•^WSBJmWi^s 
 
 
 
 
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 IN MEMORIAM 
 FLORIAN CAJORl 
 
EIGHTH GRADE 
 MATHEMATICS 
 
 By 
 
 Harry M. Keal 
 
 Head of the Mathematics Department 
 
 Cass Technical High School 
 
 Detroit, Michigan 
 
 and 
 
 Nancy S. Phelps 
 
 Grade Principal 
 
 Southeastern High School 
 
 Detroit, Michigan 
 
 1 
 
 nfi^nius 
 
 ATKINSON, MENTZER ^ COMPANY 
 
 NEW YORK CHICAGO ATLANTA DALLAS 
 
 ^ 
 
COPYRIGHT, 1917, BY' 
 ATKINSON, MENTZER & COMPANY 
 
Introduc 
 
 tion I 
 
 THE growth of this series of Mathematics for Secondary Schools, 
 has covered a period of seven years, and has been simultaneous 
 with the growth and development of the shop, laboratory, and 
 drawing courses in Cass Technical High day school, as well as in the 
 evening and continuation classes. 
 
 The authors have had clearly in mind the necessity of first developing 
 a sequence of mathematics that would enable the student to recognize 
 fundamental principles and apply them in the shop, drawing room, and 
 laboratory; and, second to so develop the course that each year's work 
 would be a unit and not depend upon subsequent development for 
 intelligent application. 
 
 It has been assumed that the school work-shop, drawing room, and 
 laboratory would furnish opportunity to apply mathematics and that it 
 was not necessary to exhaust every possible application in the 
 mathematics class. 
 
 The authors have been aware of the popular demand for a closer union 
 of algebra and geometry, but have recognized that demand only when 
 the union came about naturally and would assist the mathematical 
 sequence desired. 
 
 Instructors in the wood shop, pattern shops, machine shop, drawing 
 rooms, chernistry, physics, and electrical laboratories, etc., have furnished 
 examples of mathematical apph cation incident to the respective 
 subjects. Hundreds of problems arising in the industries, have been 
 brought in by the machinists, sheet metal workers, carpenters, electrical 
 workers, pattern makers, draughtsmen, etc., etc., coming to the evening 
 and continuation classes. Complete charts of machine shop work and 
 electrical distribution requirements have been made, including a 
 statement of the required sequence of mathematics. All of this material 
 has been classified, with a view to the mathematical sequence. 
 
 The net result is a series of Mathematics so organized that a mastery 
 of the text makes it possible for a student to use mathematics intelli- 
 gently in the various departments of the school, in the industries, and 
 at the same time prepare for college mathematics. 
 
 E. G. ALLEN, 
 
 Director Mechanical Department, 
 Cass Technical High School, 
 Detroit, Mich. 
 
Digitized by the Internet Archive 
 
 in 2007 with funding from 
 
 IVIicrosoft Corporation 
 
 http://www.archive.org/details/eighthgradematheOOkealrich 
 
TABLE OF CONTENTS 
 
 PAGE 
 
 CHAPTER I 
 The Equation 1 
 
 CHAPTER II 
 Evaluation 15 
 
 CHAPTER III 
 The Equation Applied to Angles 25 
 
 CHAPTER IV 
 
 Algebraic Addition, Subtraction, Multiplica- 
 tion AND Division 38 
 
 CHAPTER V 
 Ratio, Proportion and Variation 79 
 
 CHAPTER VI 
 Pulleys, Gears and Speed 96 
 
 CHAPTER VII 
 Squares and Square Roots 107 
 
 CHAPTER VIII 
 
 Formulas 123 
 
 V 
 
CHAPTER I 
 THE EQUATION 
 
 % 
 
 10 
 
 ^ 
 
 Fig. 1 
 
 ^ 
 
 1 In order to find the weight of an object, it was placed on 
 one pan of perfectly balanced scales (Fig. 1). It, together 
 with a 3-lb. weight, balanced a 10-lb. weight on the other pan. 
 If 3 lbs. could be taken from each pan, the object would be 
 balanced by 7 lbs. This may be expressed by the equation, 
 x+3 = 10, in which the expressions x+3 and 10 denote the 
 weights in the pans, the sign ( = ) of equality denotes the per- 
 fect balance of the scales, arid x is to be found. 
 
 2 Equation: An equation is a statement that two expres- 
 sions are equal. The two expressions are the members of the 
 equation, the one at the left of the equality sign being called 
 the first member, and the one at the right, the second member. 
 
 3 From the explanatory problem, it will be seen that the 
 same number may be subtracted from both members of an equation. 
 
 Oral Problems: 
 Solve f or X : • 
 
 1. x+7 = 21 3. x+1. 1=3.5 
 
 2. x+2 = 3 4. x+2|=7^ 
 
 1 
 
 5. x+f = 
 
 5 _ il 
 12 
 
THE EQUATION 
 
 Fig. 2 
 
 Jj. It is required to find the weight of a casting. It is found 
 that 3 of them exactly balance a 10-lb. weight (Fig. 2). If the 
 weight in each pan could be divided by 3, one casting would be 
 balanced by 3j lbs. This may be expressed by the equation, 
 
 3x=10, 
 
 X = 3i. (dividing both members by 3) 
 
 5 From this explanatory problem, it will be seen that hoik 
 members of an equation may he divided hy the same number. 
 
 Oral Problems: 
 Solve for x : 
 
 1. 4x = 12 
 
 2. 2x=16 
 
 3. 5x = 9 
 
 4. llx = 33 
 6. l.lx=12.1 
 
 Example : Solve f or x : 5x + 1 2 = 37 
 
 5x = 25 Why? 
 x= 5 Why? 
 

 
 THE EQUATION 
 
 
 
 
 Exercise 
 
 1 
 
 
 Solve for the unknown: 
 
 
 
 
 1. 
 
 x+l = 5 
 
 
 11. 
 
 9x+8=116 
 
 2. 
 
 x+7 = 9 
 
 
 12. 
 
 7w+5f = 12f 
 
 3. 
 
 2a+6 = 16 
 
 
 13. 
 
 28t+14 = 158 
 
 4. 
 
 3x+7 = 28 
 
 
 14. 
 
 3x+4j = 9 
 
 5. 
 
 5s+17 = 62 
 
 
 16. 
 
 15s+.5 = 26 
 
 6. 
 
 9x+12 = 93 
 
 
 16. 
 
 llx+J=8_9 
 
 7. 
 
 2x+l = 6 
 
 
 17. 
 
 1.2x+2 = 14 
 
 8. 
 
 5y+3 = 15 
 
 
 18. 
 
 4.6x+8 = 100 
 
 9. 
 
 4n+3.2 = 15.2 
 
 
 19. 
 
 6.3x+2.4=15 
 
 10. 
 
 12m+8 = 98 
 
 
 20. 
 
 7.1m+.55 = 9.07 
 
 Q 
 
 A 
 
 ATA 
 
 fxibrlfsn 
 
 10 
 
 Fig. 3 
 
 ^ 
 
 6 If an apparatus is arranged as in Fig. 3, it is seen that if 
 the upward pull of 2 lbs. be removed, 2 lbs. would have to be 
 
4 THE EQUATION 
 
 put upon the other pan to keep the scales balanced. This 
 may be expressed by the equation, 
 
 4x-2=10 
 
 4x = 12 (Adding 2 to both members) 
 X = 3 (Dividing both members by 4) 
 
 7 From this problem, it will be seen that the same number 
 may he added to both members of an equation. 
 
 Oral Problems: 
 
 Solve for x: 
 
 
 
 1. 
 
 3i-4 = 8 
 
 
 
 2. 
 
 7x-l = 15 
 
 
 
 3. 
 
 4x-f = 7} 
 
 
 
 4. 
 
 5x-.l = .9 
 
 
 
 6. 
 
 2x-i = 6i 
 
 Exercise 2 
 
 
 Solve for the unknown : 
 
 
 
 1. 
 
 x-7 = 10 
 
 11. 
 
 13r-21=44 
 
 2. 
 
 2x-13 = ll 
 
 12. 
 
 12s-35 = 41 
 
 3. 
 
 5x-17 = 13 
 
 13. 
 
 7f-4 = 26 
 
 4. 
 
 4x-ll = 25 
 
 14. 
 
 4x-3 = 16 
 
 6. 
 
 3x-7=15 
 
 15. 
 
 9x-3.2=14.8 
 
 6. 
 
 12x-4 = 44 
 
 16. 
 
 3m-2=3.1 
 
 7. 
 
 7m-5 = 31 
 
 17. 
 
 14x-5 = 21 
 
 8. 
 
 4x-18=18 
 
 18. 
 
 2.1x-3.2 = 3.1 
 
 9. 
 
 17t-3i=13f 
 
 19. 
 
 .5y-4 = 5.5 
 
 10. 
 
 llx-9 = 90 
 
 20. 
 
 3x-9j = 8.5 
 

 SIMILAR 
 
 TERMS 
 
 Exercise 3. 
 
 Review 
 
 Solve for the unknown : 
 
 
 
 1. 9x-8=46 
 
 
 6. 3w-lJ = lf 
 
 2. 8x-7 = 53 
 
 
 7. 19t-.2 = 3.6 
 
 3. 5x+7 = 28 
 
 
 8. 6.37n+3.92 = 73.99 
 
 4. 28m-9 = 251 
 
 
 9. .4x+.02=.076 
 
 5. 16y+13 = 73 
 
 
 10. 2s+2i = 9f 
 
 11. Two times a number increased by 43 equals 63. Fi 
 
 the number. 
 
 12. If 10 be added to 3 times a number, the result is 50. 
 What is the number? 
 
 13. Five times a number decreased by 6 equals 39. Find 
 the number. 
 
 14. If 55 be subtracted from 7 times a number, the result 
 is 22. What is the number? 
 
 15. If to 57 I add twice a certain number, the result is 
 171. What is the number? 
 
 18. State the first five problems in this exercise in words. 
 
 How many yards of cloth are 7 yds. and 5 yds.? 
 How many dozens of eggs are 12 doz. and 3 doz.? 
 How many bushels of wheat are 8 bushels and 1 1 bushels? 
 How many b's are 4 b's and 7j b's? 
 How many x's are 3x and 9x? 
 
 In such expressions as 2a -|-3x+4+2x+7+3a, 2a and 3a 
 may be combined, 3x and 2x, and also 4 and 7, making 
 the expression equal to 5a+5x+ll. 2a and 3a, 3x and 
 2x, 4 and 7 are called similar terms. 
 
6 THE EQUATION 
 
 Example 1. Solve for x: 
 
 4x+13x— 7x = 40 
 
 lOx = 40 (combining similar terms) 
 x = 4. Why? 
 Example 2. Solve for x: 
 
 14x+7-2x = 43 
 
 12x+7 = 43 Why? 
 
 12x = 36 Why? 
 
 x = 3 Why? 
 
 9 8-7+3 = ? 8x-7x+3x = ? 
 
 8+3-7 = ? Similarly 8x+3x-7x = ? 
 3-7+8 = ? 3x-7x+8x = ? 
 
 10 These problems illustrate the principle that the value of 
 an expression is unchanged if the order of its terms is changed, 
 provided each term carries with it the sign at its left. 
 
 NOTE: If no sign is expressed at the left of the first term, the sign 
 (+) is understood. 
 
 Example 1. 15-3x+llx = 39 
 
 8x+15 = 39 Why? 
 
 8x = 24 Why? 
 
 x = 3 Why? 
 
 Example 2: lly-4+21 = 50 
 
 lly+17 = 50 Why? 
 
 lly = 33 Why? 
 
 y = 3 Why? 
 
ORDER OF TERMS / 
 
 Exercise 4 
 
 Solve: 
 
 1. 4x-x = 12 
 
 2. llx+3x = 35 
 
 3. 14x-3x=44 
 
 4. 3x+7x=90 
 
 5. 9y-9y+8y = 40 
 
 6. 4s+3s-2s = 17 
 
 7. 3.2x+2.3x = 110 
 
 8. 1.3y-2.7y+3.3y = 57 
 
 9. 11.2x+7.8x = 57 
 
 10. l.ls-1.4s+lls = 26.75 
 
 Exercise 5 
 Solve: 
 
 1. x-18 = 17 9. 12x-8x+6+3x = 8+12 
 
 2. x+18 = 21 10. 25x+20-7x-5+5x = 56+5 
 
 3. 2y-16 = 30 11. 8x+60+4x-50+3x-7x = 20 
 
 4. 3m-m = 21 12. 2-2x+7x=42.5 
 6. 3m-l = 23 13. 3yH-1.2+2y=46 
 
 6. 6.5x-l.l = 50.9 14. x-1.25x+12.7+3.5x = 38.7 
 
 7. 4x+3x-3 = 25 16. 2x+ 15.8 -2.3x+14.5x = 186.2 
 
 8. lly-4y-7 = 28 16. 6.15y-1.65y+7.8 = 57.3 
 
 17. 8y+6.875+2y=46.875 
 
 18. z-8.73+5.37z = 61.34 
 
 19. 5t-8.75t+6.87+8t = 57.87 
 
 20. 3.73x-9.23+15x = 65.69 
 
8 THE EQUATION 
 
 11 Equations often arise in which the unknown appears in 
 both members. In that case, aim to make the term containing 
 the unknown disappear from one member, and the one contain- 
 ing the knowrij from the other member. 
 
 Example 1: 3x — l=x+3 
 
 3x = xH-4 (adding 1 to both members). 
 2x = 4 (subtracting X from both members). 
 X = 2 Why? 
 
 Note that in adding or subtracting a term from both members, it must 
 be combined with a similar term. 
 
 Example 2: 
 
 5x+4-3x-l = 7-x+2 
 
 2x4-3 = 9 — X (combining similar terms in each member). 
 2x = 6-x Why? 
 3x = 6 Why? 
 x = 2 Why? 
 
 Exercise 6 
 
 Solve: 
 
 1. 2x-6 = x 
 
 2. 2x+3 = x+5 
 
 3. 13x-40 = 8+x 
 
 4. 7y-7 = 3y+21 
 6. 9x-8 = 25-2x 
 
 6. 20+10x = 38+4x 
 
 7. 3x+9+2x+6 = 18+4x 
 
 8. 5x+3-x = x+18 
 
 9. 7m-18+3m=12+2m+2 
 
 10. 18+6m-f-30+6m = 4m-|-8H-12+3m+3+mH-29 
 
CLEARING OF FRACTIONS 9 
 
 11. 25x+20-7x-5 = 56-5x+5 
 
 12. 10x-61-12x+27x = 8x-41+20+4x+25 
 
 13. 25f+5x+6x+9|-2x=180-8x-8| 
 
 14. 2.8x+39.33+x = 180-1.2xH-32.09-7.16 
 
 15. 5x+26f+9x = 360-5x-143f 
 
 12 If an object in one pan of scales will balance a 4-lb. weight 
 in the other, it will be readily seen that 5 objects of the same 
 kind would need 20 lbs. to balance them. This may be 
 expressed by the equation, x = 4 
 
 5x = 20 (multiplying both members by 5). 
 
 13 From this problem, it will be seen that both members of an 
 equation may be multiplied by the same number. 
 
 This principle is needed when the equation contains frac- 
 tions. The process of making fractions disappear from an 
 equation is called clearing of fractions. 
 
 tJf. RULE: To clear an equation of fractions, multiply both members 
 by the lowest common denominator (L. C. D.) of all the fractions 
 contained in the equation. 
 
 Example 1 
 
 Example 2: -— == — 
 
 ^»= 
 
 = 4 
 
 
 x+6 = 
 
 = 8 (multiplying both members by 2). 
 
 x = 
 
 = 2 Why? . 
 
 r r _ 
 3 Y 
 
 _16 
 ' 3 
 
 
 7r-3r = 
 
 = 112 
 
 (multiplying both members by 21). 
 
 4r = 
 
 = 112 
 
 Why? 
 
 r = 
 
 = 28 
 
 Why? 
 
10 THE EQUATION 
 
 „ ,^ m_3,m7m 
 
 Example 3: — — 3t7t+ — = - — — 
 4 1^ 5 5 3 
 
 15m- 198+ 12m = 84 -20m Why? 
 
 27m -198 = 84 -20m Why? 
 
 47m -198 = 84 Why? 
 
 47m = 282 Why? 
 
 m = 6 Why? 
 
 15 The four principles used thus far may be more generally 
 stated as follows : 
 
 1. // equals are added to equals, the results are equal. 
 
 2. // equals are subtracted from equals, the results are equal, 
 
 3. // equals are multiplied by equals, the results are equal. 
 
 4 . // equxds are divided by equals, the results are equal. 
 
 Solve: 
 
 
 
 Exercise 7 
 
 
 1. 
 
 5^-^ = 10 
 3 6 
 
 6. 
 
 x_2_x 
 2 3 6 
 
 2. 
 
 ?+? = 9 
 5 4 
 
 7. 
 
 y = ?+16 
 3 7 
 
 3. 
 
 ?i'+?r=i7 
 
 3 4 
 
 8. 
 
 ?x+3 = ^+4 
 
 4. 
 
 x+Jx = 6 
 
 9. 
 
 2x x_x 1 
 9 6 18 3 
 
 6. 
 
 x-|x = 7 
 
 10. 
 
 3x 1 X ,_ 
 7"3 = 27+' 
 
PRINCIPLES OF EQUATIONS 11 
 
 11. l|s+fs = s+13 13. -+4r--=26+lJr 
 
 ^7 4 3 
 
 ^ ^^ 4 2 3 4 10 
 
 15. 7x+y+-+23=-+5ix+113 
 
 16 Sometimes it is convenient to make the term containing 
 the unknown disappear from the first member, and the one 
 containing the known, from the second. 
 
 Example 
 
 1: 
 
 x+6 = 
 
 = 3x- 
 
 -2 
 
 
 
 
 6 = 
 
 = 2x- 
 
 -2 
 
 Why? 
 
 
 
 8 = 
 
 = 2x 
 
 
 Why? 
 
 
 
 x = 
 
 = 4 
 
 
 Why? 
 
 Example 
 
 2: 
 
 L^ = 4 
 3x 
 
 
 
 
 16= 12x Why? (L. C. D. is 3x) 
 x = l| Why? 
 
 Exercise 8 
 
 Solve: 
 
 
 1. 
 
 L5 = 5 
 
 
 a 
 
 2. 
 
 5=15 
 
 a 
 
 3. 
 
 1=2 
 4x 
 
 , 3x 7 
 
 4. — = - 
 
 4 2 
 
 ^ 16 2x 
 6. — = — 
 
 5 3 
 
 6. 14 = x+9 
 
 7. 17 = 2x-3 
 
12 THE EQUATION 
 
 8 x+10 = 2x-9 12. ^+47=-+4n 
 
 2 7 
 
 9. 2x-2i = 5x-17} 13. ^-l=lZ?-^-2ia 
 
 ^ ^ 2 3 3 
 
 10. 7x+20-3x = 60+4x-50+8x 14. .lx+6.2 = .3x+.2 
 
 11. 3m+60 = 15m-f3-2m+7 15. 10H-.lx = 5+|x 
 
 Exercise 9 
 
 Solve: 
 
 1. 7m_8 = 5i-— 5. 2_t 5_t^t t ^^ 
 
 6 12 3 9 6 2 
 
 2. 7x-8 = 6x+ix 6. 1+''—'^ = '^-^ 
 
 ^ 2 5 6 3 4 
 
 3. ??-^=25i-^- 7. y-^+2i = -^-?:+^ 
 
 56 3 12 8484 
 
 4. ??+3 = ^x-2 8. -x-^x+4| = 3x+- 
 
 3 6 3 5 ^ 15 
 
 9. lly^x-l^x-302 = 60+l|x+183 
 
 10. x-3|+Jx = 9j-^ 
 
PROBLEMS 13 
 
 Exercise 10 
 
 1. Five times a certain number equals 155. What is the 
 number? 
 
 2. Four times a number increased by 7 equals 43. Find the 
 number. 
 
 3. Twelve times a number decreased by .18 is equal to 
 17.82. Find the number. 
 
 4. There are three numbers whose sum is 72. The second 
 number is three times the first, and the third is four times the 
 first. What are the numbers? 
 
 5. The sum of two numbers is 12 and the first is 4 more 
 than the second. What are the numbers? 
 
 6. If 10 is subtracted from three times a number, the 
 result is twice the number. Find the number. 
 
 7. If 1^ of a number is increased by 6, the result is 30. 
 Find the number. 
 
 8. The sum of J, ^ and ^ of a number is 26. What is the 
 number? 
 
 9. Divide 19 into two parts so that one part is 5 more 
 than the other. 
 
 10. Divide 19 into two parts so that one part is 5 times 
 the other. 
 
 11. Divide $24 between two persons so that one shall 
 receive $2j more than the other. 
 
 12. A farmer has 4 times as many sheep as his neighbor. 
 After selling 14, he has 3^ times as many. How many had 
 each before the sale? 
 
14 THE EQUATION 
 
 13. Two men divide $2123 between them so that one receives 
 $8 more than 4 times as much as the other. How much does 
 each receive? 
 
 14. Three candidates received in all 1020 votes. The first 
 received 143 more than the third, and the second 49 more than 
 the third. How many votes did each receive? 
 
 16. A man spent a certain sum of money for rent, f as 
 much for groceries, $2 more for coal than for rent, and $28 
 for incidentals. In all he paid out $100.00. How much did 
 he spend for each? 
 
 16. A farmer has 24 acres more than one neighbor and 62 
 acres less than another. The three together own one square 
 mile of land. How much has each? 
 
 17. A man traveled a certain number of miles on Monday, 
 •f- as many on Tuesday, f as many on Wednesday as on Mon- 
 day, and on Thursday 10 miles less than twice as many as he 
 did on Monday. How far did he travel each day if his trip 
 covered 82 miles? 
 
 18. One man has 3 times as many acres of land as another. 
 After the first sold 60 acres to the second, he had 40 acres 
 more than the second then had. How many acres did each 
 have before the transaction? 
 
 19. One boy has $10.40 and his brother has $64.80. The 
 first saves 20 cents each day, and his brother spends 20 cents 
 each day. In how many days will they have the same amount? 
 
 20. A man after buying 27 sheep finds that he has 1^ 
 times his original flock. How many sheep had he at first? 
 
CHAPTER II 
 
 EVALUATION 
 
 17 Definite Numbers: The numerals used in arithmetic have 
 definite meanings. For example, the numeral 7 is used to 
 represent a definite thing. It may be 7 yards, 7 pounds, 7 
 cubic feet or 7 of any other unit. Also in finding the circum- 
 ference of a circle, we multiply the diameter by w which has a 
 fixed value. Numerals and letters which represent fixed values 
 are called definite numbers. 
 
 18 General Numbers: The area of a rectangle is found by 
 multiplying the base by the altitude. This may be expressed 
 by bXa, in which the value of b may be 10 ft., 6 in., 30 rds., 
 or any number of any unit used to measure length, and a may 
 be any number of a like unit. Letters which may represent 
 different values in different problems are called general numbers. 
 
 19 Signs: When the multiplication of two or more factors is 
 to be indicated, the sign of multiplication is often omitted or 
 expressed by the sign (•)• Thus 7XaXbXm is written 
 7-a-b-m or more often 7abm. 
 
 NOTE: Care should be taken in the use of the sign (•) to distinguish 
 it from the decimal point. 7-9 means 7X9, 7.9 means 7i^o. 
 
 W Coefficient: The expression 7abm may be thought of as 
 7ab-m, 7 abm, 7-abm, or 7b -am, etc. 7ab, 7a, 7, and 7b 
 are called the coefficients of m, bm, abm, and am respectively. 
 
 1. In the following, what are the coefficients of x*^ 4abx; 
 ^xyz; 17mxw. 
 
 2. Name the coefficients of ab in the following: S^axby; 
 fmabz; .Obnsa. 
 
 3. What is the coefficient of 17 in 17mxw? 
 
 15 
 
16 EVALUATION 
 
 The coefficient of a factor or of the product of any number 
 of factors, is the product of all the remaining factors. In 
 8axy, 8 is the numerical coefficient. The numerical coefficient 
 1 is n£ver written, laxy is written axy. 
 
 21 Power: If all the factors in a product are the same, as 
 x-x-x-x, the product is called a power, x-x-x-x is read ''x 
 fourth power" and is written x*- a- a- a- a- a is read "a fifth 
 power" and is written a^. b-b or b^ is *'b second power" but 
 is more often read ^'b square." In the same way b-b-b (b^) 
 is called "b-cube." 
 
 22 Exponent: The small number written at the right and 
 above a number is called its exponent and it indicates the power 
 of the number. The exponent 1 is never written, x means 
 x^ or ''x first power." 
 
 23 Base: The number to be raised to a power is called the 
 base. 
 
 Name the numerical coefficients, bases and exponents in the 
 following : 
 
 V^x^, Sjaio, 3.7m2n^ f^r^ l|m l|m^ 
 
 24 Sign of Grouping: The Sign of Grouping most commonly 
 used is the parenthesis ( ) and means that the parts enclosed 
 are to be taken as a single quantity. For example, 3(x-y) 
 means that x-y is to be multiplied by 3 making 3x - 3y. (x -y)^ 
 means (x-y) (x-y) (x-y). 
 
 25 Evaluation: Evaluation of an expression is the process of 
 finding its valu^ by substituting definite numbers for general 
 numbers in the expression, and performing the operations 
 indicated. 
 
EVALUATION OF EXPRESSIONS 17 
 
 Example 1 : Evaluate 4:ix^x^ if a = 3, x = 2. 
 4a2x3 = 4.32.2'3=4-9-8 = 288. 
 
 a2 Sb* , m^ 
 
 Example 2: Find the value of ~, ; ~^ TTt 
 
 ^ m^ c2 2a3 
 
 when a=l, b = 
 
 = 2, c = 5, m = 2. 
 
 a2 5b4 m^ 
 m^ c2 2a3 
 
 P 5-24 25 
 
 2^ 5^ ' 2V 
 
 
 1 80 32 
 ~ 8 25 2 
 
 
 
 =i-?-« 
 
 
 5-128+640 
 
 
 40 
 
 
 40 ^^^^ 
 
 Example 3: Evaluate a(a — b+y^) when a =13, b = 3, y = 4. 
 a(a-b+y2) = 13(13 -3+42) 
 
 = 13-26 
 
 = 338 
 
 Exercise 11 
 
 Evaluate the following if a = 8, b = 6, c = 4, d = 2, x = 9: 
 
 1. 2x 7. 3x2 
 
 2. x2 8. (3x)2 
 
 3. 3x 9. llax 
 
 4. x3 10. 2abcd 
 6. 4x 11. 2a2x3 
 6. x^ 12. x2-a2 
 
18 EVALUATION 
 
 13. x(a+b) 17. —:-+:; 
 
 X b d 
 
 14. 4b(x-c) 18. (x+a)(c-d) 
 
 15. a2+2ab+b2 19. Viad 
 
 16. c2-2cd+d2 20. ab(c-3) 
 
 Exercise 12 
 
 Find the value of the following, when a = 2, b = 3, c = 7, 
 d = 4, m=l, x = 5: 
 
 1. iaVc 11. (3x+7)(c-2) 
 
 2. x3-a3 12. Vb2+d2 
 
 3. x'+d^ 3a2 
 
 13. — (x2-c2+25) 
 
 4. 3b2-4m2 bd 
 
 6. xM-a^m 14. a3(x-c+3m)(c2+d2) 
 e! 2a2x3(c-d) 15. ^^ 
 
 7. 4+^ 16- ^(x2+a2-b2)(c2-d2-m2) 
 a^ d 2d 
 
 8. ^(x+a)c 17. Vx(a+b) 
 
 9. ^a3x2c(b3-d2) 18. ^d(a+b)+c 
 c2 x2 
 
 ^^- {:^ + l^— 7 ^^- ^5x(a+b) 
 
 b^ d^ a^ 
 
 20. (a+b)(b+c)-(b+c)(x+d) + (x+d)(d+m) 
 
PERIMETER FORMULAS 
 
 19 
 
 Evaluation of Formulas 
 
 26 A Formula is the statement of a rule or principle in terms 
 of general numbers. For example, distance traveled is equal to 
 rate times time. 
 
 Formula, d = r-t 
 
 Iwt 
 
 Example 1 : Evaluate b = — (Formula for board feet) 
 
 whenl = 16', w = 8", t = 2" 
 168. 2 
 
 b = 
 
 12 
 
 = 21 J 
 
 Example 2: Evaluate A = ^h(b+bO (Area of a trapezoid) 
 ifh = 3A,b=12i b' = 6i 
 
 A = i.3^(12H6i) 
 
 A _ 1 q 3 .18^ 
 
 1 51 75 
 
 2*16' 4 
 
 3825 
 
 128 
 
 A= 
 
 29yV8 or 29.102- 
 
 Perimeter Formulas 
 
 27 The perimeter of a figure enclosed by straight lines is the 
 sum of its sides. 
 
 6 
 
 a 
 
 Fig. 4. Square 
 
 Fig. 5. Rectangle 
 
20 
 
 EVALUATION 
 
 Fig. 6. Triangle 
 
 Fig. 7. Quadrilateral 
 
 Exercise 13 
 
 1. The perimeter of a square (Fig. 4) is equal to 4 times 
 one side. P = 4a. Find P, if a = 9. 
 
 2. Find the value of P, in P = 4a, if a = ij. 
 
 3. Find the value of P, in P = 4a, if a = 1.175. 
 
 4. The perimeter of a rectangle (Fig. 5) is equal to 
 a+b+a+b = 2a+2b = 2(a+b). P = 2(a+b). Find P, if 
 a = 3, b = 5. 
 
 5. Find P, in P = 2(a+b), if a = |, b = |. 
 
 6. Find P, in P = 2(a+b), if a= 1.7862, b = 2.1324. 
 
 7. The perimeter of a triangle (Fig. 6) is expressed by the 
 formula, P = a+b+c. Find P, if a = 7, b = ll, c=19. 
 
 8. Evaluate P = a+b+c, if a = f, b = f, c = f. 
 
 9. Find the value of P, in P = a+b+c, if a = 7.621, b = 
 8.37, c = 1.3. 
 
PERIMETER PROBLEMS 21 
 
 10. The perimeter of a quadrilateral (Fig. 7) is expressed 
 by the formula, P = a+b+c+d. Find P, if a = 20, b = 15, 
 c=13, d=14. 
 
 11. Evaluate P = a+b+c+d, when a=lf, b = lf, c = 1y^, 
 d = li. 
 
 12. Find P, in P = a-f-b+c+d, if a = 172.32, b = 96.3, 
 c = 81.04, d = 56.2. 
 
 Exercise 14. Equations Involving Perimeters 
 
 1. The perimeter of a square is 96. Find a side. 
 
 2. The perimeter of a triangle is 114. The first side is 6 
 less than the second and 24 less than the third. Find the sides. 
 
 3. Find the dimensions of a rectangle whose perimeter is 
 48 if the length is 3 times the width. 
 
 4. Find the dimensions of a rectangle if its length is 4 
 more than the width and xts perimeter is 82. 
 
 6. The length of a rectangle is 4 more than twice the 
 width and its perimeter is ISS^^. Find the length. 
 
 6. The perimeter of a rectangle is 48.648. Find the width 
 if it is J of the length. 
 
 7. The perimeter of a rectangle is 94. The width is 11.3 
 more than ^ of the length. Find the length and the width. 
 
 8. The perimeter of a quadrilateral is 176. The first side 
 is J of the second, the third is 8 more than the second, and the 
 fourth is 3 times the first. Find the sides. 
 
22 
 
 EVALUATION 
 
 Exercise 15. Area Formulas 
 
 Fig. 8. Rectangle 
 
 Fig. 9. Parallelogram 
 
 b 
 
 Fig. 10. Triangle 
 
 I 
 
 Fig. 11. Trapezoid 
 
 1. The area of a rectangle (Fig. 8) is equal to the base 
 multiplied by the altitude. A = a-b. Find A, if a =11.5, 
 b=18.6. 
 
 2. Evaluate A = a.b, if a = 2|, b = 3f. 
 
 3. Express the result of problem 2 in decimal form. 
 
 4. The area of a parallelogram (Fig. 9) is the base times 
 the altitude. A = a-b. Find A, if a=ly^g^, b = 6.71. 
 
 6. The area of a triangle (Fig. 10) is \ the product of the 
 base and altitude. A = ib.h. Find A, if b = 12.23, h. = 6.57. 
 
 6. Evaluate A = ^b.h, if b = 9f, h = 4|. 
 
 7. The area of a trapezoid (Fig. 11) is J the product of 
 the altitude and the sum of the parallel sides. A = Jh(b+b'). 
 Find A, if h = 10f, b = 19f, b' = 12f 
 
 8. Express the result of problem 7 in a decimal correct 
 to .001. 
 
CIRCLE AND GENERAL FORMULAS 23 
 
 Exercise 16. Circle and Circular Ring Formulas 
 
 Fig. 12. Circle 
 
 Fig. 13. Circular Ring 
 
 1. C = 27rT. (Fig. 12). Find C, if ;r = 3.1416 (See art. 17) 
 r=li 
 
 2. C = ttB. Find C, if D = 5.724. 
 
 3. A = 7rr\ Find A, if r= l|. 
 
 4. A=.7854D2. Find A, if D = 5.724. 
 
 5. A = ;r(R2-r2) (Fig. 13). Find A, if R = 7i, r = 4j. 
 
 Exercise 17. General Formulas 
 
 Evaluate the following formulas for the values given: 
 
 1. P = awh, if a = 120, w = .32, h = 9|. 
 
 2. W=-.p, if 1 = 25, h = 4j, p = 60. 
 
 h 
 
 3. F = ljd+iifd = lf. 
 
 4. L=lfd+|, if d = 2i. 
 
 6. S = 2gt^, if t = 4. (g is a definite number. Its value is 32.16), 
 
 6. S = |g t^+vt, if t = 3, V = 7. 
 
 7. D= Va2+b2+c2, if_a = 3, b = 4, c=12. 
 
 8. V = |h(b'+b+ Vb-bO, if h = 2f, b=12, b' = 3. 
 uv 
 
 9. F = 
 
 u+v, 
 10. V = |^r3, if r = 2.3. 
 
 if u = 11.5, v = 6.5. 
 
24 EVALUATION 
 
 Checking Equations 
 
 28 The solution of an equation may be tested by evaluating 
 its members for the value of the unknown quantity found. 
 If its members reduce to the same number, the value of the 
 unknown is correct. 
 
 Example: 2x+?^5^^^ = 3x+l. 
 5 
 
 _ , 6x-2 _ , , 
 
 2xH ^- = 3x4-1. Why? 
 
 o 
 
 10x+6x-2= 15x+5. Why? 
 x = 7. Why? 
 
 Check: 
 
 5 
 
 14+8 = 21 + 1. 
 22 = 22. 
 
 Exercise 18 
 
 Solve and check: 
 
 1. 6y-7 = 3y+20. ^ 2(x+2) 
 
 2. ll=3x+9. ^ 
 
 = 7. 
 
 X V ^ 8. Z(?±^.^ = f+2. 
 
 3. 3-1 = 2-2. 12 6 4 
 
 4. 2(2x+5) = 13. 
 
 9. 2x-l = f(5-x)-l|. 
 
 7(5-x) 
 
 6. 
 
 6(z-6) = z+8. N^*^- i(5-x) = 
 
 6. ?^^ = 3. 10. ?(x+l)+^^-l=4i 
 
 5 5 4 5 
 
CHAPTER III 
 THE EQUATION APPLIED TO ANGLES 
 
 29 Angle: If the line OA (Fig. 14) 
 revolves about O as a center to the 
 position OB, the two lines meeting at 
 the point O form the angle AOB. The 
 point O is called the vertex of the angle 
 and the lines OA and OB are called 
 the sides of the angle. 
 
 Fig. 14. Angle 
 
 Fig. 15. Right Angle 
 
 B A 
 
 Fig. 16. Straight Angle 
 
 A 
 
 Fig. 17. Perigon 
 
 30 Right Angle: If the line turns 
 through one fourth of a complete rev- 
 olution (Fig. 15), the angle is called a 
 Right Angle. 
 
 31 Straight Angle: If the line turns 
 through one half of a complete rev- 
 olution (Fig. 16), the angle is called a 
 Straight Angle. 
 
 32 Perigon: If the line turns through 
 a complete revolution (Fig. 17), re- 
 turning to its original position, the 
 a-ngle is called a Perigon, 
 
 How many right angles in a straight angle? 
 How many right angles in a perigon? 
 How many straight angles in a perigon? 
 
 25 
 
26 
 
 THE EQUATION 
 
 Fig. 18. Protractor 
 
 S3 A Protractor (Fig. 18) is an instrument used for measuring 
 and constructing angles. On it, a straight angle is divided 
 into 180 equal parts called degrees, written 180°. 
 
 How many degrees in a right angle? 
 
 How many degrees in a perigon? 
 
 Fig. 19 
 
 Drawing Angles 
 34 Example: Draw an angle of 37°. 
 
 Using the straight edge of the protractor, draw a straight 
 line OA. Place the straight edge of the protractor along the 
 line OA, with the center point at O. Count 37° from the point 
 
THE PROTRACTOR 
 
 27 
 
 where the curved edge touches OA and mark the point B 
 (Fig. 19). Again use the straight edge of the protractor to 
 connect the points O and B. 
 
 Exercise 19 
 
 1. Draw an angle of 30°. 
 
 2. Draw an angle of 45°. 
 
 3. Draw an angle of 60°. 
 
 4. Draw an angle of 120°. 
 
 5. Draw an angle of 135°. 
 
 6. Draw an angle of 150°. 
 
 • 7. Draw an angle of 18°. 
 
 8. Draw an angle of 79°. 
 
 9. Draw an angle of 126°. 
 10. Draw an angle of 163°. 
 
 Measuring Angles 
 
 Fig. 20 
 
 35 Example: Measure the angle AOB. 
 
 Place the straight edge of the protractor along one side of 
 the angle as OA, with its center at the vertex of the angle 
 (Fig. 20). Count the number of degrees from the point where 
 the curved edge of the protractor touches OA to the point 
 where it crosses the line OB. The angle AOB contains 54°. 
 
28 
 
 THE EQUATION 
 
 F/GZl 
 
 F/G 28 
 
 Exercise 20 
 
 1. Measure the angle in Fig. 21. 
 
 2. Measure the angle in Fig. 22. 
 
 3. Measure the angle in Fig. 23. 
 
 4. Measure the angle in Fig. 24. 
 
 6. Measure the angle in Fig. 26. 
 
 7. Measure the angle in Fig. 27. 
 
 8. Measure the angle in Fig. 28. 
 
 9. Measure the angle in Fig. 29. 
 
 6. Measure the angle in Fig. 25. 10. Measure the angle in Fig. 30 
 
 Reading Angles 
 
 36 Reading Angles: An angle is 
 read with the letter at the vertex 
 between the two letters at the 
 ends of the sides. The angle 1 
 in Fig. 31 is read BAG or CAB 
 and is written Z BAG or Z GAB. 
 ReadtheangleZ2;Z3. (Fig.31). 
 
 Fig. 31 
 
READ NG ANGLES 
 
 29 
 
 A r/G34 S 
 
 A r/G 35 
 
 Exercise 21 
 
 1. Read the Zs 1, 2, 3, (Fig. 32). 
 
 2. Read the Zs 1, 2, 3, 4, (Fig. 33). 
 
 3. Read the Zs 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, (Fig. 34). 
 
 4. Read the Zs 1, 2, 3, 4, 5, (Fig. 35). 
 
30 THE EQUATION 
 
 Exercise 22 
 
 1. Measure the ZCAD (Fig. 31). 
 
 2. Measure the Z ACB (Fig. 32). 
 . 3. Measure the ZCDA (Fig. 33). 
 
 4. Measure the ZEFA (Fig. 34). 
 
 5. Measure the ZBGF (Fig. 35). 
 
 Fig. 36 
 
 37 Zl +Z2 + Z3 +Z4 = ZAOB 
 (Fig. 36). If AGE is a straight Hne, 
 the Z AOB contains 180°. Therefore 
 
 Zl+Z2+Z3+Z4 = 180°. 
 
 ^ 38 The sum of all the angles about a 
 point on one side of a straight line is 180°. 
 
 Fig. 37 
 
 Exercise 23 
 
 Fig. 38 
 
 Fig. 39 
 
 1. Find X in Fig. 37. Check with a protractor. 
 
 2. Find x in Fig. 38. Check. 
 
 3. Find the unknown angle in Fig. 39. Check. 
 
 4. Three of the four angles about a point on one side of 
 
 a straight line are 16°, 78°, 51°, respectively, 
 angle. 
 
 Find the fourth 
 
ANGLE EQUATIONS 31 
 
 5. Find the three angles about a point on one side of a 
 straight line if the first is twice the second, and the third is 
 three times the first. 
 
 6. Draw with a protractor the angles of problem 5 a^ in 
 Figs. 37, 38, 39. 
 
 7. Find the three angles about a point on one side of a 
 straight line if the first is twice the third, and the second is 
 a right angle. 
 
 8. Draw the angles of problem 7. 
 
 9. Find the four angles about a point on one side of a 
 straight line if the second is 5° less than the first, the third is 
 6° more than the first, and the fourth is 68°. 
 
 10. Draw the angles of problem 9. 
 
 Exercise 24 
 
 Example : . 
 
 The three angles about a point on one side of a straight line 
 
 4 X 
 
 are represented by x+6°, ^x — 12°, and 78° — ^. Find x and 
 
 the angles. 
 
 x+6+|x-12+78-|=180°. Why? 
 
 o o 
 
 3x+18+4x-36+234-x = 540. Why? 
 6x+216 = 540. Why? 
 6x = 324. Why? 
 x = 54. Why? 
 
 x+6 = 54+6 = 60° 1st angle. 
 |x - 12 = 72 - 12 = 60° 2nd angle. 
 
 78 - 1 = 78 - 18 = 60° 3rd angle. 
 o 
 
 NOTE: The fact that the sum of the angles found is 180° checks the 
 problem. 
 
32 THE EQUATION 
 
 If tne angles about a point on one side of a line are repre- 
 sented by the following, find x and the angles: 
 
 1. |x, x+4, lix+2. 
 
 2. fx-2, iVx-f 7, 3(x+7) Jx+19. 
 
 3. 4(x+l), 7(2x-ll), 127-6X. 
 
 4. 3x-i, 2x, 2f (2x+l), |(x+6). 
 6. ix+40, 2x-9, 129.18-2X. 
 
 6. Find the angles about a point on one side of a straight 
 line if the first is 25° more than the second, and the third is 
 three times the first. 
 
 7. Find the angles about a point on one side of a straight 
 line if the first is 6 times the second, plus 16°, and the third 
 is J of the first, minus 4°. 
 
 8. Find the five angles about a point on one side of a 
 straight line if the second is J of the first, the third is 5° more 
 than f of the first, the fourth is 10° less than twice the first, 
 and the fifth is 22^°. 
 
 Fig. 40 
 
ANGLES ABOUT A POINT 33 
 
 39 Z1+ Z2+ Z6=180° Why? 
 
 Z7+Z4+Z5 = 180° Why? 
 
 Therefore, Z1+ Z2+ Z3+ Z4+ Z5 = 360°. 
 
 40 The sum of all the angles about a point is 360°. 
 
 Exercise 25 
 
 If all the angles about a point are represented by the fol- 
 lowing, find X and the angles: 
 
 1. |x,88-jx,ljx-13, 4(^+11). 
 
 DO 6 
 
 2. 23+-, 136- -,-+93, -+17. 
 
 4 5' 3 2 
 
 3. 4(x-5), ?+5li, 3x+47|. 
 
 4. i(3x-36), i(2x+15), - +30, 82-|x, x+48j. 
 
 6 
 
 5. |x+3.15, 3(x+1.75), J(x+94.05). 
 
 6. The sum of four angles is a perigon. One is 18° more 
 than three times the smallest, another is 59° more than the 
 smallest, and the last is 18° less than twice the smallest. Find 
 the four angles. 
 
 Supplementary Angles 
 
 41 Supplementary Angles: If the sum of two angles is a 
 straight angle or 180°, they are called supplementary angles. 
 Each is the supplement of the other. 
 
 Exercise 26 
 
 1. What is the supplement of 16°; 92°; 24°; 13|°; 15lf°? 
 
 2. x is the supplement of 80°. Find x. 
 
34 THE EQUATION 
 
 3. X is the supplement of x+32°. Find x and its supple- 
 ment. 
 
 4. 2x — 20° and 7x+47° are supplementary angles. Find 
 X and the angles. 
 
 6. One of two supplementary angles is 24° larger than the 
 other. Find them. 
 
 6. The difference between two supplementary angles is 
 98°. Find them. 
 
 7. One of two supplementary angles is 4 times the other. 
 Find the angles. 
 
 8. How many degrees in an angle which is the supplement 
 of 3j times itself? 
 
 9. One of two supplementary angles is 27° less than 3 
 times the other. Find the angles. 
 
 10. One of two supplementary angles is y of the sum of the 
 other and 63°. Find the angles. 
 
 4^ The supplement of an unknown angle may be indicated 
 by 180° -X. 
 
 Indicate the supplement of y°; d°; fx°; |^y°. 
 
 When a problem involves two supplementary angles, but is 
 such that one is not readily expressed in terms of the other, let 
 X equal one angle, and 180— x the other. 
 
 Exercise 27 
 
 1. 1^ of an angle, plus 55° is equal to ^ of its supplement, 
 plus 4°. Find the supplementary angles. 
 
 Let X = one angle 
 180 — X = other angle 
 then fx4-55 = -|(180-x)+4. 
 
 2. The sum of double an angle and 12j° is equal to | the 
 supplement of the angle. Find the supplementary angles. 
 
COMPLIMENTARY ANGLES 35 
 
 3. If an angle is trebled, it is 30° more than its supplement. 
 Find the supplementary angles. 
 
 4. If an angle is added to J its supplement, the result is 
 128°. Find the supplementary angles. 
 
 5. If f of an angle, minus 16°, is added to f of its supple- 
 ment, plus 72°, the result is 190°. Find the supplementary 
 angles. 
 
 Complementary Angles 
 
 4S Complementary Angles: If the sum of two angles is a right 
 angle or 90°, they are called complementary angles. Each is 
 the complement of the other. 
 
 Exercise 28 
 
 1. What is the complement of 82°; 9°; 71°; 10^°; 43|°? 
 
 2. X is the complement of 32°. Find x. 
 
 3. X is the complement of x+76°. Find x and its com- 
 plement. 
 
 4. fx+ 12°, and §xH- 10° are complementary angles. Find 
 x and the angles. 
 
 5. One of two complementary angles is 25° larger than 
 the other. Find them. 
 
 6. The difference between two complementary angles is 
 37f °. Find them. 
 
 7. One of two complementary angles is three times the 
 other. Find the angles. 
 
 8. How many degrees in an angle that is the complement 
 of 2| times itself? 
 
 9. One of two complementary angles is 7° more than twice 
 the other. Find the angles. 
 
 10. One of two complementary angles is f of the sum of 
 the other and 23°. Find the angles. 
 
36 THE EQUATION 
 
 44 The complement of an unknown angle may be indicated by 
 90 — X. Indicate the complement of y°; m°; fx°; ^y° 
 
 When a problem involves two complementary angles, but 
 is such that one is not readily expressed in terms of the other, 
 let X equal one angle, and 90 — x the other. 
 
 Exercise 29 
 
 1. The sum of an angle and \ of its complement is 46°. 
 Find the angle. 
 
 2. The complement of an angle is equal to twice the angle 
 minus 15°. Find the angle. 
 
 3. If 20° is added to five times an angle, and 20° sub- 
 tracted from ^ of the complement, the two angles obtained, 
 when added, will equal 114°. Find the angle. 
 
 4. f of an angle is equal to f of its complement, minus 
 14°. Find the angle. 
 
 5. f of the complement of an angle, plus 15° is equal to 
 treble the angle. Find the angle. 
 
 Exercise 30 
 
 1. The sum of J, ^, and f of a certain angle is 126°. Find 
 the number of degrees in the angle. 
 
 2. The supplement of an angle is equal to four times its 
 complement. Find the angle, its supplement and complement. 
 
 3. The sum of the supplement and complement of an 
 angle is 98° more than twice the angle. Find the angle. 
 
 4. The complement of an angle is 20° more than \ of its 
 supplement. Find the angle. 
 
 5. The sum of an angle, J of the angle, its supplement, 
 and its complement is 243°. Find the angle. 
 
REVIEW OF ANGLES 37 
 
 6. The complement of an angle is equal to the sum of the 
 angle and J of its supplement. Find the angle. 
 
 7. An angle increased by ^ of its supplement is equal to 
 twice its complement. Find the angle. 
 
 8. ^ the supplement of an angle is equal to 3 times its 
 complement, plus 20°. Find the angle. 
 
 9. The sum of treble an angle, f of its complement, and 
 -| of its supplement is equal to 62° less than a perigon. Find 
 the angle. 
 
 10. Y^Y of the complement of an angle is equal to J the 
 supplement, plus 3°. Find the angle. 
 
 11. The three angles about a point on one side of a straight 
 line are such that the second is 89° more than ^ of the sup- 
 plement of the first, and the third is f of the complement of 
 the first. Find the three angles. 
 
 12. The sum of four angles is 223°. The second is twice 
 the first, the third is ^ the supplement of the second, and the 
 fourth is the complement of the first. Find the four angles. 
 
 13. There are four angles about a point. The second is 
 ^ the first, the third is the supplement of the second, and the 
 fourth is the complement of the second, plus 30°. Find the 
 four angles. 
 
 14. There are five angles about a point on one side of a 
 straight hne. The second is ^ of the first, the third is ^ the 
 supplement of the second, the fourth is f the complement of 
 the second, the fifth is 10°. Find the five angles. 
 
 15. Express by an equation that the supplement of an angle 
 is equal to its complement, plus 90°. 
 
 Does 41° for x check the equation? 
 
 Does 25°? Does 153°? What values may x have? 
 
CHAPTER IV 
 
 ALGEBRAIC ADDITION, SUBTRACTION, 
 MULTIPLICATION AND DIVISION 
 
 Positive and Negative Numbers 
 
 45 1. The top of a mercury column of a thermometer stands 
 at 0°. During the next hour it rises 4°, and the next 5°. 
 What does the thermometer read at the end of the second hour? 
 
 2. The top of a mercury column stands at 0°. During the 
 next hour it falls 4°, and the next, it falls 5°. What does it read 
 at the end of the second hour? 
 
 3. If the mercury stands at 0°, rises 4°, and then falls 5°, 
 what does the thermometer read? 
 
 4. If the thermometer stands at 0°, falls 4°, and then 
 rises 5°, what does the thermometer read? 
 
 6. If the mercury stands at 0°, rises 4°, and then falls 4°, 
 what does the thermometer read? 
 
 6. A traveler starts from a point and goes north 17 miles, 
 and then north 15 miles. How far and in which direction is he 
 from the starting point? 
 
 7. A traveler starts from a point and goes south 17 miles, 
 and then south 15 miles. How far and in which direction is he 
 from the starting point? 
 
 8. A traveler goes 17 miles south, and then 15 miles north. 
 How far and in which direction is he from the starting point? 
 
 38 
 
POSITIVE AND NEGATIVE NUMBERS 39 
 
 9. A traveler goes 17 miles north, and then 15 miles south. 
 How far and in which direction is he from the starting point? 
 
 10. A traveler goes 17 miles south, and then 17 miles north. 
 How far is he from the starting point? 
 
 11. An automobile travels 35 miles east, and then 40 miles 
 east. How far and in which direction is it from the starting 
 point? 
 
 12. An automobile travels 35 miles west and then 40 miles 
 west. How far and in which direction is it from the starting 
 point? 
 
 13. An automobile travels 35 miles west, and then 40 miles 
 east. How far and in which direction is it from the starting 
 point? 
 
 14. An automobile travels 35 miles east, and then 40 miles 
 west. How far and in which direction is it from the starting 
 point? 
 
 16. An automobile goes 35 miles east, and then 35 miles 
 west. How far is it from the starting point? 
 
 16. A boy starts to work with no money. The first day he 
 earns $.75, and the second $.50. How much money has he at 
 the end of the second day? 
 
 17. A boy has to forfeit for damages $.75 more than his 
 wages the first day, and $.50 more the second day. What is his 
 financial condition at the end of the second day? 
 
 18. A boy earns $.75 the first day, and forfeits $.50 the 
 second day. How much money has he? 
 
 19. A boy forfeits $.75 the first day, and earns $.50 the 
 second. How much money has he? 
 
 20. A boy earns $.75 the first day, and forfeits $.75 the 
 second. How much money has he? 
 
40 ADDITION 
 
 46 Such problems as these show the necessity of making a 
 distinction between numbers of opposite nature. This can be 
 done conveniently by plus (+) and minus ( — ). If a number 
 representing a certain thing is considered positive (plus) , then 
 a thing of the opposite nature must be negative (minus). Thus, 
 if north 10 miles is written +10, south 10 miles must be written 
 — 10. If east 25 feet is written +25, west 25 feet must be 
 written —25. 
 
 47 If such numbers as these are to be combined, their signs 
 must be considered. Thus a rise of 19° in temperature fol- 
 lowed by a rise of 9° may be expressed as follows: (+19°) + 
 (+9°) = +28°. A trip 15 miles south followed by one 25 
 miles south may be expressed: ( — 15) + ( — 25) = — 40. A 
 trip 42 miles east followed by one 26 miles west is expressed: 
 (+42) +(-26) = +16. A saving of $1.75 followed by an 
 expenditure of $2.00 is expressed: (+1.75) +(-2.00) = -.25. 
 
 These four problems may also be written: 
 1. 19+9 = 28 
 
 2. -15-25= -40 
 
 or 
 
 3. 42-26 = 16 
 
 4. 1.75-2.00= -.25 
 
 - .25 
 
 This combination of positive and negative numbers is called 
 Algebraic Addition. 
 
 1. 
 
 + 19 
 + 9 
 
 +28 
 
 2. 
 
 -15 
 -25 
 -40 
 
 3. 
 
 +42 
 -26 
 + 16 
 
 4. 
 
 + 1.75 
 -2.00 
 
ADDITION OF SIGNED NUMBERS 41 
 
 ADDITION 
 
 j^S RULE: To add two numbers with like signs, add the numbers 
 as in arithmetic, and give to the result the common sign. 
 
 To add two numbers with unlike signs, subtract the smaller number 
 from the larger, and give to the result the sign of the larger. 
 
 NOTE: If no sign is expressed with a term, + is always understood. 
 Care should be taken not to confuse this with the absence of the sign of 
 multipHcation. (See Art. 19.) 
 
 
 
 
 Exercise 31 
 
 
 Add 
 
 .: 
 
 
 
 
 1. 
 
 + 19, 
 
 + 10 
 
 16. 
 
 -if 
 
 2. 
 
 -19, 
 
 -10 
 
 17. 
 
 -li 
 
 3. 
 
 -19, 
 
 + 10 
 
 18. 
 
 -3i 2i 
 
 4. 
 
 + 19, 
 
 -10 
 
 19. 
 
 6!, -8| 
 
 5. 
 
 -10, 
 
 + 19 
 
 20. 
 
 -7h +7f 
 
 6. 
 
 + 10, 
 
 -19 
 
 21. 
 
 13|, -23f 
 
 7. 
 
 -75, 
 
 +25 
 
 22. 
 
 -llf,8f 
 
 8. 
 
 +38, 
 
 + 19 
 
 23. 
 
 -2.32, -1.68 
 
 9. 
 
 + 11, 
 
 -26 
 
 24. 
 
 3.47, 5.43 
 
 10. 
 
 + 10, 
 
 -10 
 
 25. 
 
 8.44, -7.25 
 
 11. 
 
 -40, 
 
 +39 
 
 26. 
 
 8.75, -11.25 
 
 12. 
 
 -4, +26 
 
 27. 
 
 5.732, -4.876 
 
 13. 
 
 i-i 
 
 1 
 
 28. 
 
 -18.777, -3.333 
 
 14. 
 
 il 
 
 
 29. 
 
 -173.29, 239.4 
 
 15. 
 
 1 1 
 
 16> ~ 
 
 -1 
 
 30. 
 
 -208.21, 171.589 
 
42 ADDITION 
 
 /fd 1. Add -19, -10 
 
 2. Add +11, +26. 
 
 3. Add the results of problems 1 and 2. 
 
 How does the result of problem 3 compare with the result if 
 — 19, —10, +11, +26, were to be added in one problem as 
 follows? 
 
 -19-10+11+26=? 
 
 -19+11 -10 +26 =? 
 
 -19+26+11 -10 =? 
 
 +26 -10 -19 +11 =? (See Art. 10.) 
 
 50 RULE: To add several numbers, add all the positive numbers and 
 all the negative numbers separately, and combine the two results. 
 
 Exercise 32 
 
 Add: 
 
 1. +50, +41, -23, -7. 
 
 2. +47, -49, +2, -35. 
 
 3. +3, -40, -17, 4. 
 
 4. 82, 18, -100. 
 
 6. -79, -21, -100. 
 
 6. -119, +1, -21, -14, +101. 
 
 7. -2.36, +4.24, 5.73, -8.66. 
 
 8. -3f , 5f , -4yV 
 
 10. 23|, -19|, 17f, -111,5^. 
 
ADDITION OF SIMILAR TERMS 43 
 
 61 Term: A term is an expression whose parts are not sepa- 
 rated by plus (+) or minus ( — ). llx^ — 14abxy, +23f are 
 terms. 
 
 NOTE: Such expressions as 8(x+y), 3(a— b), etc., are terms because 
 the parts enclosed in the parenthesis are to be treated as a single quantity. 
 (See Art. 24.) 
 
 52 Similar Terms: Similar or like terms are those which differ 
 in their numerical coefficients only; as 2x3yz2, —^x^yz"^. 
 
 53 Only similar terms can he combined. 
 
 Exercise 33 
 Add: 
 
 1. -16r, 18r, 8r. 
 
 2. 4.2s, ~5.7s, 2s. 
 
 3. 7|x, -4fx, -2ix, X. 
 
 4. 2jab, 4|ab, — 3|ab, ab. 
 
 5. 24abc, — 36abc, lOabc, +4abc, — abc. 
 
 6. -32a2b, 40a2b, -Qa^b, 2a2b. 
 
 7. 3vV, vV, -9v2y3, -4v2y^. 
 
 8. -3|xVz, 5fxVz, -4yVxVz. 
 
 9. 3.16xy2z5, -4.08xy2z5, QmxyHK 
 
 10. 8(x-y), -6(x-y), +4(x-y). 
 
 11. -12(x+y), -7(x+y), -(x+y). 
 
 12. -6|(c-d),3f(c-d),4|(c-d). 
 
 13. -8(x2+y2), 24(x2+y2), 17(x2+y2), +(x2+y2). 
 
 14. 8(x+y+z), 14(x+y+z), -2(x+y+z). 
 
 15. Il(x2+y)^ -5(x2+y)4, 24(x2+y)^ 
 
44 ADDITION 
 
 54 Monomial: An expression containing one term only is 
 called a monomial. 
 
 55 Polynomial: An expression containing more than one term 
 is called a polynomial. A polynomial of two terms is called a 
 binomial, and one of </iree terms a trinomial. 
 
 Addition of Polynomials 
 
 56 Example : Add 2a3 - 2a2b - b^, - Tab^ - 1 la^, 
 and b3+7a3+3ab2+2a2b. 
 
 Since only similar terms can be combined, it is convenient 
 to arrange the polynomials, one underneath the other with 
 similar terms in the same vertical column, and add each column 
 separately as follows: 
 
 2a3-2a2b-b3 
 -lla^ -7ab2 
 
 +7a^+2a^b+b^+3ab^ 
 -2a3 -4ab2 
 
 Exercise 34 
 
 Add: 
 
 1. 4a+3b-5c, -2a-m+3c, 2m-9c+2b, 5a+3m-4b. 
 
 2. pq+3qr+4rs, — pq+4rs— 3qr, st— 4rs. 
 
 3. 2ax2+3ay2-4z2, ax2+7ay2-4z2, 2z2+ay2-a2x. 
 
 4. fa2-|ab-Jb2, 2b2-a2-fab, -ab-Sb^+fa^. 
 
 5. 3|m-4|x+2jf, 2iVx-f+2jm. 
 
 6. 8.75d-3.125r, 2.873r+7.625f-10d, 4.29f-r+1.25d. 
 
ADDITION OF POLYNOMIALS 45 
 
 7. 3(x+y)-7(x-y), 5(x+y)+5(x-y), -2(x+y)- 
 
 3(x-y). 
 
 8. f(a2-b2)-f(b2-c2)+f(c2-a2), f(a2-b2)-|(c2-a2), 
 
 4(b2_c2)_2i(a2-b2). 
 
 9. 5(x+y)-7(x2+y2)+8(x3+y3), -4(x'+y')+5{x'^+y')- 
 
 4(x+y), 2(x2+y2)-4(x3+y3)-(x+y). 
 
 10. 6(ab+c)+7(a-m)+a2bc3m, Sa^bc^m-SCab+c)- 
 5(a— m), 3(ab+c) — (a — m)— 4a2bc^m. 
 
 SUBTRACTION 
 
 57 1. If a man is five miles north (+5) of a certain point, 
 and another is 12 miles north (+12) of the same point, what 
 is the difference between their positions (distance between 
 them), and in what direction is the second from the first? 
 
 2. If the first man is 5 miles south ( — 5) of a point, and 
 the second 12 miles south ( — 12), what is the difference between 
 their positions, and in what direction is the second from the 
 first? 
 
 3. The first man is at ( — 5), and the second is at (+12). 
 What is the difference between their positions, and in what 
 direction is the second from the first? 
 
 4. The first is at (+5), and the second at ( — 12). What 
 is the difference between their positions, and in what direction 
 is the second from the first? 
 
 58 To find the difference between the positions of the men in 
 the above problem, the signs of their positions must be con- 
 sidered. Finding the difference between such numbers is 
 called Algebraic Subtraction, 
 
46 
 
 SUBTRACTION 
 
 Find the difference between the positions and the direction 
 of the second man from the first in each of the following: 
 
 Second man 
 First man 
 
 + 12 
 + 5 
 
 -12 
 - 5 
 
 + 12 
 - 5 
 
 -12 
 
 + 5 
 
 + 5 
 
 + 12 
 
 - 5 
 -12 
 
 + 5 
 -12 
 
 - 5 
 
 + 12 
 
 lowing: 
 
 + 12 
 - 5 
 
 -12 
 
 + 5 
 
 + 12 
 + 5 
 
 -12 
 - 5 
 
 + 5 
 -12 
 
 - 5 
 
 + 12 
 
 + 5 
 
 + 12 
 
 - 5 
 -12 
 
 How do the results of the corresponding problems in the two 
 groups compare? 
 
 59 RULE: To subtract one nximber from another, change the sign of 
 the subtrahend mentally and add. 
 
 
 
 Exercise 35 
 
 
 
 Subtract: 
 
 
 
 
 
 
 1. +27 
 
 + 12 
 
 4. 
 
 -32 
 
 +21 
 
 7. 
 
 + 16 
 -42 
 
 10. -llf 
 + 15i 
 
 2. -13 
 
 - 8 
 
 5. 
 6. 
 
 + 15 
 
 +82 
 
 - 81 
 -127 
 
 8. 
 9. 
 
 - 39 
 + 100 
 
 12| 
 -4i 
 
 11. -fax 
 -fax 
 
 3. +21 
 - 5 
 
 12. 7ibV 
 6fbV 
 
 13. 
 
 by2 
 Ifby^ 
 
 
 16. 
 
 3a-b+c 
 4a — c 
 
 14. 
 
 -5 (a+y) 
 
 -li(a+y) 
 
 17. 
 
 3ix2+5fy3- z 
 2ix2-2|y3-2z 
 
 15. 
 
 14.92(m2 
 149.2 (m2 
 
 
 18. 
 
 -a^- 
 
 a^b+ab^ 
 a^b -b^ 
 
SUBTRACTION OF POLYNOMIALS 47 
 
 19. .3(x+y)-4.8(x2+y2) 
 
 -5.7(x+y)+4.8(x^+y^) 
 
 20. -5 (ab+c)-10.7(x+y+z)+51a2bxVz+19ab2xy3z. 
 -3|(ab+c) + l.Q7(x+y+z)-17a^bxVz+20abV^z. 
 
 60 A problem in subtraction is often written in the form 
 ( — 19) — (+7). In that case it is better to actually change the 
 sign of the subtrahend and then the problem is one of addition 
 instead of subtraction and is written: 
 
 (-19) + (-7)or -19-7= -26. 
 
 Exercise 36 
 
 Subtract: 
 
 1. (-28)-(-36) 6. (8.91abc)-(-3fabc) 
 
 2. (+35)-(-2li) 6. (-2yVVz)-(3.1416x2yz) 
 
 3. (-|)-(+f) 7. (3a+2b)-(2a+3b) 
 
 4. (+2jmx)-(-f5mx) 8. (5x2-7y2)-(-2x2+y2) 
 9. (-9|m3n+3|mn3) - (4f m^n-Sfmn^) 
 
 10. (3ja+2b)-(3ia+7c) 
 
 11. (-1.7a2b2-2.9b4)-(-3.3ab3-4.16b4) 
 
 12. (4x-5§y+3fz2)-(2ix+6.25y-5jz2) 
 
 13. (- 11.23a2b2+4jbV- Ijc2a2) - (- 11.22a2b2+ 
 
 4jbV-1.875c2a2) 
 
 14. (6x2-9mx~ 15m2) - (Qx^- IGm^) 
 
 15. (-a^b^+a^b^+ab^) - (a^b-a^b^+b^) 
 
48 ADDITION AND SUBTRACTION 
 
 Exercise 37. (Review) 
 
 1. From the sum of x2-2hx+h2 and x^-Ghx+Qh^, sub- 
 tract 3x2+ 2hx-4h2. 
 
 2. Simplify (x2-2xy+y2) + (2x2-3xy+y2)-(3x2-5xy+ 
 2y^). 
 
 3. From 6x^ — 7x— 4, subtract the sum of Qx^ — 8x+x' 
 and 5 — x^+x. 
 
 4. Simphfy (im-in+fp)-(|m-fn+|p)-f (- iVm- 
 |n-|p). 
 
 6. Subtract the sum of 6— 4x^ — x and 5x — 1 — 2x2, fj-om 
 the sum of 2x3+7-4x-5x2 and 3x2-6x3-2+8x. 
 
 6. Find the difference between (12x4+6x5-2) + (6x4- 
 8x+14-8x3), and (0)-(- 10x3+2- 15x2+llx5-4x). 
 
 7. Subtract: 3x2-5xy+2y2-2x+ 7y 
 
 2x2- xy+8y2-9x-14y 
 
 8. Simphfy: (5a3-2a2b+4ab2) + (-9a2b+7ab2+8b3) + 
 (-8a3-ab2+2b3). 
 
 9. Add: fgt2 +v- Jt 
 
 gt2 -|v+ 6t 
 -1.3gt2 + llt+.02 
 
 -10v+1.2t+lJ 
 6gt2+ 11.625V -2.25 
 
 Solve and check: 
 
 10. 12a+l-(3a-4) = 2a+8+(4a+4). 
 
 11. (3x-4)-6=(x-l)-(2x-3). 
 
 12. -25-(-5+2p) = (13p-50). 
 
 13. (3x-15)-(2x-8)=0 
 
ADDITION AND SUBTRACTION 49 
 
 14. 2k-(— -^)=--(--4|) 
 
 3 6 2 2 ^ 
 
 15. ix+(?x-?)-(^x+l) = lf. 
 3 5 5 6 3 ^ 
 
 Signs of Grouping 
 
 61 Removal of Parentheses: By Art. 47, parentheses connected 
 by plus signs may be used to express a problem in addition, 
 and the parentheses can be removed without affecting the 
 signs. For example: 
 
 (3a-2b) + (2a-3b) = 3a-2bH-2a-3b-5a-5b. 
 
 By Art. 62, two parentheses connected by a minus sign may 
 be used to express a problem in subtraction, and the paren- 
 theses can be removed by actually changing the sign of each 
 term enclosed in the parenthesis preceded by the minus sign 
 (the subtrahend). For example: 
 
 (3a-2b)-(2a+3b)=3a-2b-2a-3b = a-5b. 
 
 62 RULE: Parentheses preceded by minus signs may be removed if 
 
 the sign of each term enclosed is changed. 
 
 Parentheses preceded by plus signs may be removed without any 
 change of sign. 
 
 NOTE: The sign preceding a parenthesis disappears when the paren- 
 thesis is removed. 
 
 63 Other signs of grouping often used, are the brace { } the 
 bracket [ ] and the vinculum . These have the same 
 meaning as parentheses and are used to avoid confusion when 
 several groups are needed in the same problem. 
 
 64 When several signs of grouping occur, one within the other, 
 they are removed one at a time, the innermost one first each 
 time. 
 
50 REMOVAL OF PARENTHESIS 
 
 Example : Simplify 4x — {3x + ( — 2x — x - a) } 
 
 4x-|3x+(-2x-x-a) | 
 
 = 4x — |3x+ ( — 2x — x-fa) } (removing the vinculum) 
 
 = 4x— (Sx — 2x — x+a } (removing the parenthesis) 
 
 = 4x — 3x + 2x + X — a (removing the brace) 
 
 = 4x — a (combining like terms) 
 
 NOTE: In the case of the vinculum, special care must be taken. 
 — X — a is the same as — (x— a). The minus sign preceding the vinculum 
 is not the sign of x. 
 
 Exercise 38 
 Simplify : 
 
 1. -6+{5-(7+3) + 12} 
 
 2. 10-[(7-4)-(9-7)] 
 
 3. 4x-[2x-(x+y)+y] 
 
 4. -llb+[8b-(2b-fb)-3b] 
 
 5. 8kz-[7kz-3kz-5kz] 
 
 6. x2-{'3x2-2F+l} 
 
 7. a' - ( - 6a2 - 12a + 8) - (a^^- 12a) 
 
 8. [6mn2- (8P-mn2+3n3 - mn^) - (22mn2-8P)] 
 
 9. 3a-(5a-{-7a+[9a-4]}) 
 
 10. c-[2c-(6a-b)-{c-5a+2b-(-5a+6a-3b)}] 
 Solve and check: 
 
 11. 4x-(5x-[3x-l])=5x-10 
 
 12. 12x-{8--(8x-6)-(12-3x)} = 
 
 13. -(12-20x)-{l92-(64x-36x)-12} = 96 
 
 14. 5x-[8x-{48-18x-(12-15x)}] = 6 
 
 15. 12x-[-81-(-27x-4+ 10x)] = 61- 
 
 {8x-(-20-29-4x)} 
 
MULTIPLICATION 
 
 Multiplication of Monomials 
 
 zs^ 
 
 Fig. 41 
 
 65 Two boys of equal weight are on a teeter-board at equal 
 distances from the turning point (as at A and B, Fig. 41). 
 The board balances. If one boy weighed one-half as much as 
 the other, he would have to be twice as far from the turning 
 point in order to balance the other. Similarly, if one weighed 
 one-third as much as the other, he would have to be three 
 times as far from the turning point in order to balance the other. 
 
 From these illustrations, it is readily seen that a weight of 
 one pound, four feet from the turning point, will turn the 
 board with four times as much power as a weight of one pound, 
 one foot from the turning point. A weight of three pounds, 
 four feet from the turning point, will turn the board with three 
 times as much power as a weight of one pound, four feet from 
 the turning point, and therefore with twelve times as much 
 power as a weight of one pound, one foot from the turning 
 point. The tendency of the board {lever) to turn under such 
 conditions is called the leverage, the weights acting upon it are 
 called forces, and the distance of the forces from the turning 
 point {fulcrum) are called arms. From this explanation it is 
 evident that: 
 
 66 The leverage caused hy a force is equal to the force times the arm. 
 
 This law affords a very convenient means of working out 
 the law of signs for multiplication of positive and negative 
 numbers. 
 
 51 
 
52 MULTIPLICATION 
 
 67 Let it be required to represent the product of (+5) (+4). 
 If the result is to be thought of as a leverage, the (+5) will be 
 the force, and the (+4) the arm. In discussing positive and 
 negative numbers in Arts. 45, 46, and 47, measurements up- 
 ward and to the right were represented by (+), and measure- 
 ments downward and to the left by ( — ). Then the (+5) will 
 be considered an upward pulling force, and the (+4) an arm 
 measured to the right of the fulcrum (Fig. 42) . An upward 
 
 Fig. 42 
 
 force on a right arm causes the lever to turn in a counter- 
 clockwise (opposite the hands of a clock) direction. To be 
 consistent with arithmetic, (+5) (+4) must be (+20). There- 
 fore, in determining the sign of the result of multiplication, a 
 counter-clockwise motion of the lever must be positive, and a 
 clockwise motion, negative. 
 
 68 Let it be required to represent the product of ( — 5) (—4). 
 If the result is to be thought of as a leverage, the ( — 5) will be 
 a downward pulling force, and the (—4), a left arm (Fig. 43). 
 
 P^ 
 
 Fig. 43 
 
 It is seen that the lever turns in a counter-clockwise direction 
 which is positive. Therefore ( — 5) (—4) = +20. 
 
LAW OF SIGNS 53 
 
 69 Let it be required to represent the product of (+5) (—4). 
 In this case there is an upward-puWing force (+5), on a left 
 
 t-^ 
 
 Fig.. 44 
 
 arm (—4) (Fig. 44). The lever turns in a clockwise direction 
 which is negative. Therefore (+5)(— 4)= — 20. 
 
 70 Let it be required to represent the product of ( — 5) (+4). 
 In this case there is a downward-pulling force ( — 5), on a right 
 
 ^ 
 
 Fig. 45 
 
 arm (+4) (Fig. 45). The lever turns in a clockwise direction 
 which is negative. Therefore ( — 5) (+4)= —20. 
 
 From the four preceding articles: 
 
 1. (+5)(+4) = +20 
 
 2. (-5)(-4) = +20 
 
 3. (+5)(-4)=-20 
 
 4. (-5)(+4) = -20 
 
 From these the law of signs for multiplication can be derived. 
 
 71 Law of Signs for Multiplication: If two factors have like 
 signs, their product is plus. 
 
 If two factors have unlike signs, their product is minus. 
 
54 MULTIPLICATION 
 
 Exercise 39 
 
 Multiply: 
 
 1. (+|)(+i) 8. (+3f)(+li) 
 
 2. (-f)(-i) 9. (-6f)(-6j) 
 
 3. (+!)(-i) 10. (7|)(-A) 
 
 4. (-f)(+i) 11. (-i.i)(+i.i) 
 
 6. (-§)(+■!) 12. (-2.03)(-4.2) 
 
 6. (+|)(-|) 13. (+.3)(-.03) 
 
 7. (-|)(-t\) 14. (+8.75)(+3i) 
 
 15. (-8.66)(-2i) 
 
 72 By Art. 21, x^ means x-x-x-x-x and x' means x-x-x. 
 
 Therefore (x^)(x^) = (x-x-x- x-x) (x-x-x) =x^ 
 
 From this the law of exponents for multiplication can be 
 derived. 
 
 73 Law of Exponents for Multiplication: To multiply powers 
 of the same base, add their exponents. 
 
 NOTE: The product of powers of different bases can be indicated 
 only. (x*)(y3)=x^y'. 
 
 Example: Multiply (Ta^bx^) by (-3ab3y2) 
 7a2bx3 = 7-a2.b-x3 
 -3ab3y2=-3-a-b3-y2 
 (7a2bx3)(-3abV)=7.a2-b-x3.(-3).a.b3.y2 
 which may be arranged 7-( — 3)-a2-a-b-b^-x^-y2= — 21a^b^xy. 
 
 7^ RULE: To multiply monomials, multiply the numerical coefficients, 
 and annex all the different bases, giving to each an exponent 
 equal to the sum of the exponents of that base in the two factors. 
 

 MULTIPLICATION 
 
 OF MONOMIALS 
 
 
 Exercise 40 
 
 
 Multiply: 
 
 
 
 1. 
 
 (3a3)(7a«) 
 
 11. 
 
 (+2ic)(-2id) 
 
 2. 
 
 (4x4) (-6x5) 
 
 12. 
 
 (-9m2n3)(-7mV) 
 
 3. 
 
 (-5jmio)(-2m) 
 
 13. 
 
 (-7)(+m2n2) 
 
 4. 
 
 (-13a2b)(-2ab3) 
 
 14. 
 
 (+7|)(-gt2) 
 
 5. 
 
 (+5a2bc2)(-4abV) 
 
 15. 
 
 (3xy)(xy) 
 
 6. 
 
 (-6a2b)(+3b2c) 
 
 16. 
 
 (6r2)(-|r2) 
 
 7. 
 
 (+2ab)(-3cd) 
 
 17. 
 
 (_x2)(-x3) 
 
 8. 
 
 (+3)(-x) 
 
 18. 
 
 (-6.241)(+3.48m) 
 
 9. 
 
 (+5)(+|) 
 
 19. 
 
 (x + y)3.(x+y)4 
 
 10. 
 
 ("DC-pq) 
 
 20. 
 
 (m2-n2)5.(m2-n2)2 
 
 55 
 
 Solve and check: 
 
 21. (x)(3) = (-3)(-6) 
 
 22. (r)(-2) + (r)(+3) = (-4)(-9) 
 
 23. (-3)(-6) = (w)(+2) + (0)(-5) 
 
 24. (-2)(d) + (d)(+3) + (2)(-3)=0 
 
 25. (+3)(-8) + (3k)(4) + (-2)(4k) = (0)(4) 
 
 26. (3s)(-6)-(-90)(-2) = (6s)(-6) 
 
 27. (3x)(+3)-(800)(+13) = (-5x)(19) 
 
 28. (5y)(-4) + (-56)(-l) = (-3y)(4) 
 
 29. (-9)(-2i)-(8x)(2) = (4i)(6) + (-7)(4x) + (6x)(0) 
 
 30. (-ll)(-3x)-(4|)(+6) = (20j)(-2)-(-3)(17x) 
 
56 MULTIPLICATION 
 
 75 1. What is the leverage caused by a force +7 on an arm 
 
 -3? 
 
 2. What is the leverage caused by a force —16 on an arm 
 +4? 
 
 3. What is the leverage caused by a force +3f on an 
 arm +6? 
 
 4. What is the leverage caused by a force —27 on an 
 arm -2i? 
 
 5. What is the leverage caused by a downward force of 
 6, on a right arm of 3? 
 
 6. What is the leverage caused by a downward force of 
 12, on a left arm of 7? 
 
 7. What is the leverage caused by an upward force of 
 16, on a left arm of 3 J? 
 
 8. What is the leverage caused by an upward force of 
 3 J, on a right arm of 1^? 
 
 76 Suppose two or more forces are acting on the lever at the 
 same time as in Figs. 46, 47, 48. 
 
 -« — 2 — *-+-* 3 *- 
 
 ; i 
 
 6 4 
 
 Fig. 46 
 
 What is the leverage caused by the force ( — 6), Fig. 46? 
 
 What is the leverage caused by the force (—4)? 
 
 In which direction will the lever turn? 
 
 This may be expressed by (-6)(-2)H-(-4)(+3) = 
 
 + 12-12 = 
 
LAW OF LEVERAGES 
 
 
 
 
 
 '^ ! 
 
 J 
 
 57 
 
 Fig. 47 
 
 What is the leverage caused by the force ( — 6), Fig. 47? 
 What is the leverage caused by the force (+4)? 
 In which direction' will the lever turn? 
 
 This may be expressed by (-6)(+2) + (+4)(+3) = 
 -12+12 = 0. 
 
 ^ 
 
 Fig. 48 
 
 What is the leverage caused by ( — 2), Fig. 48? 
 What is the leverage caused by (+3)? 
 What is the leverage caused by (—3)? 
 In which direction will the lever turn? 
 
 This may be expressed by(-2)(-6) + (+3)(+l) + (-3)(+5) 
 = 12+3-15 = 0. 
 
 From these illustrations the law of leverages may be derived. 
 
 77 Law of Leverages: For balance, the sum of all the leverages 
 must equal zero. 
 
58 
 
 MULTIPLICATION 
 
 Exercise 41 
 
 1-10. Find the unknown force or arm required for balance 
 in the levers shown in Figs. 49, 50, 51, 52, 53, 54, 55, 56, 57, 58. 
 
 See Art. 16. 
 
 ^ 
 
 Fig. 49 
 
 7 i 
 
 ^ 
 
 Fig. 54 
 
 r^-^ rr^^i — i 
 
 Fig. 60 
 
 Fig. 55 
 
 F 
 
 /^ 
 
 r^i* 
 
 Fig. 61 
 
 A5" 
 
 -J* 
 
 r~^ — AT 
 
 -« 10 
 
 ^0 
 
 Fig. 56 
 
 6 — -4— J 
 
 -^ /^ JW 
 
 Fig. 52 
 
 260 
 
 Fig. 67 
 
 n 
 
 ^ 
 
 r 
 
 'i^-2i-\ 3i-*\ 
 
 
 Fig. 53 
 
 Fig. 68 
 
 zyc 
 
 11. What weight 12" to the left of the fulcrum will balance 
 a weight of 10 lbs., 9" to the right of the fulcrum? (Draw a 
 figure). 
 
MULTIPLICATION OF SEVERAL MONOMIALS 59 
 
 12. Two boys weighing 75 lbs. and 105 lbs. play at teeter. 
 If the larger boy is 5' from the fulcrum, where would the 
 smaller boy have to sit to balance the board? 
 
 13. A crowbar is 6' long. What weight could be raised by 
 a man weighing 165 lbs., if the fulcrum is placed 8" from the 
 other end of the bar? 
 
 14. A lever 12' long has the fulcrum at one end. How 
 many pounds 3' from the fulcrum can be lifted by a force of 
 80 lbs. at the other end? 
 
 15. A man uses an 8' crowbar to lift a stone weighing 
 1600 lbs. If he thrusts the bar 1' under the stone, with what 
 force must he lift to raise it? 
 
 Multiplication of three or more Monomials 
 
 78 Example: Multiply (-2a)(-3a2)(+4a^)(-7a3) 
 (-2a)(-3a2) = +6a3 
 (+6a3)(+4a5) = +24a8 
 (+24a8)(-7a3)=-168aii 
 or (-2a)(-3a2)(+4a5)(-7a3)= -168a" 
 
 Exercise 42 
 
 Multiply: 
 
 1. (-3)(-4)(+5) 
 
 2. (-|)(+3i)(-i) 
 
 3. (+6)(-li)(-ft)(-7) 
 
 4. (Ila)(-7ab)(+4abc)(-9b2c2) 
 
 5. (aV)(-4a2b)(-lla3bO 
 
60 
 
 MULTIPLICATION 
 
 6. (-4jab)(-3fac)(-3^bc)(Jabc) 
 
 7. (1 .25m2x) ( - 2.4m3x2y) ( - 4.63mxy2) 
 
 8. (-3.57)(+a^2)(_i|.aV). 
 
 9. 4(x-y)3. (x-y)2.{-7(x~7)} 
 
 10. 3 (m2-n3) {-4(m2-n3)4} . (m2-n'^)2 
 
 Multiplication of Polynomials by Monomials 
 
 a ..J-. — 1~ 
 
 2D 
 
 2b 
 
 Fig. 59 
 
 19 The product of 2(a+b) may be represented by a rectangle 
 (Fig. 59) having a+b for one dimension and 2 for the other. 
 The area of the entire rectangle is equal to the sum of the two 
 rectangles, 2a and 2b, or 2(a+b) = 2a+2b. 
 
 80 RULE: To multiply a polynomial by a monomial, multiply each 
 term of the polynomial by the monomial, and write the result as 
 a polynomial. 
 
 Example: Multiply Sm^ — 5mH-7 by — 9m^ 
 -9mH3m2-5m+7)= -27m5-l-45m4-63m^ 
 
 
 Exercise 43 
 
 1. 
 
 a3-7a2b+9ab2 by 
 
 3a2b3 
 
 2. 
 
 6x5- 5x6 -7x* by 
 
 -7x» 
 
 3. 
 
 -3m2-n2+5mn by 
 
 4m2n3 
 
 .4. 
 
 a3-2a2x+4ax2--8x3 by 
 
 -2ax 
 
 6. 
 
 ia2-iab+ib2 by 
 
 -ib 
 
MULTIPLICATION OP POLYNOMIALS BY MONOMIALS 61 
 
 6. 3x^-15x2+24 by -^x^ 
 
 7. ^+1-1 by 12 
 
 2^3 6 
 
 8. -L-1+- by -30 
 10 5 6 
 
 ^ 2m 3p 7 , ^^ 
 
 9. — - — — by 10 
 
 5 2 15 
 
 Simplify : 
 
 10. -2.4a3zH2ja2x-3|xz+.125z3) 
 
 11. 5(3x+2y)+4(2x-3y) 
 
 12. 3x(4a-2y)-5x(3y-5a) 
 
 13. -5(3a-2)-3(a-6)+9(2a-l) 
 
 14. x(3x-l)-2x(7-x)-5(x2+2x-l) 
 
 ,5. 16(^+1) -24(|+| 
 
 Exercise 44 
 
 Solve and check: 
 
 1. 2(5m+l)=3(m+7)-5 
 
 2. 3(5x+l)-4(2x+7) = 3 
 
 3. 3-(x-3)=7-2x 
 
 4. 10(m-6) = 3(m-2)-5 
 
 5. 15y2+lly(2-5y)+4(10y2-9) = 19 
 
 6. ^_^ = 3 
 
 SUGGESTION: 2(x+5)-(x+l)=12 (clearing of fractions). The 
 line of the fraction has the same meaning as a parenthesis. See Art. 62. 
 
62 
 
 MULTIPLICATION 
 
 „ x+5 x-10 , 
 
 7. — =4 
 
 3 4 
 
 8. 
 
 1 
 
 x-1 
 
 9. 6a-— -3(2a-l)=i 
 a a 
 
 10. 5x-+2x+6_,,i 
 5x 
 
 Multiplication of Polynomials by Polynomials 
 —Cf — M- d 
 
 za 
 
 ac 
 
 2b 
 
 be 
 
 Fig. 60 
 
 81 The product of (a+b)(c+2) may be represented by a 
 rectangle whose dimensions are a+b and c+2 (Fig. 60). The 
 area of the entire rectangle is equal to the sum of the four 
 rectangles, ac, be, 2a, and 2b, or (a+b)(c+2) = acH-bc+2a+2b. 
 It will be seen that the first two terms are obtained by multi- 
 plying a+b by c, and the last two terms by multiplying a+b 
 by 2. It is convenient to arrange the work thus: 
 
 a+b 
 
 c+2 
 
 ac+bc 
 
 +2a+2b 
 ac+bc+2a+2b 
 
MULTIPLICATION OF POLYNOMIALS BY POLYNOMIALS 
 
 63 
 
 S2 RULE: To multiply a polynomial by a polynomial, multiply one 
 polynomial by each term of the other emd combine like terms. 
 
 Example : Multiply x^— x+lbyx— 3 
 
 X +1 
 
 x -3 
 
 x^- 
 
 3? 
 
 - X2+ X 
 
 -3x2+3x- 
 
 -4x2+4x- 
 
 -3 
 -3 
 
 
 Exercise 45 
 
 Multiply: 
 
 
 
 1. x+2 
 
 by 
 
 x+7 
 
 2. a-3 
 
 by 
 
 a-5 
 
 3. m+2 
 
 by 
 
 m— 4 
 
 4. 3m -5n 
 
 by 
 
 4m+3n 
 
 5. 2a2+3b5 
 
 by 
 
 5a?+4:V> 
 
 6. a+1 
 
 by 
 
 a^-l 
 
 7. a-3 
 
 by 
 
 b+7 
 
 8. 2x2-5x+7 
 
 by 
 
 3x-l 
 
 9. 4m2-3ms- 
 
 s^ by 
 
 m2-3s2 
 
 10. x^-x^+x^-x+l by x+1 
 
 Exercise 46 
 Simplify : 
 
 1. (a — b+c)(a — b — c) 
 
 2. (2n2+m2+3mn)(2n2-3mn+m2) 
 
 3. (ix-iy)(|x+iy) 
 
 4. (x+3)(x-4)(x+2) 
 
 6. (x2+xy+y2)(x2-xy+y2)(x2-y2) 
 
64 
 
 MULTIPLICATION 
 
 6. (2a+3b)(6a-5b) + (a-4b)(3a-b) 
 
 7. 5(x-4)(x+l)-3(x-3)(x+2) + (x+l)(x-5) 
 Solve and check: 
 
 8. (y-5)(y+6)-(y+3)(y~4) = 
 
 9. (m+3)(m+2) = (m+7)(m-5)+50 
 
 10. 3 (2x-4)(x+7)-2 (3x-2)(x+5) = 5-(3x-l) 
 4(x2H-3x+7) 
 
 11. 
 
 12. 
 
 13. 
 
 14. 
 
 16. 
 
 = 2x 
 
 2x4-7 
 
 x+3 "^""^^ 
 
 (3x4-2) (2x+3) ^ 
 (2x-l)(x-f4) 
 (x-2)(x-3) _ (x4-3)(x- 4) (x-4)(x-5) 
 
 3 4 ~ 12 
 
 1 x(2x+l) _(2x -3)(3x-f4) 2x+17 
 4+2 6 ~ 4 
 
 
 Exercise 47 
 
 Mu] 
 1. 
 
 itiply: 
 2m2-m-l 
 
 by 
 
 3m2+m-2 
 
 2. 
 
 2p3-3p2q+7pq2+4q5 
 
 by 
 
 4p-3q 
 
 3. 
 
 a^+b^+ab^-f-a^b 
 
 by 
 
 a^b-ab^ 
 
 4. 
 
 5-3a+7a2 
 
 by 
 
 4+12a2 
 
 6. 
 
 -4mn4-3m2-lln2 
 
 by 
 
 2m2-5n2+7mr 
 
 6. 
 
 -5m2+9+2m3-4m 
 
 by 
 
 5m2-l+6m 
 
 7. 
 
 3ax2-4ax3-5ax5 
 
 by 
 
 l-x+2x2 
 
 8. 
 
 p3_6p2_^12p-8 
 
 by 
 
 p3+6p2+12p+8 
 
 9. 
 
 s3-2s2-s-l 
 
 by 
 
 s3+2s2-s+l. 
 
 10. 
 
 a-l+a^-a^ 
 
 by 
 
 1+a 
 
MULTIPLICATION OF POLYNOMIALS BY POLYNOMIALS 65 
 
 11. 4x3-3x^+2x2-6 by x-x^+l 
 
 12. 3b3-7b2c+8bc2-c3 by 2b3H-8b2c-7bc2+3c3 
 
 13. a^+b^+c^+ab— bc+ac by a — b — c 
 
 14. a3-3+2a2-a by 3-a+a3-2a2 
 
 15. ia-|b+fc-|d by ^a+fb-Jc+|d 
 
 16. |a2-fab-|b2 by ija^-fb^ 
 
 17. 2fm2n2-4jn3 by Im^-fn 
 
 18. 1.25a+2.375bH-3.5c by 8a-8b+8c 
 
 19. .35a2+.25ab+3.75b2 by 4.1a2-.02ab-.57b2 
 
 20. 3.5x2-2.1xy-1.05y2 by 4x-f 
 
 Exercise 48 
 
 Simplify : 
 
 1. (a-l)(a-2)(a-3)(a-4) 
 
 2. (a-b)(a2+ab+b2)(a3+b3) 
 
 3. (3x-4y)(2x+3y)(4x-5y)(x-7y) 
 
 4. (m+n)(m-n)(5^+^) 
 
 6. (x+y)(x3+y3){x2-y(x-y)} 
 
 6. (x+yH-z)(x-y+z)(x+y-z)(y+z-x) 
 
 7. (2a+5b-c-4d)2 
 
 8. (fa3-|b2)3 
 
 2 3 4 
 
 10. (x+y)(x2-y2)-(x-y)(x2+y2) 
 
66 MULTIPLICATION 
 
 11. (3a-2b)(2a2-3ab+2b2)-3a(2a2-3ab) 
 
 12. 6(m-n)(m+n)-4(m2+n2) 
 
 13. 12(x-y)-(x2+x-6)(x2+x+y) 
 
 14. 15ab-3(2a2+4b2) + (3a-2b)(5a-3b) 
 
 15. 6(a+2b-2c)2-(2a+2b-c)2 
 
 16. (x2+l)(x-l)-(x-3)(2x-5)(x+7)-(x+2)3 
 
 17. (a+b+c)3-3(a+b+c)(a2+b2+c2) 
 
 18. (2in2-3mn+4n2)K^-^)^ 
 
 a+b+c a-b+c a+b-c b+c-a 
 2*2*2 2~ 
 
 (x-2)(2x-3) (x+2)(2x+3) (x^+ 4) (4x^^+9) 
 3*7*2 
 
 19. 
 
 20. 
 
 Exercise 49. (Review) 
 Solve and check: 
 1. (-16)(-x) + (-13)(+12) + (-2)(+2x)=0 
 
 |) + (-14)(-|) + (-10f)(+i|; 
 
 2. (+15)(-^) + (-14)(-^) + (-10f)(+±|) = 
 
 3. (-4f)(5x) + (+7|)(7x) + (-8f)(0) + (7)(-12)=0 
 
 4. 3x-3(ix-7)=35 
 
 5. (2x-l)(3x+7)-3x2=(x-l)(3x-12)+20 
 
 ^ 3x+5 , x-7 
 
 ^- "^— ^"~6~ 
 
 7. (4x+f)(fx-i)=|i 
 
 ^ 3(3 -2x) 2(x-3) 2 4(x+4) . 1 
 
 ^' 10 ~ 5 '^'^^-~^ +10 
 
 9. t(x+5)-|-(x+7)+V(x+l)-|(2x-5)=i(x+22) 
 
EQUATIONS INVOLVING MULTIPLICATION 67 
 
 (x-l)(x+2) _ (2x+l)(x+2) (2x+l)(x-l) 
 2 12 ~ 6 
 
 11. If I" the supplement of an angle is subtracted from the 
 angle, the result is 27°. Find the angle. 
 
 12. If f the complement of an angle is subtracted from 
 three times the angle, the result is 39°. Find the angle. 
 
 13. If -f of the supplement of an angle is decreased by f 
 of the complement, the result is 53°. Find the angle. 
 
 14. J the supplement of an angle is equal to the angle 
 diminished by f of its complement. Find the angle. 
 
 15. Find three consecutive numbers such that the product 
 of the second and third exceeds the product of the first and 
 second by 40. 
 
 16. The difference of the squares of two consecutive num- 
 bers is 43. Find the numbers. 
 
 17. The length of a rectangle is three times its width. If 
 its length is diminished by 6, and its width increased by 3, 
 the area of the rectangle is unchanged. Find the dimensions. 
 
 18. Two weights, 123 and 41 respectively, are placed at 
 the ends of a bar 24 ft. long. Where should the fulcrum be 
 placed for balance? (Suggestion: Let x = one arm, 24— x = 
 the other.) 
 
 19. A man weighing 180 lbs. stands on one end of a steel 
 rail 30 ft. long, and finds that it balances with a fulcrum placed 
 2 ft. from the center. What is the weight of the rail? (Sug- 
 gestion: The weight of the rail may be considered a down- 
 ward force at the middle point of the rail.) 
 
 20. An I-beam 32 ft. long weighing 60 lbs. per foot, is 
 being moved by placing it upon an axle. How far from one 
 end shall the axle be placed, if a force of 213^ lbs. at the other 
 end will balance it? 
 
DIVISION 
 
 Division of Monomials 
 
 83 To divide positive and negative numbers, a law of signs and 
 a law of exponents are necessary. Tiiese may be derived from 
 the same laws for multiplication, from the fact that the product 
 divided by one factor equals the other factor. 
 
 By Art. 70: 1. (+5)(+4) = +20 
 
 2. (-5)(-4) = +20 
 
 3. (+5)(-4) = -20 
 
 4. (-5)(+4) = -20 
 (+5) = +4 
 (+4) = ? 
 (-5)= -4 
 (-4) = ? 
 (+5) =-4 
 (-4) = ? 
 (-5) = +4 
 (+4) = ? 
 
 Therefore, from 1. 
 
 from 2 
 
 l(+20)- 
 l(+20)- 
 /(+20)- 
 l(+20)- 
 
 (-20)- 
 
 (-20)- 
 
 (-20)- 
 
 (-20)- 
 
 84 Law of Signs for Division: 
 their quotient is plus. 
 
 If two numbers have unlike signs, their quotient is minus. 
 
 from 3 
 
 from 4 
 
 // two numbers have like signs. 
 
 Divide : 
 
 1. (+f)-(+i) 
 
 2. (-i)-(+i) 
 
 3. (+i)^(-i) 
 
 4. (-l)-(-l) 
 
 (-f+)-(+3V) 
 
 Exercise 50 
 
 6. 
 7. 
 8. 
 9. 
 10. 
 68 
 
 (+2i) 
 ( 
 
 16\ 
 2 l) 
 
 (+3f) 
 ■( 
 
 (+7i)-( 
 
 10) 
 
 9) 
 
 (-42) + (-fi) 
 (+72) ^(-41) 
 
DIVISION OF SIGNED NUMBERS 69 
 
 11. (-8.5)-^(-1.7) 16. (+3.6)^(-2i) 
 
 12. (+3.2)-(-.8) 17. (-3i-56)^(-6.25) 
 
 13. (+2.65) ^(+100) 18. (-34.56) -^ (-.288) 
 
 14. (-.008)-^(-.02) 19. (+26i)-^(-llf) 
 
 15. (-15)^(+.003) 20. (-.0231)-^(-6|) 
 
 Exercise 51 
 Solve and check: 
 
 1. 3x+14-5x+15 = 4x+ll 
 
 2. 20x+15+32x+193-12 = 36x+100-32x 
 • 3. s(2s-3)-2s(s-7)+231 = 
 
 4. (x-5)(x-6) = (x-2)(x-3) 
 
 5. (3x-l)(4x-7) = 12(x-l)2 
 e 12 3x i9_4x 172 
 
 x+3 _x-2 3x-5 1 
 2 ~ 3 ~ 12 "^"4 
 
 2x±5 x-3 x_ 1 
 
 ®- 9 ~ 5 -3-^^+173 
 
 9. f(x+2)-T'5(x+5) + 10 = 2-i(x+l) 
 
 s(s-2) _ s(s-9) -2s^-91 
 "•5 3 ~ 15 
 
 85 By Art. 72, {x^){x^) = x^ 
 
 Therefore (x^) 4- (x^) = x^ 
 (x8)^(x3) = ? 
 
 86 Law of Exponents for Division: To divide powers of the same 
 base, subtract the exponent of the divisor from that of the dividend. 
 
/ U DIVISION 
 
 NOTE: The quotient of powers of different bases can be indicated 
 only. 
 
 (x^)-(y3)=^ 
 
 Example: Divide 48 a^b^c^x^V by -Sa^b^c^x^ 
 
 — •; = -6.a5.b.l.x3.y4 
 
 — Sa^b^c^x^ "^ 
 
 = — 6a%xy 
 
 c* c" 
 
 NOTE : — = 1 • Also— = c" by the law of exponents. 
 
 Therefore c° = 1 and may be omitted as a factor in problems like the 
 above example. 
 
 S7 RULE: To divide a monomial by a monomial, divide the numerical 
 coefficients, and annex all the different bases, giving to each an 
 exponent equal to the difference of the exponents of that base 
 in the two monomials. 
 
 Exercise 52 
 
 Divide: 
 
 1. -91a 
 
 2. -32x6 
 
 3. +22a^b3c7 
 
 4. +6jm3n3 
 
 6. -8jpi3qi'r'^ 
 
 6. -4.24xV^z7 
 
 7. 1.75an)«x5 
 
 8. -.SSm^n^ 
 
 9. -.OOlxVm^ 
 10. +3.1416a2b3mi« 
 
 by +13 
 
 by -8x^ 
 
 by -lla^b^c^ 
 
 by — l^mn^ 
 
 by Sfqior^ 
 
 by — .4xy z^ 
 
 by -.35x^ 
 
 by 17n* 
 
 by — lOOxym 
 
 by +4a2b3mi« 
 
DIVISION OF MONOMIALS 71 
 
 Simplify : 
 
 1155a^x7z5 3.1416r^ 
 
 -231a2x«z .7854r2 
 
 -1.732t^uV +a^bVQ 
 
 +2u6s2 -2a3bc» 
 
 3.1416xyV7 -a^(x+y)« 
 
 ". _iix26y225 "• +2-fa(x+y)4 
 
 27m^n^x7 (a+b)^(a+b)^(x+y) ^ 
 
 1.125m2n3x7 .0625(aH-b)3(x+y)^ 
 
 32.16t^ 18.75a(m^-n^)^Q 
 
 -.08t2 ^"* 2i(m2-n^)7 
 
 Exercise 53 
 Solve for x and check: 
 
 1. -ax=-ab 6. 3a2b3x=-12a3b3 
 
 2. +|bx=-8b 7. 7a-3ax = 28a 
 
 3. — .3inx = 2.4m 8. 4mx— 7mx=12m — 18m 
 
 4. -4x=-12(a+b) 9. 6a2b-7ax= -29a2b 
 
 .n X 3 1 
 
 5. -fx = 2m 10. 3^-- = e^ 
 
 lOx . 5x 5m 
 
 12. ^b_3(x-2b)^^^ 
 
 o o 
 
 13. (x-5y)(x+4y)=x2+y2 
 
 14. (x+m)(x-2m)-x(x-7m)=m(3x-5m) 
 (x-3s)(x-2s) x(x-5s) x(x-3s) 
 
 15. 7Z 71 = :; • 
 
72 DIVISION 
 
 Division of a Polynomial by a Monomial 
 
 88 By Art. 79, 2(a+b) = 2a+2b 
 
 Therefore, — '- = a+b 
 
 89 RULE: To divide a polynomial by a monomial, divide each term of 
 
 the polynomial by the monomial, and write the result as a poly- 
 nomial. 
 
 Example: Divide 2 lm«-35m^+7m3 by -7m2 
 21m«-35m4+7m2 
 
 -7m2 
 
 = -3m*+5m2-m 
 
 Exercise 54 
 Divide: 
 
 a^x'c* — ax^c^y^ + a^xc^z 
 
 axc^ 
 4xVV- 12xVz^-24xVz^H-16xyz 
 
 — 4xyz 
 2.31m2n2+7.7m3n3- .33m%^ 
 
 l.lm^n^ 
 
 - ijt V - 9.81tV - . 378tv^ 
 
 -9tv2 
 1 ■ 125a^x^z^ - .375aVz^ - 4.2a^x^z^ 
 .25aVz2 
 
 -3fabcd+2|bcdeH-7|acde 
 -l|cd 
 
 Solve for x and check: 
 
 7. ax = 2ab — 3ac+4ae 
 
 8. 3a2m3x = I.lla3m3-3.3a2m* 
 
 9. 4m2s2x - 3.2m3s2 = ISam^s^ 
 
 6. 
 
EQUATIONS INVOLVING DIVISION 73 
 
 10. 3jxyz-1.4y2z = .35yz2-70yz 
 
 11. 4m2x-7m3n4-3m2x+8m2n5 = 5m3n4-2m2x+2m2n5 
 
 12. il^-T^B^^^.lOlm^ 
 
 o u 
 
 ^^ X a2b+2bx 
 
 13. - — ^ = 3ab2 
 
 a ab 
 
 nx 3(n2x-m2n2) 4n 
 
 14. 8mn + -^ = hm^n 
 
 m mn m 
 
 2mx+a2m^ 5(b^n^+nx) _^ 2m2n -3mn^ 
 
 15. — — <jX 
 
 m n mn 
 
 Division of Polynomials by Polynomials 
 
 90 By Art. 81, (a+b)(c+2) =ac+bc+2a+2b 
 
 a+b 
 
 In multiplying a+b by c + 2, the first two terms were obtained 
 by multiplying a+b by c, and the last two by multiplying 
 a+b by 2. In dividing, the c may be obtained by dividing ac 
 by a, and the 2 may be obtained by dividing 2a by a. It is 
 convenient to arrange the work as follows: 
 c + 2 
 a+b) ac+bc+2a+2b 
 ac+bc 
 
 +2a+2b 
 + 2a+2b 
 
 9t RULE: To divide a polynomial by a polynomial, divide the first 
 term in the dividend by the first term in the divisor to obtain the 
 first term of the quotient. Multiply the divisor by the first term 
 of the quotient, and subtract the result from the dividend. To 
 obtain the other terms of the quotient, treat each remainder as 
 a new dividend and proceed in the same way. 
 
74 DIVISION 
 
 Example (1): Divide a3-6a2-19a+84 by a-7. 
 a^+a-12 
 a-7)a3-6a2-19a+84 
 a^-7a^ 
 +a2-19a 
 +a-— 7a 
 
 -12a+84 
 -12a+84 
 
 Example (2): Divide 24+26x3+120x4- 14x- 11 Ix^ 
 
 by -x-6+12x2 
 
 NOTE: Arrange the terms according to the powers of x, in both 
 dividend and divisor. 
 
 10x2+3x-4 
 
 
 12x2-x-6)120x4+26x3- 
 
 - 111x2- 14x -1-24 
 
 
 120x^-10x3- 
 
 -60x2 
 
 
 
 +36x3- 
 
 - 51x2- 14x 
 
 
 
 +36x3- 
 
 - 3x2 -18x 
 
 
 
 - 
 
 - 48x2+ 4x+24 
 
 
 
 - 
 
 - 48x2+ 4x+24 
 
 Exam 
 
 ple (3): Divide 
 
 i x^+xV- 
 
 f y4 by x2+xy+y' 
 
 
 x2- 
 
 xy +y2 
 
 
 
 x2+xy+y2)x* 
 
 +xV 
 
 +y^ 
 
 
 x4+ 
 
 x3y+xV 
 
 
 
 - 
 
 x3y 
 
 +y^ 
 
 
 - 
 
 X3y_x2y2 
 
 — xy3 
 
 
 
 +xy 
 
 +xy3+y4 
 
 
 
 +xV 
 
 +xy3+y4 
 
 
 
 Exercise 55 
 
 Divide the following: 
 
 
 
 1. 
 
 x2-7x+12 
 
 
 by x-3 
 
 2. 
 
 a2-2a-15 
 
 
 by a — 5 
 
 3. 
 
 a2-3ab-28b2 
 
 
 by a+4b 
 
DIVISION OF POLYNOMIALS BY POLYNOMIALS 
 
 75 
 
 4. 
 
 6m4-29m2+35 
 
 by 2m2-5 
 
 6. 
 
 12x2+31xy-15y2 
 
 by x+3y 
 
 6. 
 
 m3+3m2-13m-15 
 
 W m+1 
 
 7. 
 
 10x3- 19x2y+26xy2-8y3 
 
 by 2x2-3xy+4y2 
 
 8. 
 
 3m4 - lOm^ - 16m2 - 10m - 3 
 
 by 3m2+2m+l 
 
 9. 
 
 2x^-x3y+4xy+xy3+12y4 
 
 by x2-2xy+3y2 
 
 10. 
 
 4x4-24x3+51x2-46x+15 
 
 by 2x2-7x+5 
 
 11. 
 
 9xV - 15x3y2+ 13xy - 3xy4 
 
 by 3x2y-xy2 
 
 12. 
 
 Sm^n - 22m6n2 - 7m^n3+ 
 
 
 
 53m4n4-30m3n5 
 
 by 4Tn4+3m3n-5m2n2 
 
 13. 
 
 10x3 _ 29.3x2y +37xy2 - 20y3 
 
 by 2.5x2 -4.2xy+4y2 
 
 14. 
 
 14x3+17x2y+39xy2+17y3 
 
 by 3.5x2+2.5xy+8.5y2 
 
 15. 
 
 18x3-53x2+27x+14 
 
 by 4x-8 
 
 16. 
 
 13.12m5n+1.36m%2_ 
 
 
 
 7.15m3n3+2.35m2n4-.125mn 
 
 ' by 3.2in2n-2.4mn2+.5n3 
 
 17. 
 
 a2+ab+2ac+bc+c2 
 
 by a+c 
 
 18. 
 
 a^bx + abcx + a^cx + ab V + 
 
 
 
 b^cy+abcy 
 
 by ax+by 
 
 19. 
 
 x2+8-10x+x3 
 
 by 2+x2-3x 
 
 20. 
 
 m^ + n^ — 4m3n — 4mn3 + Gm^n^ 
 
 ' by m2+n2 — 2mn 
 
 21. 
 
 a5-9a3+7a2-19a+10 
 
 by a2+3a-2 
 
 22. 
 
 16m4-72m2n2+81n4 
 
 by 4m2-12mn+9n2 
 
 23. 
 
 m^— 64n3 
 
 by m— 4n 
 
 24. 
 
 32as+243b5 
 
 by 2a+3b 
 
 25. 
 
 a2+2ab+b2-c2 
 
 by a+b— c 
 
 26. 
 
 a2-_x2-2xy-y2 
 
 by a— x-y 
 
 27. 
 
 a^+4+3a2 
 
 by a2+2-a 
 
76 DIVISION 
 
 28. 4a2-b2-6b-9 by 2a+b+3 
 
 29. a2-b2+x2-y2+2ax+2by by a+x+b-y 
 
 30. m2-2mn+n2+3m-3n+2 by m-n+1 
 
 Solve and check: 
 
 31. (a+3)x = ab-fa+3b+3 
 
 32. (a2-4ab+3b2)x = a3-8a2b+19ab2-12b' 
 
 33. (2m-3n)x = 8m3-22m2n+mn2+21n3 
 
 34. (a+bH-2)x = a2+2ab+b2+4a+4b+4 
 
 35. (y-f-2)x = y3-y2-34y-56 
 
 Exercise 56 
 
 1. One number is 6 more than another, and the difference 
 of their squares is 144. Find the numbers. 
 
 2. One number is 3 less than another, and the difference 
 of their squares is 33. Find the numbers. 
 
 3. Divide 42 into two parts such that J of one is equal to 
 ^ of the other. 
 
 4. Divide 57 into two parts such that the sum of ^ of 
 the larger and \ of the smaller is 12. 
 
 5. The difference of two numbers is 11, and, if 18 is sub- 
 tracted from f of the larger, the result is yof the smaller 
 number. Find the numbers. 
 
 6. Divide 24 into two parts such that if ^ of the smaller 
 is subtracted from f of the larger, the result is 9. 
 
 7. If the product of the first two of three consecutive 
 numbers is subtracted from the product of the last two, the 
 result is 18. Find the numbers. 
 
REVIEW PROBLEMS 77 
 
 8. If the square of the first of three consecutive numbers 
 is subtracted from the product of the last two, the result is 
 41. Find the numbers. 
 
 9. I paid a certain sum of money for a lot and built a 
 house for 3 times that amount. If the lot had cost $240 less 
 and the house $280 more, the lot would have cost i as much 
 as the house. What was the cost of each? 
 
 10. A boy has 2 J times as much money as his brother. 
 After giving his brother $25.00, he has only 1^ times as much. 
 How much had each at first? 
 
 11. The sum of J a certain angle, J of its complement and 
 Y^Q- of its supplement is 48°. Find the angle. 
 
 12. Three times an angle, minus 4 times its complement, is 
 equal to -j^y of its supplement + 131°. Find the angle. 
 
 13. If 3 times an angle is subtracted from J its supplement, 
 the result is yt of its complement. 
 
 14. A certain rectangle contains 15 sq. in. more than a 
 square. Its length is 7 in. more and its width 3 in. less than 
 the side of the square. Find the dimensions of the rectangle. 
 
 15. The altitude of a triangle is 4 in. more than the base, 
 and its area exceeds one half the square of the base by 16. 
 Find the base and altitude. (Suggestion: See Exercise 15, 
 problem 5.) 
 
 16. A wheelbarrow is loaded with a barrel of flour weighing 
 196 lbs. The center of the load is 2' from the axle of the 
 wheel. What force at the handles, 4j' from the axle of the 
 wheel, will be required to raise the load? 
 
 17. A wheelbarrow is loaded with 5 bars of pig iron weigh- 
 ing 77 lbs. each. How far from the axle of the wheel should 
 the center of the load be placed, if a force of 154 lbs. 4 ft. from 
 the axle will raise it? 
 
78 DIVISION 
 
 18. A timber 12" X 18" X 24' is balanced on wheels and an 
 axle by a force of 120 lbs. at one end. How far from the center 
 shall the axle be placed if the timber weighs 45 lbs. per cu. ft.? 
 
 19. A lever 12' long weighs 24 lbs. If a weight of 30 lbs. 
 is hung at one end and the fulcrum is placed 4' from this end, 
 what force is needed at the other end for balance? 
 
 20. A piece of steel 1' long, weighing 15 lbs. per foot, is 
 resting upon one end. A weight of 1400 lbs. is placed l|' 
 from that end. What force at the other end is necessary to 
 balance the load? 
 
CHAPTER V 
 RATIO, PROPORTION, AND VARIATION 
 
 Ratio 
 
 92 Ratio: The relation of one quantity to another of the same 
 kind is called a ratio. It is found by dividing the first by the 
 second. For example: the ratio of $2 to $3 is f , written also 
 2:3; the ratio of 7" to 4" is |; the ratio of 18" to 6' is +| = i. 
 
 93 Terms of Ratio: The numerator and denominator of a ratio 
 are respectively the first and second terms of a ratio. The 
 first term of a ratio is called its antecedent, and the second, its 
 consequent. 
 
 Exercise 57 
 
 1. Find the ratio of 85 to 51. 
 
 2. Find the ratio of 27 to 243. 
 
 3. Find the ratio of 2j to 3f . 
 
 4. Find the ratio of 6.25 to 87.5. 
 
 5. Find the ratio of y\ to .3125. 
 
 6. Find the ratio of 8" to 6'. 
 
 7. Find the ratio of 12a to 16a. 
 
 8. Find the ratio of 577- to Stt. 
 
 9. Find the ratio of a right angle to a straight angle. 
 
 10. Find the ratio of a right angle to a perigon. 
 
 11. Find the ratio of a straight angle to a perigon. 
 
 12. Find the ratio of f of a perigon to -| of a right angle. 
 
 79 
 
80 RATIO, PROPORTION, AND VARIATION 
 
 13. Find the ratio of 55° to its complement. 
 
 14. Find the ratio of 55° to its supplement. 
 
 15. Find the ratio of 45° to § its supplement. 
 
 16. Find the ratio of the supplement of 48° to its comple- 
 ment. 
 
 17. A door measures 4' X 8'. What is the ratio of the 
 length to the width? 
 
 18. There were 25 fair days in November, while the rest 
 were stormy. What was the ratio of the fair to the stormy 
 days? 
 
 19. The dimensions of two rectangles are 5" X 8", and 6" 
 X S''. Find the ratio of their lengths, widths, perimeters, and 
 areas. 
 
 20. The bases of two triangles are 3.9 and 2.4, and their 
 altitudes are respectively .8 and .7. Find the ratio of their 
 areas. 
 
 21. Find the ratio of the circumferences of two circles whose 
 diameters are respectively 5j" and 2f ". (See Exercise 16, 
 problem 2.) 
 
 22. Find the ratio of the areas of two circles whose diame- 
 ters are respectively IV and 13". (See Exercise 16.) 
 
 23. Find the ratio of the two values of P in the formula P = 
 awh, when a = 120, w = .32, h = 9j, and when a = 48, w = .38, 
 and h = 24. 
 
 24. Find the ratio of the two values of F in F=l§d+J, 
 when d = if, and when d = 2j. 
 
 25. Find the ratio of the two values of S in S = Jgt2, when 
 t = 3j, and when t = 10j. (See Exercise 17.) 
 
RATIO AS DECIMALS 
 
 81 
 
 94 To Express Ratios as Decimals: It is often convenient to 
 have results in decimal rather than in fractional form. For 
 example : the ratio -I- is often written .875. 
 
 
 
 Exercise 
 
 58 
 
 
 
 
 Find the decimal equivalents of the following 
 
 ratios: 
 
 
 1. A 
 
 8 
 
 3. 1 
 25 
 
 5. ?-^ 
 
 32 
 
 
 7. '1 
 64 
 
 9. 
 
 .72 
 
 1* 
 
 2. ^ 
 16 
 
 4. 1? 
 20 
 
 6. !i 
 
 30 
 
 
 8 2f 
 
 10. 
 
 3.24 
 129.e 
 
 95 Sometimes it is sufficiently accurate to express the decimal 
 to two places only. In this case it is necessary to determine the 
 third place, and, if this is 5 or more, it is customary to increase 
 the second place by 1. For example: the ratio y|^=.946 +, 
 which would be written .95 if two places only are desired. 
 
 Exercise 59 
 
 Find the decimal equivalents of the following ratios, correct 
 to .01 : 
 
 1. 
 
 9 
 
 10 ^19 ^25 , 37.5 
 — 3. — 4. — 6. 
 
 11 16 7| 5.15 
 
 Percentage is found by reducing a ratio to a decimal correct to 
 .01, and multiplying it by 100. 
 
 For example: ^?^ = 6.688 = 669%. 
 .0369 
 
82 RATIO, PROPORTION, AND VARIATION 
 
 6. In a class of 27 students, 22 passed an examination. 
 Find the percentage of successful students. 
 
 7. A base ball player made 89 hits out of 321 times at bat. 
 Find his batting average (percentage). 
 
 8. The total cost of manufacturing an article is $5.36 of 
 which $2.79 represents labor. What per cent of the total cost 
 is the labor? 
 
 9. If 62j tons of iron are obtained from 835 tons of ore, 
 what per cent of the ore is iron? 
 
 10. In a class of students, 25 passed, 2 were conditioned, and 
 6 failed. Find the percentage of failures. 
 
 11. Babbitt metal is by weight 92 parts tin, 8 parts copper, 
 and 4 parts antimony. Find the percentage of copper. 
 
 12. Potassium nitrate is composed of 39 parts of potassium, 
 14 parts of nitrogen, and 48 parts of oxygen. Find the per- 
 centage of potassium. 
 
 13. Potassium chloride is composed of 39 parts of potassium 
 and 35.5 parts of chlorine. Find the percentage of chlorine. 
 
 14. Baking powder is composed of 3j parts of soda, if parts 
 of cream of tartar, and 6.5 parts of starch. Find the percentage 
 of cream of tartar. 
 
 16. If 12 quarts of water are added to 25 gallons of alcohol, 
 what per cent of the mixture is alcohol? 
 
 16. If 5 lbs. of a substance loses 5 oz. in drying, what per 
 cent of its original weight was water? 
 
 17. If 5 lbs. of a dried substance has lost 5 oz. in drying, 
 what per cent of its original weight was water? 
 
 18. If a dried substance absorbs 5 oz. of water and then 
 weighs 5 lbs., what per cent of its original weight is water? 
 
SPECIFIC GRAVITY ^ 83 
 
 19. The itemized cost of a house is as follows: 
 
 Masonry . . $ 750 Plumbing . . . $350 
 
 Carpenter Work $ 900 Furnace . . . $150 
 
 Lumber . . $1200 Painting . . • . . $300 
 
 Plastering . . $ 250 
 
 What per cent of the total cost is represented by each 
 item? 
 
 Check by adding the per cents. 
 
 20. The population of Detroit in 1900 was 285,704, and in 
 1910, it was 465,776. Find the percentage of increase. 
 
 96 Specific Gravity: The specific gravity of a substance is the 
 
 ratio of the weight of a certain volume of the substance to the 
 
 weight of the same volume of water. For example: if a cubic 
 
 inch of copper weighs .321 lbs., and a cubic inch of water weighs 
 
 321 
 .0361 lbs., the specific gravity of copper is = 8.88. 
 
 Example. The dimensions of a block of cast iron are 3j"X 
 2f "X 1", and its weight is 37.5 oz. Find its specific gravity. 
 
 3jX2f X 1 =8.94 cu. in. (the volume of the block) 
 
 .0361 lbs. = .5776 oz. (weight of 1 cu. in. of water) 
 .5776 X 8 . 94 = 5 . 16 (weight of 8.94 cu. in. of water) 
 
 — '— = 7 . 27, (specific gravity of iron) 
 5.16 
 
 NOTE: Specific gravity is usually found correct to .01. 
 
 Exercise 60 
 
 1. A cubic inch of aluminum weighs .0924 lbs. Find its 
 specific gravity. 
 
84 KATIO, PROPORTION, AND VARIATION 
 
 2. A cubic inch of tungsten weighs .69 lbs. Find its specific 
 gravity. 
 
 3. A cubic inch of cast steel weighs .282 lbs. Find its 
 specific gravity. 
 
 4. A cubic inch of lead weighs 6.56 oz. Find its specific 
 gravity. 
 
 5. A cubic foot of bronze weighs 550 lbs. Find its specific 
 gravity. 
 
 6. A cubic foot of cork weighs 240 oz. Find fts specific 
 gravity. 
 
 7. A brick 2" X 4" X 8" weighs 4.64 lbs. Find its specific 
 gravity. 
 
 8. A cedar block 5" X 3" X 2" weighs 10.5 oz. Find its 
 specific gravity. 
 
 9. Each edge of a cubical block is 2' . If it weighs 4450 
 lbs., what is its specific gravity? 
 
 10. A man weighing 185 lbs., displaces when swimming 
 under water, 5760 cu. in. of water. Find the specific gravity of 
 the human body. 
 
 97 Separating in a given ratio. 
 
 Example: Divide 17 into two parts which shall be in the ratio f . 
 Let 2x = one part. 
 
 3x = other part. mott? ^'^-^ 
 
 Then2x+3x = 17 ^^ ii^S 
 
 5x = 17 
 x = 3f 
 2x = 6|-, one part. 
 3x=10|-, other part. 
 
 Check: 6|+10i=17, T^ = ^ = f 
 
SEPARATING IN A GIVEN RATIO 85 
 
 Exercise 61 
 
 1. Divide 20 in the ratio f . 
 
 2. Divide 18 in the ratio ^. 
 
 3. Divide 100 in the ratio -f-. 
 
 4. Divide 200 in the ratio ^. 
 
 6. Two supplementary angles are in the ratio ^. Find 
 them. 
 
 6. Two complementary angles are in the ratio -J. Find 
 them. 
 
 7. A board 18" long is to be divided in the ratio ^. How 
 far from each end is the point of division? 
 
 8. If a line 4' 6" long is divided in the ratio ^, what is the 
 length of each part? 
 
 9. Divide a legacy of $25,000 between two persons so that 
 their shares shall be in the ratio ^. 
 
 10. The sides of a rectangle are in the ratio -J, and its 
 perimeter is 100. Find the dimensions of the rectangle. 
 
 11. Bronze is composed of 11 parts tin and 39 parts copper. 
 Find the number of pounds of tin and copper in 625 lbs. of 
 bronze. 
 
 12. A gold medal is 18 carats fine (18 parts of pure gold in 24 
 parts of the whole alloy). Find the amount of pure gold in the 
 medal if it weighs 2.7 oz. 
 
 13. Two men purchase some property together, one paying 
 $750 and the other $450. If the property is sold for $2,000, 
 what will be the share of each? 
 
 14. Two men agree to do a piece of work for $45. The work 
 is completed in 10 days, but one of them was absent 2 days. 
 How should the pay be divided? 
 
86 RATIO, PROPORTION, AND VARIATION 
 
 16. How much copper would there be in 208 lbs. of Babbitt 
 metal? (See Exercise 59, problem 11.) 
 
 16. Divide a perigon into three angles in the ratio 7:8:9. 
 
 17. Divide a line 5' 3" long into four parts in the ratio 
 5:6:7:3. 
 
 18. The sides of a triangle are in the ratio 5:8:9, and its 
 perimeter is 6' 5". Find the sides. 
 
 19. Divide the circumference of a circle whose diameter is 
 16" into three parts in the ratio 3:5:7. 
 
 20. Five angles about a point on one side of a straight line 
 are in the ratio 1:2:3:4:5. Find them. 
 
 Proportion 
 
 98 Proportion. A proportion is an equation in which the two 
 members are ratios. For example : y^ = ii is a proportion, and 
 may be read 8 is to 12 as 16 is to 24. The first and fourth terms 
 of a proportion are called the extremes, and the second and third 
 are called the means. In the proportion -^ = ^^, 8 and 24 are 
 the extremes, and 12 and 16, the means. 
 
 Example: Solve 1^ = 9 
 
 15 = 4x (clearing of fractions.) 
 x = 3f 
 
 Check: A=— 
 ^^ 9 
 
 5 _ 5 
 T2— T2 
 

 PROPORTION 
 
 
 
 
 Exercise 62* 
 
 
 
 Solve and check : 
 
 
 
 
 1. ^ = i? 
 25 14 
 
 
 6. 
 
 11 18 
 
 12 X 
 
 2. ^ = i? 
 
 7 17 
 
 
 6. 
 
 125 206 
 X 305 
 
 3 ^-^ 
 ^- x'll 
 
 
 7. 
 
 144 3x 
 195 25 
 
 4 ^ = ^ 
 
 9 14 
 
 
 8. 
 
 X 3j 
 24 41 
 
 87 
 
 9. The ratio of x+1 to 9 is equal to the ratio of x+5 to 15. 
 Find x. 
 
 10. The ratio of the complement of an angle to the angle is 
 equal to the ratio y. Find the angle. 
 
 11. The ratio of the supplement of an angle to the angle is 
 equal to the ratio y^ . Find the angle. 
 
 12. The ratio of an angle to 84° is equal to the ratio of its 
 complement to 96°. Find the angle. 
 
 13. One number is 5 larger than another, and the ratio of the 
 larger to the smaller is equal tof . Find the two numbers. 
 
 14. The length of a rectangle is 6 more than its width, and 
 the ratio of the length to the width is ^. Find the dimensions 
 of the rectangle. 
 
 16. Two numbers are in the ratio f . If 2 is added to the 
 smaller, the ratio of that number to the larger is f . Find the 
 numbers. (See Example, Art. 97.) 
 
 16. If the scale of a drawing is J" to 1', how long should a 
 line be made in the draving to represent 32'? 
 
88 RATIO, PROPORTION, AND VARIATION 
 
 17. If the scale of a drawing is f " to 1', how long should a 
 line be made to represent 10"? 
 
 18. If the scale of a drawing is ij" to 1', what line would be 
 represented by a line 3|" on the drawing? 
 
 19. If a drawing is to be reduced to f its size, what would be 
 the length on the new drawing, of a dimension 3^" on the 
 original drawing? 
 
 20. If a dimension line f " on a drawing represents a line 4j" 
 long, what is the scale of the drawing? 
 
 99 It is often necessary in shop practice to express a fraction or 
 decimal in halves, fourths, eighths, sixteenths, etc. A proportion 
 is a convenient means of changing to these denominators. 
 
 Example : How many g^'s in y^. 
 
 3^ 2^ s in Y 5^. 
 
 Let X = number of -oV's in — — 
 
 32 15 
 15x = 352 
 x = 23y5^, approximately 23^. 
 
 Exercise 63 
 
 1. How many J's in -^q? 
 
 2. How many y^'^ i^ i'^ 
 
 3. How many -^'s in .3? 
 
 4. Reduce 1.312 to eighths. 
 
 5. Reduce 1^%- to sixty-fourths. 
 
PROPORTION 89 
 
 X 4 
 
 100 Example: 1. Solve — rT = "r 
 
 x-j-l 5 
 
 Check: 
 
 4^4 
 4+l~ 5 
 
 5 5 
 
 5x = 4x+4 iL. C. D. is5(x+l)> 
 X = 4 Why? 
 
 X 1 
 
 Example: 2. Solve ^, ^ ,, =^ 
 
 o(x — 1) 6 
 
 2x = x-l |l. C. D. is6(x-l)| 
 x=-l Why? 
 
 Check; 
 
 -1 1 
 
 3(-l-l) 6 
 
 -6 6 
 
 6 6 
 
 Example: 3. Solve 
 
 x+l x-3 
 
 ::heck: 
 
 x+2 x-4 
 x2-3x-4 = x2-x-6 I L. C. D. is (x+2) (x-4) 
 — 2x=-2 Why? 
 X = 1 Why? 
 
 1 + 1 ^ 1-3 
 
 1 + 2 1-4 
 
 3-3 
 
 1 = 1 
 3 3 
 
90 RATIO, PROPORTION, AND VARIATION 
 
 Exercise 64 
 Solve and check: 
 
 1 _^=1 7 -^ = i 
 
 x-1 4 3(5x-6) 9 
 
 _x ^4 7(3x-7) ^23 
 
 ^* 3(x-l) 9 4(x+3) 12 
 
 y ^ 1 2 ^ 3 
 
 ^* 5(y+2)"l0 3x+l 5x+2 
 
 4. ^-±?4 10. ' ' 
 
 x-5 6 4x-3 3x+4 
 
 X 2 ^^ x+5 x+25 
 
 3x+l 7 x-4 x-2 
 
 2y+3 ^3 5x-7 ^ 10x+ll 
 
 3y+7 4 3x-5 6x+ 7 
 
 13. 
 
 2x-3 3x 
 
 2(x-3) 3x-4 
 
 14. The ratio of an angle to its supplement is J. Find the 
 angle. 
 
 15. The ratio of an angle to its complement is y. Find the 
 angle. 
 
 16. The ratio of the supplement of an angle to the comple- 
 ment is f . Find the angle. 
 
 17. If an angle is increased by 3° and its complement de- 
 creased by 13°, the ratio of the two angles will then be -|. Find 
 the original angle. 
 
 18. The base of one rectangle is 3 less than the base of 
 another. The altitude of the first is 3, and that of the second 
 is 5. The ratio of the areas is f . Find the bases of the 
 two rectangles. 
 
 19. The ratio of 3° to the complement of an angle is equal to 
 the ratio of 21° to the supplement of the same angle. Find the 
 angle. 
 
DIRECT VARIATION 91 
 
 20. Find three consecutive numbers such that the ratio of 
 the first to the second is equal to the ratio of 5 times the third 
 to 5 times the first plus 16. 
 
 Variation 
 
 101 Direct Proportion: If a train travels 120 miles in 3 hours, it 
 would travel 240 miles in 6 hours, -f is the ratio of the two 
 times, and -J^f ^ is the ratio of the two distances, taken in the same 
 order. Both ratios reduce to J and therefore the problem may 
 be expressed by the proportion, -f = i^%. An increase in time 
 produces an increase in distance. 
 
 If the train travels 120 miles in 3 hours, it would travel 80 
 miles in 2 hours because |- = ^-^-^ . A decrease in time produces a 
 decrease in distance. 
 
 When two quantities are so related that an increase or de- 
 crease in one produces the same kind of a change in the other, 
 one is said to be directly proportional to the other, or to vary 
 directly as the other. 
 
 Example: If a piece of steel 3 yds. long weighs 270 lbs., how 
 much will a piece 5 yds. long weigh? 
 
 Let X = weight of the 5-yd. piece. 
 
 3 270 
 
 - = — — (the weight is directly proportional to the length.) 
 5 X 
 
 x= ? 
 
 Exercise 65 
 
 1. If 60 cu. in. of gold weighs 42 lbs., how much will 35 cu. 
 in. weigh? 
 
 2. If the interest on a certain sum of money is $84.20 for 5 
 yrs., what would be the interest for 8 J yrs.? 
 
 3. If a section of I-beam 10 yds. long weighs 960 lbs., how 
 long is a piece of the same material weighing 1280 lbs.? 
 
92 RATIO, PROPORTION, AND VARIATION 
 
 4. An engine running at 320 revolutions per minute 
 (R.P.M.) develops 8^ horsepower. How many horsepower 
 would it develop at 365 R. P. M.? 
 
 5. At 40 lbs. pressure per square inch, a given pipe dis- 
 charges 180 gallons per minute. How many gallons per minute 
 would be discharged at 55 lbs. pressure? 
 
 6. What will be the resistance of a mile of wire if the 
 resistance of 500 yds. of the same wire is .65 ohms? 
 
 7. A steam shovel can handle 900 cu. yds. of material in 
 8 hrs. At the same rate how many cu. yds. can be handled 
 in 7 hrs.? 
 
 8. A 12 pitch gear 10" in diameter has 120 teeth. How 
 many teeth would a 6" gear with the same pitch have? 
 
 9. An engine running at 185 R. P. M. drives a line shaft 
 at 210 R. P. M. At what R. P. M. should an engine run to 
 give the line shaft a speed of 240 R. P. M.? 
 
 10. If a machine can finish 65 pieces in 75 minutes, how 
 long will it take it to finish 104 pieces? * 
 
 102 Inverse Proportion: K a train travels a given distance in 
 4 hrs. at the rate of 40 miles per hour, it would take 8 hrs. 
 to travel the same distance if the rate were 20 miles per hour. 
 ■^ is the ratio of the two times, and -|^ is the ratio of the two 
 rates, taken in the same order. |- = J but ■f§ = |-. Therefore 
 the problem may be expressed as a proportion if one ratio is 
 first inverted. ^ = ^% or |- = -f^. An increase in time produces 
 a decrease in rate. 
 
 If a train travels a given distance in 4 hours at the rate of 
 40 miles per hour, it would take 2 hours to travel the same 
 distance if the rate were 80 miles per hour, because |^ = -|^ or 
 1^ = 1^-0^. A decrease in time produces an increase in rate. 
 
INVERSE VARIATION 93 
 
 When two quantities are so related that an increase or a 
 decrease in one produces the opposite kind of a change in the 
 other, one is said to be inversely proportional to the other, or 
 to vary inversely as the other. 
 
 Example: If 6 men can do a piece of work in 10 days, 
 how long will it take 5 men to do it? 
 
 Let X = time it will take 5 men 
 
 6 X^ (the number of men is inversely proportional to 
 
 ^ ~ jQ the number of days.) 
 
 x = ? 
 
 Exercise 66 
 
 1. A train traveling at the rate of 50 miles per hour covers 
 a distance in 5 hrs. How long would it take to cover the same 
 distance if it traveled at 40 miles per hour? 
 
 2. A man walking at 4 miles per hour can travel a dis- 
 tance-in 3 hrs. At what rate would he have to walk to cover 
 it in 2 hrs.? 
 
 3. If 40 men can do a piece of work in 10 days, how long 
 will it take 25 men to do it? 
 
 4. 12 men can do a piece of work in 28 days. How many 
 men could do it in 84 days? 
 
 5. The number of posts required for a fence is 42 when 
 they are placed 18 ft. apart. How many would be needed 
 if they were placed 14 ft. apart? 
 
 6. One investment of $6,000 at 3|% yields the same 
 income as another at 3%. What is the amount of the second 
 investment? 
 
 7. A man has two investments, one of $15,900, and the 
 other $21,200. The first is invested at 6%. At what rate 
 must the other be invested to produce the same income as 
 the first? 
 
94 RATIO, PROPORTION, AND VARIATION 
 
 8. A man planned to use 36 posts spaced 9 ft. apart in 
 building a fence. His order was 6 posts short. How far 
 apart should he place them? 
 
 Exercise 67. (Review) 
 
 1. The circumference of a circle is directly proportional 
 to its diameter. If the circumference of a circle whose diam- 
 eter is 6" is 18.8496", what is the circumference of a circle 
 whose diameter is 4"? 
 
 2. If the circumference of a circle whose diameter is 5" 
 is 15.708", what is the diameter of a circle with a circumference 
 of 28.2744"? 
 
 3. The area of a circle varies directly as the square of its 
 diameter. If the area of a 2" circle is 12.5664 sq. in., find the 
 
 12.5664 4 
 area of a 4" circle. (Suggestion : = — ) 
 
 4. The volume of a quantity of gas varies inversely as 
 the pressure when the temperature is constant. If the volume 
 of a gas is 600 cubic centimeters (c. c.) when the pressure is 
 60 grams per square centimeter, find the pressure when the 
 volume is 150 c. c. 
 
 5. A quantity of gas measures 423 c. c. under a pressure 
 of 815 millimeters (m. m.). What will it measure under 760 
 m. m.? 
 
 6. The volume of a cube varies directly as the cube of the 
 edge. If the volume of an 11" cube is 1331 cu. in., what is 
 the volume of a 7" cube? (See suggestion, problem 3.) 
 
 7. The volume of a sphere is directly proportional to the 
 cube of its diameter. Find the volume of a 6" sphere if a 
 10" sphere contains 523.6 cu. in. 
 
 8. The volume of a quantity of gas varies directly as the 
 absolute temperature when the pressure is constant. If a 
 
REVIEW 95 
 
 quantity of gas occupies 3.25 cu. ft. when the absolute tem- 
 perature is 287°, what will be its volume at 329°? 
 
 9. The velocity of a falling body varies directly as the 
 time of falHng. If the velocity acquired in 4 seconds is 128.8 
 ft. per sec, what would be the velocity acquired in 7 seconds? 
 
 10. The weight of a disk of copper cut from a sheet of 
 uniform thickness varies as the square of the diameter. Find 
 the weight of a circular piece of copper 12" in diameter if one 
 7" in diameter weighs 4.42 oz. 
 
 11. A wheel 28" in diameter makes 42 revolutions in going 
 a given distance. How many revolutions would a 48" wheel 
 make in going the same distance? 
 
 12. If 3 men can build 91 rods of fence in a certain time, 
 how much could 7 men build in the same time? 
 
 13. If 25 men can do a piece of work in 30 days, how long 
 would it take 27 men to do the same work? 
 
 14. If the pressure on 230 c. c. of nitrogen is changed 
 from 760 m. m. to 665 m. m., what will be its new volume? 
 
 15. The absolute temperature of 730 c. c. of hydrogen is 
 changed from 353° to 273°. What is its new volume? 
 
 16. If the circumference of a circle 3 J" in diameter is 
 10.9956, what is the diameter of a circle whose circumference 
 is 23.562? 
 
 17. A sum of money earns S1750 in 3j yrs. How long will 
 it take it to earn $2750? 
 
 18. An investment of $1125 at 5% earns the same amount 
 as another of $1250. What is the rate of the second investment? 
 
 19. If an investment at 2j% produces an income of $400, 
 what would it produce if invested at 3f %? 
 
 20. The diameter of a sphere which contains 47.71305 cu. 
 in. is 4|". What will a sphere contain whose diameter is 3"? 
 
CHAPTER VI . 
 
 PULLEYS, GEARS, AND SPEED 
 
 103 An important problem in the running of lathes is the cal- 
 culation of the speed at which the work should be turned, in 
 order to complete the work in the shortest time possible, with- 
 out injury to the work or the tools used. Similar problems 
 arise in the use of fly-wheels, emery wheels, grindstones, etc. 
 
 104 Rim Speed: When the work in a lathe is turned through 
 one complete revolution, a point upon the surface of the work 
 travels a distance equal to the circumference of the work. In 
 one minute, it would travel a distance equal to the circum- 
 ference of the work multiplied by the number of revolutions 
 per minute (R. P. M.). 
 
 The distance in feet traveled by a point on the circumference 
 of a wheel in one minute is called Rim Speed or Surface Speed. 
 
 105 RULE: To find the rim speed, multiply the circumference of the 
 
 revolving object by the number of revolutions per minute (R. P. 
 M.)f and express the result in feet. 
 
 Example 1: The diameter of a wheel is 2". If it makes 
 2500 R. P. M., what is the rim speed? 
 
 C = ;r.D = 3.1416 -2 = 6.2832". 
 
 6.2832 
 
 ^^ = .5236 (circumference in feet.) 
 
 .5236X2500= 1309, rim speed. 
 
 Example 2: The surface speed of a wheel is 3000. If the 
 diameter is 4", what is its R. P. M.? 
 
 3.1416.4 = 12.5664" 
 
 12.5664 
 
 ^fy = 1.0472 (circumference in feet.) 
 
 96 
 
RIM SPEED 97 
 
 Letx = R. P. M. 
 
 then 1.0472x = 3000 
 
 x = 2865, R. P. M. 
 
 Example 3 : What is the diameter of a wheel if its R. P. M. 
 is 2500 and its surface speed is 1500 ft. per minute? 
 
 Let x = diameter of the wheel. 
 
 Then 3. 1416x = circumference of the wheel. 
 3.1416X. 2500 = 1500 
 7854x=1500 
 
 X = .19, diameter in feet. 
 .19 • 12 = 2.28 diameter in inches. 
 
 Exercise 68 
 
 1. What would be the rim speed of a 12' fly wheel running 
 at 75 R. P. M.? 
 
 2. An emery wheel 15" in diameter runs at 1400 R. P. M. 
 Find the surface speed. 
 
 3. A pulley 5i" in diameter runs at 1250 R. P. M. What 
 is its rim speed? 
 
 4. A 12" circular saw runs at 2450 R. P. M. What is 
 its cutting speed (rim speed)? 
 
 5. A 10" emery wheel has a rim speed of 5000 ft. per 
 minute. How many R. P. M. does it make? 
 
 6. A grindstone will stand a surface speed of 800 ft. per 
 minute. At how many R. P. M. can it run if its diameter 
 is 4' 8"? 
 
 7. At how many R. P. M. should a 9|" shaft be turned 
 in a lathe to give a cutting speed of 60 ft. per minute? 
 
 8. A fly wheel having a rim speed of a mile a minute 
 runs at 120 R. P. M. What is its diameter? 
 
98 PULLEYS, GEARS, AND SPEED 
 
 9. An emery wheel runs at 950 R. P. M. If its surface 
 speed is 5500 ft. per minute, what is its diameter? 
 
 10. A line shaft runs at 186 R. P. M. A pulley on this 
 shaft has a riih speed of 1350 ft. per minute. What is the 
 diameter of the pulley? 
 
 11. The splicing of a belt connecting two equal pulleys 
 travels through the air at the rate of 2000 ft. per minute. 
 At what speed must the pulleys run if they are 20" in diameter? 
 
 12. A band saw runs over two pulleys each 32" in diameter. 
 If the band saw is 16' long, and the speed of the wheels 600 
 R. P. M., what is the cutting speed of the band saw? 
 
 Pulleys 
 
 Fig. 61. Pulleys 
 
 106 When two pulleys are connected by a belt, the rim speeds 
 of the two pulleys must be the same if there is no slipping of 
 the belt. Suppose pulley I (Fig. 61) is 6" in diameter and 
 pulley II is 12". The circumference of I is J as large as the 
 circumference of II, and therefore I will revolve twice while 
 II revolves once. In other words, the ratio of the diameter 
 of I to the the diameter of II is J, while the ratio of the R. P. M. 
 of I to the R. P. M. of II is y. This may be expressed as a 
 proportion if one ratio is inverted and therefore : 
 
PULLEYS 99 
 
 107 When two pulleys are connected hy a belt, the size of the pulley 
 varies inversely as its R. P. M. 
 
 Example: One of two pulleys connected by a belt is 12" 
 in diameter, and its R. P. M. is 400. What is the R. P. M. 
 of the other pulley if it is 3" in diameter? 
 
 Let x = R. P. M. of the second pulley. 
 
 12 X 
 
 — = (the size varies inversely as the R. P. M.) 
 
 3 400 
 
 x=1600. R. P. M. 
 
 Exercise 69 
 
 1. A 12" pulley running at 200 R. P. M. drives an 8" 
 pulley. Find the R. P. M. of the 8" pulley. 
 
 2. A 14" pulley drives a 26" pulley at 175 R. P. M. What 
 is the R. P. M. of the 14" pulley? 
 
 3. A 30" pulley running 240 R. P. M. is belted to a 12" 
 pulley. Find the R. P. M. of the 12" pulley. 
 
 4. A pulley on a shaft running at 120 R. P. M. drives a 
 24" pulley at 200 R. P. M. What is the diameter of the pulley 
 on the shaft? 
 
 Lineshaft: The line shaft is the main shaft which drives 
 the machinery of a shop by means of pulleys and belts. 
 
 Counter Sfiaft: A counter shaft is an auxiliary shaft placed 
 between the line shaft and a machine to permit a convenient 
 location of the machine. 
 
 5. A line shaft runs at 250 R. P. M. Determine the size 
 of the pulley on the line shaft in order to run a 6" pulley on 
 a machine at 1550 R. P. M. 
 
100 PULLEYS, GEARS, AND SPEED 
 
 6. It is found necessary to run a counter shaft at 310 
 R. P. M. If driven by an 18" pulley running at 175 R. P. M., 
 what must be the diameter of the pulley on the counter shaft? 
 
 7. A counter shaft for a grinder is to be driven at 375 
 R. P. M. by a line shaft that runs at 210 R. P. M. If the 
 pulley on the counter shaft is 12" in diameter, what size pulley 
 should be put on the line shaft? 
 
 8. A motor running at 875 R. P. M. has a 10|" driving 
 pulley. If the motor drives a line shaft at 180 R. P. M., 
 what must be the size of the line shaft pulley? 
 
 9. The diameters of two pulleys connected by a belt are 
 in the ratio f . If the R. P. M. of the larger pulley is 966, 
 what is the R. P. M. of the smaller? 
 
 Fig. 62 
 
 10. Pulley I is belted to II, and III to IV (Fig. 62). II and 
 III are on the same shaft. If the diameter of I is 18" and its 
 R. P. M. is 240, find the R. P. M. of II if its diameter is 8". 
 Find the R. P. M. of IV if it has a diameter of 6", and III has 
 one of 20". 
 
PULLEYS 
 
 Step-Cone Pulleys 
 
 101 
 
 Fig. 63, Step-Cone Pulleys 
 
 108 To secure different speeds on the same machine, step-cone 
 pulleys (Fig. 63) are used on both the driving shaft and the 
 driven shaft. The large step of the driving pulle> may be 
 belted to the small one of the driven for high speed, the medium 
 one to the medium one for middle speed, and the small one to 
 the large one for low speed. 
 
 Example: A step-cone pulley having diameters 11", 8 J", 
 and 6", running at 120 R. P. M., drives a step-cone pulley 
 having diameters 4", 6^", and 9". Find the three speeds. 
 
 Let x = R. P. M. at high speed. 
 
 Why? 
 
 1320 = 4x 
 
 X = 330, R. P. M. at high speed. 
 
 T'>-T=i|-0 
 
102 PULLEYS, GEARS, AND SPEED 
 
 Let y = R. P. M. at middle speed. 
 
 Then -4=-^ Whv? 
 6 J 120 
 
 1020 = 6^y. 
 
 y = 157 — , R. p. M. at middle speed. 
 
 Let z = R. P. M. at low speed. 
 
 6 z 
 
 Then -= — • Why? 
 
 9 120 
 
 720 = 9z. 
 
 Z = 80 R. P. M. at low speed. 
 
 Exercise 70 
 
 
 
 1 
 
 
 it 
 
 
 
 
 
 
 
 "'"""III"' 
 
 o 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 J* 
 
 
 
 t 
 
 Fig 64 
 
 Fig. 65 
 
 1. The steps of a pair of cone pulleys are 7", 5", 3", and 
 4", 6", 8" in diameter (Fig. 64). If the lower pulley has a 
 speed of 1050 R. P. M., find the three speeds of the upper 
 pulley. 
 
 2. The diameters of the steps of a step-cone pulley on a 
 machine are 10", 8|" and 7", and the corresponding counter 
 shaft diameters are 5^'', 7", and 8j". Find the speed for each 
 step on the machine if the counter shaft runs at 1190 R. P. M. 
 
GEARS 103 
 
 3. The steps of the cone pulley on a wood-turning lathe 
 are 7|", 5f ", and 4". The corresponding diameters of the 
 driving pulley on the motor are 2f ", 4|", and 6j". Find the 
 three speeds on the lathe if the motor speed is 1165 R. P. M. 
 
 4. The smallest steps on a pair of cone pulleys are 2|" 
 and 2f ". The increase in diameter of each succeeding step is 
 1^' (Fig. 65). The first pulley has a speed of 1000 R. P. M. 
 Find the three speeds of the second pulley. 
 
 Gears 
 
 Fig. 66. Gears 
 
 109 In machines where absolute accuracy in the speed of the 
 work is required, gears are used instead of belts to eliminate 
 slipping. When two gears are meshed as in Fig. 66, it is evi- 
 dent that their rim speeds are the same. Sizes of gears are 
 measured by the number of teeth rather than their diameters. 
 Suppose a 48-tooth gear drives one with 24 teeth. The smaller 
 one will revolve twice, while the larger one revolves once. 
 The ratio of the numbers of teeth is f, while the ratio of the 
 speeds is -J. Therefore : 
 
 110 When one gear drives another, the speed is inversely propor- 
 tional to the number of teeth. 
 
104 
 
 PULLEYS, GEARS, AND SPEED 
 
 Exercise 71 
 
 1. A 38-tooth gear is driving one with 72 teeth. If the 
 first gear runs at 360 R. P. M., what is the speed of the second 
 gear? 
 
 2. A 14-tooth gear running at 195 R. f*. M. is to drive 
 another gear at 105 R. P. M. What must be the number of 
 teeth in the second gear? 
 
 3. Two gears are to have a speed ratio of 3 to 4. If the 
 first gear has 36 teeth, how many will the second have? 
 
 4. The ratio of the numbers of teeth in two gears is y. 
 The R. P. M. of the first is 350. What is the speed of the 
 second? 
 
 ^6T 
 
 Fig. 67 
 
 6. In Fig. 67 gear I has 72 teeth, II has 40, III has 56, 
 and IV has 32. The R. P. M. of gear I is 60. Find the 
 R. P. M. of II. If gear III is on the same shaft as II, find 
 the R. P. M. of IV. 
 
REVIEW PROBLEMS 
 
 Exercise 72. (Review) 
 
 lO" 
 
 Fig. 68 
 
 1. The gear with 72 teeth has a speed of 35 R. P. M. 
 Find the speed of the 32-tooth gear. (Fig. 68.) 
 
 2. If the 32-tooth gear (Fig. 68) is to be replaced by one 
 which is to have a speed of 280 R. P M., what size gear must 
 be used? 
 
 Fig. 69 
 
 3. In Fig. 69 what must be the size of the line shaft 
 pulley (I) to run the emery wheel (V) at 1215 R. P. M., if the 
 R. P. M. of the line shaft is 150? 
 
106 PULLEYS, GEARS, AND SPEED 
 
 4. What would be the R. P. M. of the emery wheel (V), 
 Fig. 69, if the line shaft pulley (I) is replaced by a 48" pulley? 
 
 6. Find the grinding speed of the emery wheel in problem 
 4, if its diameter is 12". 
 
 6. A wood-turning lathe is driven by a motor running at 
 1200 R. P. M. The smallest step of the cone pulley on the 
 motor shaft is 2" in diameter, and its mate on the lathe is 7". 
 All increases in the diameters of succeeding steps are 2". If 
 the work being turned is 3" in diameter, find the cutting speed 
 on high speed. 
 
 7. Find the cutting speed in problem 6 on middle speed. 
 
 8. Find the cutting speed in problem 6 on low speed. 
 
CHAPTER VII 
 
 SQUARES AND SQUARE ROOTS 
 
 111 Square of a binomial: A few kinds of multiplication prob- 
 lems are used so often that it is a saving of time to be able to 
 write the result without performing the actual multiplication. 
 One of these is the square of a binomial. 
 
 Find the value of the following by multiplying, and write 
 the results as in part 1: 
 
 1. (a+3)2 = a2H-6a4-9. 3. (m+n)2 = 
 
 2. (b+5)2= 4. (x+7y)2 = 
 In each result, observe the following: 
 
 I. There are 3 terms in the result. 
 
 II. The first term of the result is the square of the first 
 term of the binomial, and the third term of the result is the 
 square of the second term of the binomial. 
 
 III. The second term of the result is 2 times the product 
 of the two terms of the binomial. 
 
 Find the value of the following by actual multiplication and 
 write the results as in part 1 : 
 
 1. (a-3)2 = a2-6a+9. 3. (-m+n)2 = 
 
 2. (b-10)2= 4. (-x-7y)2 = 
 
 In each result observe that the same facts hold true as in 
 the preceding case, and that the law of signs for multiplication 
 must be used. 
 
 112 RULE: To square a binomial, square the first term, take 2 
 
 times the product of the two terms, square the second term, 
 and write the result as a trinomial. 
 
 107 
 
108 SQUARES AND SQUARE ROOTS 
 
 Example: (2a-3bx)2 = (+2a)2+2(+2a)(-3bx) + (-3bx)^ 
 = (-f4a2) + ( - 12abx) + (+9bV) 
 = 4a2-12abx+9b2x2. 
 
 
 
 Exercise 73 
 
 
 Wri 
 
 te the results without written multiplication: 
 
 1. 
 
 (a+l)2 
 
 16. 
 
 (a2+b2)2 
 
 2. 
 
 (t+u)^ 
 
 17. 
 
 (2m2-3n)2 
 
 3. 
 
 (d-4)2 
 
 18. 
 
 (4t3-3u2)2 
 
 4. 
 
 (x-y)^ 
 
 19. 
 
 (a4+4a)2 
 
 6. 
 
 (2a+b)2 
 
 20. 
 
 (7-3m2)2 
 
 6. 
 
 (3x-5)2 
 
 21. 
 
 (m-J)2 
 
 7. 
 
 (a-3b)2 
 
 22. 
 
 (y+*)^ 
 
 8. 
 
 (x+4y)2 
 
 23. 
 
 (2x-i)2 
 
 9. 
 
 (2m+3n)2 
 
 24. 
 
 (3m+f)2 
 
 10. 
 
 (5t-4u)2 
 
 26. 
 
 (4x+iy)2 
 
 11. 
 
 (6ab-5xy)2 
 
 26. 
 
 (|x^-fy)^ 
 
 12. 
 
 (5ab+4bx)2 
 
 27. 
 
 (Ift2-fu3)2 
 
 13. 
 
 (m2+5)2 
 
 28. 
 
 (ff+sO^ 
 
 14. 
 
 (x2-8)2 
 
 29. 
 
 (2.3 l-5.1m)2 
 
 15. 
 
 (a2-2)2 
 
 30. 
 
 (.3125m3n+3|mn3)2 
 
 31. Square 32 mentally. 
 Suggestion 322= (30+2)2 
 
 = 900+120+4 = 1024 
 
 Square the following mentally: 
 
 32. 21 36. 34 
 
 33. 22 36. 37 
 
 34. 29 (Suggestion 29 = 30-1) 37. 49 
 
 38. 
 
 19 
 
 39. 
 
 35 
 
 40. 
 
 43 
 
SQUARE ROOT OF MONOMIALS 109 
 
 SQUARE ROOT 
 
 Square Root of Monomials 
 
 113 Square Root: Problems often arise in which the reverse of 
 squaring is necessary. For example: what must be the side 
 of a square whose area is 25 sq. in.? The side must be such 
 that, if multiplied by itself, the result will be 25. It is evident 
 that 5 is the side of the square since 5^ = 25. 
 
 The square root of a number is a number which if squared, will 
 produce the given number. 
 
 Finding such a number is called extracting square root, and 
 the operation is indicated by the radical sign, V 
 
 ( 1 4)2 1 16 } .-. V"l6 = +4, or - 4, written +4. 
 
 (-3a)2 = 9a2\ .-^-^ , „ 
 (+3a)2 = 9a2/-'-^^^ =±^^' 
 
 (Ila2b4)2 = 121a%8.-. 
 
 V121a%« = 
 
 ^lla^b^. 
 
 Exercise 74 
 
 
 Find the square root of: 
 
 
 
 1. 81 
 
 4. 
 
 144m2n2 
 
 2. 121 
 
 5. 
 
 25xV 
 
 3. 4a2 
 
 
 
 Find the value of: 
 
 8. 
 9. 
 
 
 6. V49xy 
 
 V 196xV' 
 
 7. V64a2b^ 
 
 V256ci«d«e^ 
 
 10. V400a2b4c8 
 
 The square root of a negative number cannot be found since, by the 
 law of signs for multiplication, the square of either a positive or a negative 
 number is positive. 
 
no 
 
 SQUARES AND SQUARE ROOTS 
 
 Square Root of Trinomials 
 
 114 (a+b)2 = a2+2ab+b2 \ . . , , ^ , , , , , , , , 
 
 (_a_b)2 = a2+2ab+b/-- ^^ +2ab+b2=+aH-b,or-a-b, 
 
 written + (a +b). 
 (a-b)2 = a2-2ab+b2 \ . 
 
 (_a+b)2 = a2-2ab+bV •*• Va2-2ab+b2= +a-b,or-aH-b, 
 
 written +(a — b). 
 
 115 Trinomial Square: A trinomial in which two of the terms 
 are squares and positive, and the other term is 2 times the 
 product of the square roots of those terms, is called a trinomial 
 square, and is the square of a binomial. 
 
 Exercise 75 
 Select the trinomial squares in the following: 
 
 11. 64a^-176a+121 
 
 12. 49m4n2+112m2+8n4 
 
 13. x2+x+i 
 
 14. m2+fm+^ 
 
 1. x2+2xy+y2 
 
 2. m^— 4m+4 
 
 3. m2-4m+6 
 
 4. a2-4a-4 
 
 5. x2-6xy4-9y2 
 
 6. 4t2+6tu+9u2 
 
 7. 16x2+25y2 
 
 8. 169m6-26m3n+n2 
 
 9. 25x2+16y2-40xy 
 10. 49V-70xyz-25z2 
 
 15. x2+|x+| 
 
 16. 4a2+ab+YVb' 
 
 17. 9m2-24mn + 16 
 
 18. x2 + |x + y% 
 
 19. y^+fy+A 
 
 20. t^-it-aV^e 
 
SQUARE ROOT OF TERMINALS 111 
 
 116 By Art. 114, Va2+2ab+b2= -f(a+b). 
 
 Va2-2ab+b2=-|-(a-b). 
 
 Observe the following facts in each result: 
 
 I. The two terms of the binomial are the square roots of 
 the two terms of the trinomial which are squares. 
 
 II. If the sign of the other term of the trinomial is plus, 
 the terms of the binomial have like signs, and if it is minus, 
 the terms of the binomial have unlike signs. 
 
 117 RULE: To find the square root of a trinomial square, extract the 
 
 square root of the two terms which are squares, connect them 
 with the sign of the other term of the trinomial, and prefix the 
 Sign + to the binomial thus formed. 
 
 Example : Find the square root of 25x2 _j_ j gy2 _ 40xy . 
 
 V25x2+16y2-40xy=+(V25x2- V16y2). 
 = +(5x-4y). 
 
 Exercise 76 
 Find the square root of: 
 
 1. 9x2-24xy+16y2 
 
 2. 9+6x+x2 
 
 3. 49m2+14mn+n2 
 
 4. t2-10tu+25u2 
 
 5. a^^-2aV+y^^ 
 
 6. 4a6-4a3b2c+bV 
 
 7. 4a2-20ay+25y2 
 
 8. 9m2+42mx+49x2 
 
 9. -72xy+81x2H-16y2 
 10. 25x64-49a^b2-70a2bx3 
 
112 SQUARES AND SQUARE ROOTS 
 
 Find the value of: 
 
 11. V30m+25+9m2 
 
 12. V -60m2nV+25m4n4+36p8 
 
 13. V 49a4x2+ 1 12a3x3+64a2x* 
 
 14. Vx2+x-hi 
 
 15. yJisi^-\-25h^-\^sih 
 
 16. VaV'+Vtu+Qu^ 
 
 17. Vf|m^+2m2n24-||n* 
 
 18. V x^-fx^+yfo 
 
 19. Via%2_|.a2bc3H-2^3— c« 
 
 20. V|fx«+iV'z'-ixVz 
 
 Square Root of Numbers 
 i^5 By problem 31, Exercise 73. 
 
 322= (30+2)2 = 900+1204-4 = 1024. 
 .•.V1024= V 900+ 120+4 =+(30+2) = +32. 
 
 To extract the square root of such numbers as 1024, it is 
 necessary to separate them into the form of a trinomial square. 
 •This can not be done by inspection. Therefore it is convenient 
 to use the simplest form of trinomial square, t2+2tu+u2, as 
 a formula. In that case, t2+2tu+u2 corresponds to 1024, 
 and its square root, t+u, corresponds to the square root of 
 1024, or 32. The work may be arranged as follows: 
 
 t+u 
 t2+2tu+u2 = t2+u(2t+u)= 1024 1 30+2 
 
 t2 = 
 
 900 
 
 2t = 60 
 u= 2 
 
 2t+u = 62 
 
 124 = u(2t+u) 
 124 
 
 ".V 1024 =+32. 
 
SQUARE ROOT OF NUMBERS 
 
 113 
 
 Example 1 : Find the square root of 5625. 
 
 In order to find how many digits there are in the square root 
 of a number, observe the following: 
 
 92 = 81. 
 992 = 9801. 
 9992 = 998001. 
 
 The square of a number of one digit can not contain more 
 than two digits, the square of a number of two digits can not 
 contain more than four digits, etc. Therefore, the number of 
 digits in the square root of a number may be determined by 
 separating the given number into groups of two digits each, 
 beginning at the decimal point. 
 
 t2+u(2t+u) = 56'25 170+5 
 t2 = 4900 
 
 2t = 140 
 u= 5 
 
 725 = u(2t+u) 
 
 2t+u=145 
 
 725 
 
 
 /.V 5625 =+75. 
 
 Observe that t is found by extracting the square root of the 
 greatest square in the first group, and u is the integral number 
 found by dividing the remainder by the number equal to ^t. 
 
 Example 2: Find the value of V 289. 
 
 t2+u(2t+u) = 
 t2 = 
 
 2'89 
 100 
 
 t+u 
 10+9 
 
 2t = 20 
 u = 9 
 
 2t+u = 29 
 
 189 = u(2t+u) 
 
 2 61 
 
114 
 
 SQUARES AND SQUARE ROOTS 
 
 
 
 t+u 
 
 t2+u(2t+u) = 
 
 = 2'89 
 
 |10+8 
 
 t2 = 
 
 a 00 
 
 
 2t = 20 
 
 189 = 
 
 = u(2t+u) 
 
 u= 8 
 
 
 
 2t+u = 28 
 
 2 24 
 
 
 
 ? 
 
 t+u 
 
 t2+u(2t+u) = 
 
 = 2'89 
 
 |10+7 
 
 P = 
 
 100 
 
 
 2t = 20 
 
 189 = 
 
 = u(2t+u) 
 
 u= 7 
 
 
 
 2t+u = 27 
 
 189 
 
 
 .-•V 289 =+17. 
 Observe that in finding u, it is not always possible to take 
 the largest integral number found by dividing the remainder 
 by the number equal to 2t. 
 
 
 Exercise 77 
 
 
 Extract the square root of: 
 
 
 1. 1849 5. 2916 
 
 8. 4624 
 
 2. 3136 6. 961 
 
 9. 1521 
 
 3. 576 7. 256 
 
 10. 4489 
 
 4. 5184 
 
 
 cample: Find the square root of 60516. 
 
 
 t+u 
 t2+u(2t+u)=6'05'16 1200+40 
 t2 = 4 00 00 
 
 
 2t = 4 00 
 
 u= 40 
 
 2t+u = 4 40 
 
 2 05 16 = ur2t+u 
 1 76 00 
 
 
 29 16 
 
SQUARE ROOT OF NUMBERS 
 
 115 
 
 The square root of 60516 will contain three digits. The 
 first two are found in the usual way. The root is evidently 
 240+ ? and the amount that has been subtracted from 60516 
 (40000+17600) is 240^. Therefore 240 may be considered a 
 new value of t, and 2916 a new value of u(2t+u), in finding 
 the third digit of the root. The problem then becomes: 
 
 t+u 
 t2+u(2t+u) = 6'05'16 1240+6 
 t2 = 5 76 00 
 
 2t = 480 
 u= 6 
 
 2t+u = 486 
 
 29 16 = u(2t+u) 
 
 29 16 
 
 These two operations may be combined into one problem as 
 follows: 
 
 t+u t+u 
 
 t2+u(2t+u)=6'05'16 1200+40 240+6 
 t2 = 4 00 00 
 
 2t = 400 
 
 2 05 16 = u(2t+u) 
 
 u= 40 
 
 
 2t+u = 440 
 
 1 76 00 
 
 2t = 480 
 
 29 16 = u(2t+u) 
 
 u= 6 
 
 
 2t+u = 486 
 
 29 16 
 
 .-. V 60516 = ±246. 
 
 Find the value of 
 
 1. V 37636 
 
 2. yllSUi 
 
 3. V 54756 
 
 Exercise 78 
 
 4. V 173889 
 
 5. V 98596 
 
 6. V 233289 
 
 7. V 94249 
 
 8. V 648025 
 
 9. V 9778129 
 10. V 1022121 
 
116 
 
 SQUARES AND SQUARE ROOTS 
 
 119 The operation of extracting square root may be abridged as 
 follows: 
 
 Find the the value of: 
 
 V2, 
 t2+u(2t-hu) = 
 
 t2 = 
 
 35.6225 
 
 r 
 
 t + u 
 1 5. 
 
 t + 
 + u 
 
 3 
 
 u 
 5 
 
 -2' 
 = 1 
 
 3 5' 
 
 .6 2' 2 
 
 = u(2t+u) 
 
 5 
 
 2t= 20 
 
 u= 5 
 
 2t+u= 25 
 
 1 
 1 
 
 3 5 = 
 2 5 
 
 
 2t= 300 
 
 u= 3 
 
 2t+u= 303 
 
 
 1 
 9 
 
 6 2 = u(2t+u) 
 9 
 
 2t = 3060 
 
 u= 5 
 
 2t+u = 3065 
 
 
 1 
 1 
 
 5 3 2 
 5 3 2 
 
 5 = u(2t+u) 
 5 
 
 
 
 
 
 
 /. V 235.6225 =±15.35 
 
 NOTE: In pointing off the given number into groups of two digits 
 each, begin" at the decimal point and proceed both right and left. 
 
 Exercise 79 
 Find the square root of: 
 
 1. 2323.24 
 
 2. .120409 
 
 3. 2.6569 
 
 4. 32.1489 
 
 5. 123.4321 
 
 6. .07557001 
 
 7. .00003481 
 
 8. 1621.6729 
 
 9. 1040400 
 10. 1624.251204 
 
SQUARE ROOT OF NUMBERS 
 
 117 
 
 120 If a number is not a perfect square, the operation may be 
 continued to as many decimal places as is desired by annexing 
 a sufficient number of ciphers. 
 
 Example: Find the value correct to .001 of: 
 
 V4 
 
 .329< 
 
 94. 
 
 t + u 
 
 
 t 
 
 + u 
 
 
 t 
 
 + u 
 
 8 
 
 
 t 
 2. 
 
 + u 
 
 
 8 = 2.081 
 
 t2+u(2t+u)=4. 
 t2 = 4 
 
 32' 
 
 99' 
 
 = u(2t4 
 
 40' 00 
 
 2t = 40 
 u= 
 
 32 = 
 00 
 
 -u) 
 
 2t = 400 
 u= 8 
 
 32 
 32 
 
 99 = 
 64 
 
 = u(2t+u) 
 
 2t+u = 408 
 
 
 2t = 4160 
 u= 
 
 
 35 
 00 
 
 40 = u(2t+u) 
 00 
 
 2t =41600 
 
 u= 8 
 
 2t+u = 41608 
 
 
 35 
 33 
 
 40 00 = u(2t+u) 
 28 64 
 
 11 
 
 36 
 
 /. V 4. 32994 =+2. 081 
 
 Observe that if 3 decimal places in the result are required, 
 it is necessary to determine the digit in the 4th place, and if 
 it is 5 or more, to add 1 to the digit in the 3rd place. 
 
118 SQUARES AND SQUARE ROOTS 
 
 Exercise 80 
 
 Find the square root of the following correct to 4 decimal 
 places : 
 
 1. 15 3. 126 
 
 2. 38 4. 2.5 
 
 5. 634.125 
 
 Find the value of the following correct to .0001: 
 
 6. V2 8. V5 
 
 7. V3 9. V.5 
 
 10. V14.4 
 V36= v"9^= VV- V4 = 3-2 = 6. 
 121 From this it is evident that : 
 
 The square root of a number is equal to the product of the square 
 roots of its factors. 
 
 This law may be used to simplify the process of finding the 
 square roots of numbers which contain one or more factors 
 that are squares. For example: 
 
 VT2 = vT V3 = 2 V3 = +3.4642. 
 
 Exercise 81 
 
 Given V2^= 1.4142, V^= 1.7321, v'5 = 2.2361, 
 Find the value of the following correct to .001 : 
 
 1. V~8 6. V45 
 
 2. VTS 7. V48 
 
 3. V20 8. V50 
 
 4. V~27 9. V72 
 
 5. V32 10. V108 
 
SQUARE ROOT OF FRACTIONS 119 
 
 /'2\2_2 2_£ 
 \3J — 3*3 
 
 11. 
 
 V180 
 
 
 16. 
 
 V 98 
 
 12. 
 
 V 80 
 
 
 17. 
 
 V147 
 
 13. 
 
 V125 
 
 
 18. 
 
 V320 
 
 u. 
 
 V363 
 
 
 19. 
 
 V243 
 
 16. 
 
 V512 
 
 
 20. 
 
 V128 
 
 4 
 
 .-. vl= 
 
 ■X3- 
 
 
 
 The square root of a fraction is found by extracting the 
 square root of the numerator and of the denominator. 
 
 Exercise 82 
 Find the square root of: 
 lie q -i- 
 
 2. 
 
 "25 ^- 64 
 
 3.6. A 121 
 
 4 9 *• 9 
 
 O- 2 2^ 
 
 Find the value of the following correct to .001: 
 
 6. Vif 8. VsV 
 
 7. V 8T ^' ^144 
 
 in ./ 1 62 
 10- V 2 
 
 123 In fractions where the denominator is not a perfect square 
 the operation of finding the square root may be simplified by 
 multiplying both numerator and denominator by a number 
 which will make the denominator a square. 
 
 Example: vl= v"A=-^ = ±?^^^=+.6124 
 
 V16 4 — 
 
120 SQUARES AND SQUARE ROOTS 
 
 Exercise 83 
 Find the value of the following correct to .001 : 
 
 1. 
 
 v§ 
 
 2. 
 
 v4 
 
 3. 
 
 vi 
 
 4. 
 
 vi 
 
 6. 
 
 VI 
 
 6. 
 
 VI 
 
 7. 
 
 vf 
 
 8. 
 
 vf 
 
 9. 
 
 VI 
 
 10. 
 
 vV 
 
 Quadratic Equations 
 
 124 Quadratic Equation: A quadratic equation is one which con- 
 tains the square of the unknown quantity as the highest power 
 of the unknown. 
 
 X 13 3x 40 
 
 Example: - — — = — 
 
 2 3x 2 3x 
 
 3x2-26 = 9x2-80 Why? 
 
 54 = 6x2 Why? 
 
 x2 = 9 Why? 
 
 X= +3 (extracting the square root of both 
 members) 
 
 Observe that: 
 
 I. A quadratic equation of the form x2 = 9 may be trans- 
 formed into one containing the first power of the unknown by 
 extracting the square root of both members. 
 
 II. In extracting the square root of both members of the 
 equation x2 = 9, the full result would be +x=+3, which is a 
 condensed form of: 
 
 1. +x=+3 3. -x=+3 
 
 2. +x=-3 4. -x=-3 
 
 1 and 4, 2 and 3 are the same equations and therefore x= +3 
 expresses all four equations. 
 
QUADRATIC EQUATIONS 121 
 
 Exercise 84 
 Solve (correct to .001 where necessary): 
 
 1. x2=12 7. x2 = f 
 
 2. x2 = 75 8. x2+10 = 59 
 
 9. y2-ll = 185 
 
 10. 7m2- 175 = 
 
 11. 8s2-38 = 90 
 
 12. lla2-5 = 2+2a2 
 
 13. 3(x-2)-x = 2x(l-x) 
 
 14. (2t+3)(t+2)-(t+3)(t+4)=4t2-21 
 
 15. (t-|-4)2+(t-4)2=48 
 
 ^^ 3xy-l 5(x^-l) (4x^+1) _^ 
 ^^' —5 lO 25^"" 
 
 y-l y+1 
 
 5r-3^_r+2 
 ^*** 9r+l 2r+5 
 
 2x-5 , - 3x+10 
 19. — TT- =1.5 — 
 
 3. 
 
 X2 = 
 
 = 55225 
 
 4. 
 
 X2 = 
 
 = 46 
 
 6. 
 
 X2 
 
 169 
 1225 
 
 6. 
 
 X2 = 
 
 _ 75 
 
 108 
 
 20. 
 
 3 2x+ 5 
 
 3x-l 3x+1^29 
 3x+l"^3x-l 14 
 
122 SQUARES AND SQUARE ROOTS 
 
 21. The length of a rectangle is 3 times its width, and the 
 area is 243 sq. in. Find the dimensions of the rectangle. 
 
 22. How long must the side of a square field be that the 
 area of the field may be 5 acres? 
 
 23. The dimensions of a rectangle are in the ratio f , and 
 its area is 300. Find the dimensions of the rectangle. 
 
 24. The side of one square is 3 times that of another, and 
 its area is 96 sq. in. more than that of the other. Find the 
 sides of the two squares. 
 
 25. If the area of a 3" circle is 28.2744, find the diameter 
 of a circle whose area is 78.54. (See Exercise 67, problem 3.) 
 
 26. Find the diameter of a circular piece of copper whose 
 weight is 3.01 oz. if a 10" disk weighs 9.03 oz. (See Exercise 
 67, problem 10.) 
 
 • 27. The intensity of light varies inversely as the square of 
 the distance from the source of light. How far from a lamp 
 should a person sit in order to receive one half as much light 
 as he receives when sitting 3 ft. from the lamp? 
 
 28. The distance covered by a falling body varies directly 
 as the square of the time of falling. If a ball drops 402 ft. in 
 5 seconds, how long will it take it to drop 600 ft.? 
 
 29. The weight of an object varies inversely as the square 
 of the distance from the center of the earth. If an object 
 weighs 180 lbs. at the earth's surface, at what distance from 
 the center will it weigh 160 lbs., if the radius of the earth is 
 4000 miles? 
 
 30. The surface of a sphere varies directly as the square of 
 the diameter. Find the diameter of a sphere whose surface 
 is 78.54 sq. in., if the surface of an 11". sphere is 380.1336 sq. in. 
 
. CHAPTER VIII 
 
 FORMULAS 
 
 Evaluation of Formulas Containing Square Root 
 
 Exercise 85 
 
 Evaluate the following formulas for the values given (correct 
 to .001 where necessary): 
 
 1. h = ^V3 when a = 5. 
 
 2. c=Va2+b2 whena = 4, b = 5. 
 
 3. V = 2 7r2r2R when r = J, R= l|. 
 
 4. A = |r2V3 whenr = 3|. 
 
 5. V = ^V2 when a = 3.2. 
 
 6. G=Vab whena=4, b = 5. 
 
 7. Y = — ^ --^-^TTSi^ when a = 6, r = 18, R = 24. 
 
 V g 
 
 8. t^TTA/- when 1=1, g = 32. 
 
 9. s=^(Ar5-l) whenr = 2j. 
 
 10. D=Va2+b2+c2 whena = 5, b = 6, c = 7. 
 
 11. b= Va2+c2-2a'c when a =14, a' = 5, c = 12. 
 
 12. l = 2VD2+a2 + ttD when D = 16, a = 35. 
 
 13. M = |V2(a2+c2)-b2 whena=15, c = 17, b = 19. 
 
 — b+Vb2+4ac , o u c on 
 
 14. x = when a = 3, b = 5, c = 20. 
 
 2a 
 
 123 
 
124 FORMULAS 
 
 16. x= — : >.:cn a = 3, b = 5, c = 20. 
 
 16. l = 4/(^^)Va2+7r(5±^)when D = 36, d = 6,a = 96. 
 
 17. s = -rV10-2V5 when r = 4l. 
 
 N ^„ C 
 
 18. s = 5^.;rR2-yV4R2-0 whenN = 72,R= 10,C= 13. 
 
 19. x= Vr(2r- V4r2-s2) whenr = 3,s = 2. 
 
 20. A= Vs(s-a) (s-b) (s-c) when a= 15, b= 18, c = 22, 
 
 s = |(a+b+c). 
 
 125 A formula is an equation and may be solved for any of the 
 letters involved if the values of all the other letters are given. 
 
 Example 1: Z = 4;rra. Find a, when Z = 502.656, r = 8. 
 502.656 = 4.3.1416.8-a 
 
 5. 
 
 15.708 
 62.832 
 502.656 
 
 a = 
 
 4-3.1416-8 
 
 Example 2: V = Jrfa. Find r, when V = 593.7624, a = 7. 
 
 1.0472 
 593.7624 = f 3. 1416- r2. 7 
 
 81 
 84.8232 
 „ 593.7624 
 
 1.04727 
 
 r=+9. 
 
 = 81 
 
FORMULAS INVOLVING SQUARE ROOT 
 
 125 
 
 Example 3: b2 = a2-f c^— 2ac'. Solve for c', when a = 5, 
 b = 6, c = 7. 
 
 36 = 25+49 -2- 5c' 
 10c' = 38 
 c' = 3.8 
 
 Exercise 86 
 
 Find the values (correct to .001 when necessary) : 
 
 P=4a 
 P=:a+b+c. 
 
 |bh. 
 
 P=2(a-l-b) 
 A = ab. 
 A 
 
 6. A=Jh(b+b'). 
 
 7. A=ih(b+bO. 
 
 8. C = 27rr. 
 
 9. A = 7rr^. 
 
 10. 
 
 1 
 w = £.p. 
 
 11. W = r.p. 
 
 12. W = r.p. 
 
 13. L=lfd+J. 
 
 14. S = |gt2. 
 
 16. S = |gt2+vt. 
 
 Find a, when P = 5§. 
 
 Find c, when P = 7962, a =1728, 
 b = 3154. 
 
 Find a, when P = 17|, b = 2j. 
 
 Find b, when A = 2.31, a = l.l. 
 
 Find h, when A = 3U, b = 3|. 
 
 Find h, when A = 96, b = 18, b' = 6. 
 
 Find b', when A = 12.8, b= 1.2, h = 8. 
 
 Findr, when C = 50. 
 
 Find r, when A = 50. 
 
 Find p, when w = 333 J, 1 = 25, h = 4 J. 
 Find 1, when w=320,h = 24,p = 213|. 
 
 Find h, when w = 150, 1 = 162, p = 100. 
 
 Find d, when L=4:yq' 
 Find t, when S = 196.98. 
 (See Exercise 17, problem 5.) 
 Find V, when S = 164.72, t = 3. 
 
126 FORMULAS 
 
 IIV 
 
 16. F=— -—. FmdF,whenu=ll,v = 7. 
 
 17. F=-^. Findv,whenF=liu = 3. 
 
 u+v ' ^' 
 
 18. x = — ' . Fmdx, whenb=-5, a = 3, c=-2. 
 
 19. A=-r-. Find D, when A =115. 
 
 4 ' 
 
 20. V = !rr2a. Find r, if V = 330, a = 7. 
 
 21. V = ^2a. Find a, if V = 46.9, r = 2.3. 
 
 22. V = 3nr2.?^i5. ^'^^ ^' w^^^ V=1932, H = 14.6, 
 
 23. V = ;rr2.^^±5. FindH, when V = 2246, r = 8, h = 6. 
 
 24. A = ^. Find A, when a = 2.3, b = 3.2, c = 4.L 
 
 ^ r = 2.058. 
 
 25. A = ^-. Find r, when a = 21, b = 28, c = 35, 
 
 A = 294. 
 
 26. A = |r(a+b+c). Find r, when a = 79.3, b = 94.2, c = 
 
 66.9,A = 261.012. 
 
 27. A = |r(a+b+c). Find a, when A = 27.714, r = 2.3095, 
 
 b = 8, c = 8. 
 
 28. A = J(23rR+2zrr)s. Find A, when R = 8, r = 3, s = 7. 
 
 29. A = J(2?rR+23rr)s. Find r, when A = 439.824, R = 10, 
 
 s=10. 
 
 30. A=|(2jrR-f-2;rr)s. Find s,when A = 106.029, R = 7j,r = 6- 
 
 31. l = 2/(5±^)^+a^ + .(^). 
 
 Find 1, when D = l|, d = 1 J, a = 15. 
 
 32. h = ^V3r Find a, when h = 27.7136. 
 
FORMULAS INVOLVING SQUARE ROOT 127 
 
 33. A = ^ V"3l Find h, when A = y vl". 
 
 34. A = ^ V"3^ Find a, when A = VlS. 
 
 35. v = :^V2^ Find V, when a = 6. 
 
 Qt.2 
 
 36. A = ^V3. Find r, when A = 153. 
 
 37. c2 = a2+b2. Findb, whenc = 2.1, a = 1.7. 
 
 38. b2 = a2+c2+2a'c. Find a, whenb = 8, c = 5, a' = 2.1. 
 
 39. b2 = a2+c2+2a'c. Finda', whena=18, b = 16, c = 31. 
 
 40. b2 = a2+c2-2ac'. Find c, when a = 5, b = 4, c' = 2.3. 
 
 41. b2 = a2+c2-2ac'. Find c', when a= 14, b = 15, c = 16. 
 
 42. H = r-Vs(s-a)(s-b)(s-c). 
 
 ■ FindH,whena = 2.18,b = 5,c = 3.24, 
 
 s = J(a+b+c). 
 
 43. a2+b2 = 2Qy+2m2. Find m, when a = 9, b = 12, c = 15- 
 
 44. a-+b2 = 2r|y+2m2. Findb, whena = 5, c = 13, m = 6i. 
 46. s = I(vT-l). Find r, when 8=10.50685. 
 
 46. x= ^+^^'+^^^- Findx,whena = 3,b=-7,c=+2. 
 
 ^a 
 
 47. V = 27r2r2R. Find r, when V = 98696.5056, R = 50-. 
 
 48. X = V r (2r — V 41^ — s^) . Find x, when r = s = 10. 
 
 N 
 
 49. s = ^.;rr2-- V4r2-c2. 
 
 Find N, when s = 23. 1872, c = r= 16. 
 
 — b— Vb2+4ac -o- j , ^ , 
 
 50. x = \- — ' . Findx,when a = — 6,b= — 9,c=+2. 
 
 ^a 
 
128 
 
 FORMULAS 
 
 Right Triangle 
 
 126 One of the formulas most commonly used is that of the right 
 triangle. 
 
 
 ^jb^^C 
 
 Fig. 70 
 
 127 Right Triangle: A right triangle is a triangle in which one 
 angle is a right angle. The lines including the right angle are 
 called the sides, and the line opposite the right angle is called 
 the hypotenuse. 
 
 It can be proved that; 
 
 128 The square of the hypotenuse is equxil to the sum of the squxires 
 of the two sides. 
 
THE RIGHT TRIANGLE 
 
 129 
 
 This truth is stated by the formula: 
 c2 = a2+b2 (Fig. 70). 
 
 Exercise 87 
 
 Find results correct to .001 when necessary: 
 
 1. Find c, when a = 8, b = 15. 
 
 2. Find a, when b = 9, c=41. 
 
 3. Find b, when a = 3, c = 6. 
 
 4. Find the hypotenuse of a right triangle when the sides 
 are 3.2 and 2.4. 
 
 6. The hypotenuse and one side of a right triangle are 
 respectively 2f and l|. Find the other side. 
 
 6. The sides of a right triangle are 5j and 12.5. Find the 
 hypotenuse. 
 
 7. The two sides of a right triangle are equal to each 
 other, and the hypotenuse is 18. Find the sides. (Fig. 71.) 
 
 
 
 \ 
 
 ^ 
 
 
 7 
 
 \^ 
 
 X 
 
 
 /32 
 
 
 /tf" 
 
 Fig. 71 
 
 
 Fig. 72 
 
 
 Fig. 73 
 
 8. One side of a right triangle is 3 times the other, and 
 the hypotenuse is 80. Find the sides. Draw a figure. 
 
 9. The two sides of a right triangle are in the ratio f , 
 and the hypotenuse is 225. Find the sides. Draw a figure. 
 
 10. Find the diagonal of a square whose sides are 1.32. 
 (Fig. 72.) 
 
130 FORMULAS 
 
 11. Find the perimeter of a square whose diagonal is 17. 
 Draw a figure. 
 
 12. Find the diagonal of a rectangle whose dimensions are 
 11 and 16. (Fig. 73.) 
 
 13. Find the dimensions of a rectangle whose diagonal is 
 91, if the length is 5 times the width. Draw a figure. 
 
 14. The perimeter of a rectangle is 70, and its sides are in 
 the ratio f. Find the diagonal. 
 
 15. A ladder 36 ft. long is placed with its foot 11 ft. from 
 the base of the building. How high is a window which the 
 ladder just reaches? 
 
 16. A flag staff 79 ft. long is broken 29 ft. from the ground. 
 If the parts hold together, how far from the foot of the staff 
 will the top touch the ground? 
 
 17. How long is a guy wire which is attached to a wireless 
 tower 227 ft. from the ground, and is anchored 362 ft. from 
 the foot of the tower? 
 
 18. The slant height of a cone is 12", and the radius of the 
 base is 5|". Find the altitude of the cone. (Fig. 74.) 
 
 19. One side of the base of a square pyramid is 14", and 
 the altitude is 16". Find the edge, E. (Fig. 75.) (Sugges- 
 tion : The altitude of the pyramid meets the base at the middle 
 point of the diagonal.) 
 
 20. Find the slant height, S. (Fig. 75.) 
 
INDEX 
 
 SUBJECT PAGE 
 
 Addition, Algebraic, Defini- 
 tion 40 
 
 Addition, Algebraic, Rule .- 41 
 
 Addition, Algebraic, of several 
 
 numbers 42 
 
 Algebraic Subtraction, Defini- 
 tion of 45 
 
 Algebraic Subtraction Rule.— 4(5 
 
 Angle, Definition of 25 
 
 Angle, Right 25 
 
 Angle, Straight 25 
 
 Angles, Complementary 35, 36 
 
 Angles, Drawing of 26, 27 
 
 Angles, Measuring 27, 2S 
 
 Angles, Reading 28 
 
 Angles, Sum of 30, 33 
 
 Angles, Supplementary 33, 34 
 
 Antecedent 79 
 
 Arm , 51 
 
 Base 16 
 
 Binomial, Definition 44 
 
 Binomial, Square of 107 
 
 Brace 49 
 
 Bracket 49 
 
 Checking Equations 24 
 
 Clearing of Fractions 9 
 
 Clockwise — 52 
 
 Coefficient 15 
 
 Coefficient, Numerical 16 
 
 Complement 35 
 
 Consequent 79 
 
 Counter Clockwise 52 
 
 Counter Shaft 99 
 
 Decimals, Ratios as 81 
 
 Decimal Equivalents 81 
 
 Degrees 26 
 
 Division Law of Exponents 
 
 for 68 
 
 Division Law of Sign for 68 
 
 Division of Monomials... .68, 69. 70 
 
 SUBJECT PAGE 
 
 Division of Polynomials by 
 
 Monomials '. 72 
 
 Division of Polynomials by 
 Polynomials 73, 75 
 
 Equations, Definition of 1 
 
 Equation, Principles of 10 
 
 Equation, Checking 24 
 
 Equations, Quadratic Defini- 
 tion 120 
 
 Equations, Quadratic, Solu- 
 tion of 121, 122 
 
 Formula, Definition '. 19 
 
 Formulas, Area 22 
 
 Formulas, Circle 23 
 
 Formulas, Circular Ring 23 
 
 Formulas, Evaluation of 19 
 
 Formulas, General 23 
 
 Formulas, Involving Square 
 
 Root 123, 124, 125, 126, 127 
 
 Formulas, Perimeter 20 
 
 Fractions, Clearing of 9 
 
 Fulcrum 51 
 
 Gears, Size and R.P.M. of 103 
 
 Hypotenuse 128 
 
 Law of Exponents for Divis- 
 ion 68 
 
 Law of Exponents for Multi- 
 plication 54 
 
 Law of Leverages 57 
 
 Law of Signs for Division 68 
 
 Law of Signs for Multiplica- 
 tion 53 
 
 Lineshaft 99 
 
 Lever 51 
 
 Leverage 51 
 
 Means of a Proportion.... 86 
 
 Monomial, Definition 44 
 
 Multiplication 51 
 
132 
 
 INDEX 
 
 SUBJECT PAGE 
 
 Multiplication of a Poly- 
 nomial by a Monomial GO 
 
 Multiplication of a Poly- 
 nomial by a Polynomial— .62, 63 
 Multiplication of Monomials 
 
 : 54, 59 
 
 Multiplication Law of Expon- 
 ents for 54 
 
 Multiplication Law of Signs 
 
 for 53 
 
 Multiplication Sign 15 
 
 Negative Numbers 40 
 
 Members, Definite 15 
 
 Members, General 15 
 
 Numbers, Definite 15 
 
 Numbers, General 15 
 
 Numbers, Positive and Nega- 
 tive 38, 39, 40 
 
 Numbers, Signed 40 
 
 Order of Terms 6 
 
 Parenthesis 16 
 
 Parenthesis, Kemoval of 49 
 
 Percentage 81 
 
 Perigon 25 
 
 Perimeter, Definition 19 
 
 Perimeter, Formulas 20 
 
 Perimeters, Equations involv- 
 ing 21 
 
 Polynomial, Definition 44 
 
 Polynomials, Addition of 44 
 
 Positive Numbers 40 
 
 Power 16 
 
 Proportion, Definition 86 
 
 Proportion, Direct 91 
 
 Proportion, Extremes of 96 
 
 Proportion, Inverse 92, 93 
 
 Proportion, Means of 86 
 
 Protractor 26 
 
 Pulleys, R.P.M. and Size of.... 99 
 
 Pulleys, Step, Cone 101 
 
 Quadratic Equation, Defini- 
 tion 120 
 
 Quadratic Equations, Solu- 
 tion of 121, 122 
 
 Ratio, Definition 79 
 
 Ration, Separating in a given 84 
 
 SUBJECT PAGE 
 
 Ratio, Terms of 79 
 
 Ratios, To express as Deci- 
 mals 81 
 
 Right Triangle, Definition 128 
 
 Right Triangle, Formula 129 
 
 Right Triangle, Hypotenuse of 128 
 
 Right Triangle, Sides of 128 
 
 Rim Speed 96 
 
 Separating in a Given Ratio.. 84 
 
 Sign 6f Multiplication 15 
 
 Signed Numbers 40 
 
 Signs, Law of Signs for Divis- 
 ion 68 
 
 Signs, Law of Signs for Mul- 
 tiplication 53 
 
 Signs of Grouping 16, 49 
 
 Similar Terms 5 
 
 Similar Terms, Combination 
 
 of 43 
 
 Singular Terms, Definition 43 
 
 Specific Gravity 83 
 
 Speed i 96 
 
 Speed, Cutting 97 
 
 Speed, Rim or Surface 96 
 
 Bpeed Rule 96 
 
 Square of a Binomial 107 
 
 Square Root, Definition 109 
 
 Square Root of a Negative 
 
 No 109 
 
 Square Root of Fractions 119 
 
 Square Root of Monomials 109 
 
 Square Root of Numbers.. 112, 113 
 Square Root of Numbers not 
 
 Perfect Squares ....117, 118 
 
 Square Root Trinomials 110 
 
 Square, Trinomial 110 
 
 Subtraction, Algebraic, Defini- 
 tion 45 
 
 Subtraction, Algebraic, Rule.. 46 
 Supplement 33 
 
 Terms, Definition 43 
 
 Terms, of Ratios 79 
 
 Terms, Order of 6 
 
 Trinomial, Definition 44 
 
 Trinomial Square 110 
 
 Trinomials, Square Root of.... Ill 
 
 Variation 91 
 
 Vinculium 49 
 
iv;270571 
 
 0-/A 3 9 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY