4x 2 -4xy+y* (3) A IT: (9) AlT la 2 * 2 22 16 : i6a 2 b 4 c 2 (mn) 4 ' I \(m-n) |~9(^ =: ^) 7 ' r x l(6*- 5 y) 6 (a + 3^ \2a + 6 2 \ a- C 4 2 r-5 2 2. Find approximate values, to four decimal places, of the following: 1/1; i/l; i/f; i/f. 307. To find an approximate value of v f we may ap- proximate the square root of 2, obtaining 1.4142, and the square root of 3, obtaining i .7320. Divide i .4142 by i .7320, obtaining .8165. But the following method of finding an approximate value of 1/f is shorter and gives a safer approximation : f = l / l- Why? . Why? Polygons, Congruent Triangles, Radicals 357 By the first method we obtain the square roots of two numbers, and then divide the root of the numerator by that of the denominator. By the second method, we reduce the radical number first, then find the square root of only one number, and divide by 3 i . Find an approximate value of J/f . .5773 + . 2. Find an approximate value of -%l . _ a 2 /- / -47i4a 2 /- \ \l - = \/ ~ 2X= ' V 2 ' V X*= V X, gx \ gx x \ gx 2 $x x approximately. 3. State a rule for simplifying irrational numbers like those in problems i and 2. 308. To simplify an irrational number which is the square root of a common fraction whose denominator is not a square : (1) Multiply denominator and numerator by a number that will make the denominator of the resulting fraction a square. (2) Factor and reduce to simplest form as in 307. Simplify the following, and find approximate values to four decimal places: 12. T/i4 i. V ^ 6. 2. I ^ 7- 3> , I 7 ./ ~ 8. 9- 4- -> 16 10. n. 13- 14. 358 First- Year Mathematics 1 6. Find the side and area of an equilateral triangle having an altitude 7; 2; 8; 10; 22. 17. In the equilateral triangle, Fig. 295, show that and area A B C = p 2 j/3. (2) 1 8. Using equations (i) and (2), problem 17, as formulas find the side and the area of an equilateral triangle having an, altitude 6; 9; 12; 10; 14. _^ i \ B FlG 19. In the equilateral triangle, Fig. 295, show that ^=71/3 (3) s 2 and area A B =1/3. (4) 4 20. By means of formulas (3) and (4), find the altitude and the area of an equilateral triangle having a side 6; 10; 9; 7. 21. In the equation & = -j/3, substitute 1/5 for h, and show that 5= . 1/3 2T/ / ^ To find an approximate value of - , we may find an approximate square root of 5, obtaining 2.2360, and of 3, obtaining 1.7320. Multiply 2.2360 by 2, and divide the result by i .7320, obtaining 2.5814. The following method of finding an approximate value of 2|/c - is shorter, and gives a safer approximation : 2 l/5 2 l/5 1/3 2 1/S ' 1/3 2 l/i5 > / ,0- ^ =* l -^- - 2 L - ^= ! 2 = i/ I 5- Give reasons. 1/3 1/31/33 3 Polygons, Congruent Triangles, Radicals 359 1/15 =3 -8729 + .-.^'15 = 2.5818 + . 22. In the following, multiply numerator and denominator by a number that will make the denominator of the resulting fraction a rational number. Simplify and find approximate values to four decimal places: / \ 3 r .\ v i r*.\ 1/18 SI/* <3>^ >?f W 23. Solve the equation 5 for 5 in terms of h, and find the value of 5 for h = 24. Solve the equation for A in terms of 5, and find the value of h f or s = 1/7. Problems Involving Radicals 1. Find the area A of an equilateral triangle in terms of the side s. 2. Using the equation obtained in problem i, find the side of an equilateral triangle whose area is 25 sq. in. ; 161/3 sc l- m - > A square inches. 3. Find the area A of an equilateral triangle in terms of the altitude h. 360 First- Year Mathematics 4. Using the equation obtained in problem 3, find the altitude of an equilateral triangle whose area is I7V/3 sc l- ft.; 35 sq. ft.; A sq. ft. 5. A plot of ground is to be staked off in the form of an equilateral triangle covering an area of 100 sq. ft. How long must the side be ? The altitude ? 6. How many degrees are there in each interior angle of an equilateral triangle? Show that six equal equilateral triangles may be so placed in a plane about a point that they just fill the angular magnitude about the point. How many degrees are there in each interior angle A, B, C, etc., of the polygon thus formed (Fig. 296) ? Show that the polygon is a regular hexagon. FIG. 296 7. Draw a regular hexagon starting with an equilateral tri- angle having a side 3. Circumscribe a circle about the hexagon. How long is the radius of the circle ? 8. Letting A be the area of a regular hexagon, and R the radius of the circum- scribed circle (Fig. 297), show that a side of the regular hexagon is R, and that FlG 2g7 9. Find the area of a regular hexagon whose side is 2 ; i ; 3; 45 15; s. Polygons, Congruent Triangles, Radicals 361 10. Find the radius of a circle circumscribed about a regular hexagon whose area is 31/3; 6]/f5; 91/7; A. 1 1 . Letting r be the radius of a circle inscribed in a regula hexagon (Fig. 298), find the area A of the hexagon, and s, th length of a side in terms of r. 12. Find the value of A and r, Fig. 298, in terms of 5. 13. Make problems that can be solved by the aid of the formulas ob- tained in problems n and 12. FlG - 2 9 8 14. Show that a side 5 of a square inscribed in a circle (Fig. 299) is equal to rj/2, and that the area A is equal to 2r 2 . 15. Find the values of A and R (Fig. 299) in terms of S. 1 6. Make problems that can be solved by means of the formulas obtained in problems 14 and 15. 17. Calculate the radius of the circle circumscribed about a square whose area is 625; 200; y 2 ; A. 1 8. The diagonal of a square is 2 inches longer than a side x. Find the area. 19. The diagonal of a square is b inches longer than a side. Find the length of a side. 20. Show that the area of triangle A B C in the rhombus, Fig. 300, is equal to xy. See problems 4 and 6, p. 346, and problem IQ, p. 348. 21. Show that the area of the rhombus, Fig. 300, is 2xy. The area of a rhombus is one-half the product of the diagonals. FIG. 299 362 First-Year Mathematics 22. A side of a rhombus is 15, and one diagonal is 24. Find the other diagonal and the area. 23. A side of a rhombus is 12, and one diagonal is 18. Find the area. 24. One diagonal of a rhombus is twice as long as the other, and FIG. 300" tne area i s I2 ^- Find the length of each diagonal. 25. Find the area of an isosceles right triangle, the hypote- nuse being 15; 31/7; h. Algebraic Problems Based on Geometry i. The sides and angles of the triangles being designated as in Fig. 301 , find c, x, and y. o FIG. 301 2. In the two triangles of Fig. 302, a=a', c=c*, and -' C* = ZB'; find the values of x, y, and z and of b, and ZC: Polygons, Congruent Triangles, Radicals 363 = 5# 12, ZA' = (i) If 6= = i5o-2Z, and (2) if 6=7+2!, &'=2? 15*; ZC = 23*- 3 z, ZC'= (3) If b=y(y-i), b'= 3 (y- 7 ); 5(2^-7); ZC=z(z-4), ZC / = i6(z-6). 3. With angles and sides of two triangles of values that may be. designated as in Fig. 303, find the values of a, c, and x. A A C FIG. 303 4. In Fig. 304 find r and s. B FIG. 304 (i) if 6 = 28, &' = 2 r--, a = 23, and a'=r + 1 ; (2) if 6=3r+2s i, &'=68, and a = 2r + 2-5$, a'=6o ; (3) if b=6r s 6, & / =3(r+s)+2, and a=3r + 2$ + 2, and 3 6 4 First- Year Mathematics 5. In Fig. 305 the triangles have sides and angles of values as designated. Find x, y, and z. FIG. 305 6. The triangles in Fig. 306 have sides and angles of values as designated. Find x, y, and z. FIG. 306 7. The two triangles of Fig. 307 are congruent. If the values of the angles may be designated as shown, what are the values of x, y, and z ? FIG. 307 8. The lengths of one pair of opposite sides of a parallelo- gram are Sx +6y and 12 y, and the lengths of the other pair of opposite sides are and i . Find x, y, and 3 4 the lengths of the sides of the parallelogram. 309. The corresponding sides of triangles whose angles are equal, each to each, are proportional. Polygons, Congruent Triangles, Radicals 365 For example, if in Fig. 308, ZA=ZA', ZB = ZB', and ZC=ZC', thena:a'=b:b'=c:c'. FIG. 308 The ratio of any pair of corresponding sides, as a : a', b: b', etc., is called the ratio of similarity of the figures. If the ratio of similarity is 3, find the values of x, y, and z under the following conditions: y 3. a=x-\-y + T > z and #'=4 3 b = 2X+y+6z and b' = j x c=x + 2y + gz and c' = q 3 4. a=x+y+z and a' = 3 rSz and ' = 3 2Z and c'=4 and a = ^ x ^. a == T > x ~t~ y ~r 4^ and a == 7 andc'=^ + i c=ic+';'y + ^z and c'=6. :. a=^c+z, and a' b -\xz and 6' = for a " m f f j- * ' Mend by 1 iine-R. [Unusual mending timexrhareed extra] ,.... Stab by/\...No.Sect/. < l J . Sewby^2'.. Before sewing,Score\"ress..-. Strip Sect.... [Scoring is necessary on stiff or heavy paper] Rate This book bound by Pacific Library Eluding Corn- par.y, Log Angeles, specialists in Library Bindinj,'. Work and materials furnished are guaranteed to wear indefinitely to satisfaction of purchaser, and any defects appearing in either will be made good with- out additional charge. "Bound to wear." UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles k is DUE on the last date stamped below. SEP 28 1971 SE? fiEC'D a D-URL 1373 1973 LD-URt i DEC 1 1373 MOV 2 9 1973 LD LD- 1 9 jg APR APR Q , 198Q Form L9-7(-7,'56(.C1107s4)444 3 \ V58 00573 6805