be bot ; and ab =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 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 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 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. ion and a 2 157. 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+ by by 3x2 -- 5x 7a Ex. 6. Multiply 14 Зах — 5а 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 a-X a 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 с 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 50-5x The divisor inverted, become's a+b atb 30-32 3a-3x х atb 5a-5.0 5а - 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 C a atb 1 The reciprocal of the divisor is ; hence a+b 1 (a+b)(a−b)_a-6 x +(a+b) -b x2 +aa 23-a? by at 22 -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 2 159. 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 IL |