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Full text of "Eighth grade mathematics"



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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 



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