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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
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
7 3a X 4a=120o = numerator, ?
.. the fraciion re5X7=35= denominator;)
12a quired is
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
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
3 5a+ Here, at = 5 5
3 3 Then, (5a +x) (30—2)=15a – 2ax-**= new numerator, and 5 X3=15= denominator : There15a2 --2adc
is the product 15
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.
3x2 -- 5x
7a Ex. 6. Multiply
Зах — 5а
402_-6 3a2 151 - 30 Ex. 7. Multiply by
2 2a - 2x
3ax Ex. 8. Multiply 3ab 5a - 5%
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
Ex. 12. Multiply 02-**+1 by 2-X.
Ans. 24-273442-X. To divide one fractional quantity by another
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.
58 6 Ex. 1. Divide by b
52 The divisor - inverted, becomes hence X
is the fraction required. ab
3a-3x 50-500 Ex. 2. Divide
50-5x The divisor
a+b atb 30-32
atb 5a-5.0 5а - 5 3(2-->)_3
is the quotient required. 5a-x) 5
a'_-62 Ex. 3. Divide
is the quotient
1 The reciprocal of the divisor is
x2 +aa 23-a?
; then, the X2
22 -aa becomes
the quotient required. x2 axa +cx?
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