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8. Reduce

m

n

a

and to equivalent fractions having the

X least common denominator.

9a 76 11a 7(a+8) 9. Reduce

and 8m 36m 28m'

to equivalent frac

4m tions having the least common denominator. 2 3

2x-3 10. Reduce

and x 2x-1

to equivalent fractions

4x2_1 having the least common denominator.

Addition of Fractions. 117. The denominator of a fraction shows into how many parts a unit is to be divided, and the numerator shows how many of those parts are to be taken. Fractions can only be added when they are like parts of unity; that is, when they have a common denominator. In that case, the numerator of each fraction will indicate how many times the common fractional unit is repeated in that fraction, and the sum of the numerators will indicate how many times this result is repeated in the sum of the fractions. Hence we have the following

RULE. Reduce the fractions to a common denominator; then add the numerators together, and write their sum over the common denominator.

If there are mixed quantities, we may add the entire and fractional parts separately.

EXAMPLES.

1. What is the sum of and ? Reducing to a common denominator, the fractions become

3x

2ac
and
6 6.

5x Adding the numerators, we obtain

6 It is plain that three sixths of x and two sixths of x make five sixths of x.

a

с

and ?

a

and a +227

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2. What is the sum of

to a

n

adn+bon + bdm Ans.

bdn 3. What is the sum of and

.? ato

6

2a 4. What is the sum of 5x, 3x2

4x

2ąc 5. What is the sum of 2a, 3a+ 7

,

and at
a+ ?
9

583 Ans. 6a+

45 6. What is the sum of a+a, ama

x,

and a—?
a-

X2
Ans. a+x+2+

a' - axi

a7. What is the sum of

and a=b;
?

Ans. a. 2

2 a a-2m

+ 8. What is the sum of

2 4

4

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A-X

a

a+b

and a+2m

?

ma-6

na+b?

7y2 +3?

9. What is the sum of and

mtn mtn

X-n •10. What is the sum of

Y->

2z+n and

? a+y+z' x+y+z +y+2

3y2-2 11. What is the sum of

and 772-5

4y2-1

13a--296 7-21a 12. What is the sum of

and 5a-3) 5(ab) 5(a0)

Ans. 9. 1+« 1-2 1-x+22 1+x+x2 13. What is the sum of

1-'17 1+x2 1-X and -1?

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Subtraction of Fractions. 118. Fractions can only be subtracted when they are like parts of unity; that is, when they have a common denominator. In that case, the difference of the numerators will indicate how many times the common fractional unit is repeated in the difference of the fractions. Hence we have the following RULE. Reduce the fractions to a common denominator; then subtract the numerator of the subtrahend from the numerator of the minuend, and write the result over the common denominator.

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EXAMPLES. 2.

Зх 1. From subtract 3

5. Reducing to a common denominator, the fractions become

10. 9.3

and
15 15

10x 9cc Hence we have

15 15-15 and it is plain that ten fifteenths of x, diminished by nine fifteenths of x, equals one fifteenth of x. 12

3x 2. From subtract 7

50 9a-4x

Ба— Зах 3. From

subtract 7

3 It must be remembered that a minus sign before the dividing line of a fraction affects the quotient (Art. 111); and since a quantity is subtracted by changing its sign, the result of the subtraction in this case is

9a-4x 5a - 3x

t 7

3

; which fractions may be reduced to a common denominator, and the like terms united as in addition.

2acx 4. From subtract

Ans. 0+ 0

622+73

50-6

3552-6 5. From 2+ subtract

Ans. 8

21

168

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ax

AX

b-c

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13a-56
7a-21

25a-116 8. From subtract

Ans.
4
6

12
6a-76

ба

64ab-15a__6372 9. From subtract

Ans.
3a-26
969

27ab-1862 10. From

(a+b)

subtract unity. 4ab

2x 11. From subtract lly

7x— Бу

ax 12. From subtract a

a2-22

3x-87

a

ac

Multiplication of Fractions. 119. Let it be required to multiply s by a

First let us multiply s by c. According to the first principle of Art. 109, the product must be

. But the proposed multiplier was ä; that is, we have used a multiplier d times too great. We must therefore divide the result by d; and, according to the second principle of Art. 109, we obtain bd; that is, 6 xarici

bd. Hence we have the following

ас

a с

Х b

RULE.

Multiply the numerators together for a new numerator, and the denominators for a new denominator.

Entire and mixed quantities should first be reduced to fractional forms. Also, if there are any factors common to the numerator and denominator of the product, they should be canceled.

EXAMPLES. 2.c

22 1. Multiply by

Ans.
6
99

27 xta 2. Multiply by

a+7

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812

bc

ab

X 4x

21

X-1
a +6°

a

mnx

bac
3. Multiply 6+ å by

a
x2 — 62 32+32

24-34 4. Multiply by

Ans. b+co

72c+bc2 a> +62

a> +62 5. Multiply

Ans.
by
a_62
a+b

(a+b)2

10x 6. Multiply together

and 2' 5'

2x 3ab Зас 7. Multiply together

and a

26 +1

aces 8. Multiply together x,

and

Ans.

a? +abi 4ax 9. Multiply

by Oman. (18m2nx) (81many) 10. Multiply

by a2b2c4. (a2b3c") (a+b3c") (a?b3c4)

5a3712 14aRm 5n1lm6 6am 11. Multiply together

and 7man 25n11' 6a15 03n

2mon Ans.

a-b3 13(a-b) 5(x-y) 21(m,n) 12. Multiply together

and 7(m-n' 39(a)'

3dn Bbm 5mn 11abc 13. Multiply +

by
4ac 7 6bc 13dmn

3ax a2-02 bc+bx 14. Multiply together

and 4by' c? — x2' a2 + ax'

x

3x Ans.

4y 1-22 1-ya 15. Multiply together

and 1+ 1+y' +2'

1-X

Ans. 16. Multiply

a(a + x)
(a—x)

by
a + 2ax + x2 a-2ax + x2
at-34

a+62 by

Ans.
a2 + ab
D

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55 (36-Y

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1+

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C

17. Multiply až – 2ab +62

a-6

al

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