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=b. In like manner, the result being the same, b whether the numerator be multiplied by a whole quantity, or the denominator divided by it, the latter method is to be preferred, when the denominator is some multiple of the multiplier : Thus, let ad

ad
be the fraction, and c the multiplier; then
adc ad ad ad ad
Xc=

as before. bc b be bcC Also, when the numerator of one of the fractions to be multiplied, and the denominator of the other, can be divided by some quantity which is common to each of them, the quotients may be used instead

a+b of the fractions themselves; thus, х

ab a+b ; cancelling a+b in the numerator of the one, -b and denominator of the other.

3а 4a Ex. 1. Multiply by

5

7 3a X 4a=120o = numerator, ?

.. the fraciion re5X7=35= denominator;)

12a quired is

35

33+2 8x Ex. 2. Muliply by

4 Here, (3x+2) X 8x=24x2 +16x=numerator,

and 4X7=28= denominator;

2472 +162 Therefore,

(dividing the numera28

6x2 +4. tor and denominator by 1)

the product re

quired.

O

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a? -22 7x2 Ex. 3. Multiply by

3a Here, (a2-x2) x 7x2=(a+x) x (a-) x 722 numerator (Art. 106), and 3a X(a~-~= denominator; see Ex. 15, (Art. 79).

(a+x) x(am) X 722 Hence, the product is

3a X(a-2) (dividing the numerator and denominator by amx) 72co (a +*)_7axa +7203 3a

За
Ex. 4. Multiply at: by a

5

3 5a+ Here, at = 5 5

3 3 Then, (5a +x) (30—2)=15a 2ax-**= new numerator, and 5 X3=15= denominator : There15a2 --2adc

2artfore,

is the product 15

15 required.

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157. But, when mixed quantities are to be multiplied together, it is sometimes more convenient to proceed, as in the multiplication of integral quantities, without reducing them to improper fractions. Ex. 5. Multiply xo - 2+} by x + 2.

ao -c+
*+2

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by

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by

3x2 -- 5x

7a Ex. 6. Multiply

14
3x3 - 3x

Зах — 5а
Ans.

402_-6 3a2 151 - 30 Ex. 7. Multiply by

5x-10

22

9x Ans.

2 2a - 2x

3ax Ex. 8. Multiply 3ab 5a - 5%

200 Ans.

569 Es. 9. It is required to find the continual pro

3a 202 duct of

and 5? 3

2ax + 2ab Ans.

5 Ex. 10. It is required to find the continued proa4

-X4 duct of

and a-ya? a? to?

Ans. at. Ex. 11. It is required to find the continued pro

-02 a262 duct of

and a+b

at

aa-ab Ans.

a+b

ах

aty

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a-X

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2

Ex. 12. Multiply 02-**+1 by 2-X.

Ans. 24-273442-X. To divide one fractional quantity by another

RULE.

a

158. Multiply the dividend by the reciprocal of the divisor, or which is the same, invert the divisor, and proceed, in every respect, as in multiplication of algebraic fractions; and the product thus found will be the quotient required.

When a fraction is to be divided by an integral quantity; the process is the reverse of that in multiplication; or, which is the same, multiply the denominator by the integral, (Art. 120), or divide the numerator by it. The latter mode is to be preferred, when the numerator is a multiple of the divisor.

a

с

hence

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58 6 Ex. 1. Divide by b

52 The divisor - inverted, becomes hence X

b

a 5cx

is the fraction required. ab

3a-3x 50-500 Ex. 2. Divide

by
at-6 a+b

50-5x The divisor

inverted, become's

a+b atb 30-32

3a-3x
;
hence

х

atb 5a-5.0 - 5 3(2-->)_3

is the quotient required. 5a-x) 5

a+

ja-52;

a'_-62 Ex. 3. Divide

by a+b.

is the quotient

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C

a

atb

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1 The reciprocal of the divisor is

; hence

a+b
22 -62

1 (a+b)(ab)_a-6
X
a+b

x +(a+b)
required.
a? -62

-b
Or =a-b; hence is the fraction
required.
Ex. 4. Divide

x2 +aa 23-a?

by at
atc

22 -a?
Here, at
ao +?-a?

; then, the X2

c2 fraction

22 -aa becomes

х atc

a-tec -a3

the quotient required. x2 axa +cx?

a

2

a

a

a

a2

divided by

a

a

axa

2

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159. But it is, however, frequently more simple in practice to divide mixed quantities by one another, without reducing them to improper fractions, as in division of integral quantities, especially when the division would terminate. Ex. 5. Divide x4--23+1 by x-x. 22 - x)x4 -83 +— 3x(p? -- .+1

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