But if they are reduced to a common denominator, the numerators may be subtracted. 3 ac 2 e Ans. 20c* sign -was changed to +. See Art. VI, example 6th XIX. Division of whole numbers by Fractions, and Fractions by Fractions. How many times is contained in 7? Ans. is contained in 7, 35 times, and is contained as many times; that is, 35 or 11 times. 2. How many times is contained in a? Ans. is contained in a, 8 a times, and § is contained as many times; that is, a. 3. How many times is contained in c? is contained b c times in c, and is contained 1 b a bc as many times; that is, Hence, to divide a whole number by a fraction, multiply it by the denominator of the fraction, and divide the product by the numera tor. How many times is contained in Solution. Reducing them to a common denomi^ator, is 1, and is 3. is contained in 33 as many times as 2-4 is contained in 35; that is, or 111. Ans. 1. Hence, to divide a fraction by a fraction, multiply the numerator of the dividend by the denominator of the divisor, and the denomi nator of the dividend by the numerator of the divisor. Or more generally, when the divisor is a fraction, multiply the dividend (whether whole number or fraction) by the divisor inverted. Arith. Arts. XXIII. and XXIV. XX. Sometimes division may actually be performed when both divisor and dividend are compound quantities. Since division is the reverse of multiplication, the proper method to discover how to perform it, is to observe how a product is formed by multiplication. Multiply 2a3b-3 a2 b2 c + a b3 c2 by 4 a2 b2+2abc. 8 ao b3 — 12 aˆ b* c + 4 a3 b3 c2 + 4 a* b2 c — 6 a3 b3 c2 + 2 a2 bˆc3. Observe that each term of the multiplier is multiplied separately into each term of the multiplicand. The product therefore must consist of a number of terms equal to the product of the number of terms in the multiplicand by the number of terms in the multiplier. If the product be divided by the multiplicand, the multiplier must be reproduced, and if by the multiplier, the multiplicand must be reproduced. The three terms 8 a' b3- 12 a b c +4 a b c of the product were produced by multiplying the three terms of the multiplicand by the first term of the multiplier, 4 a2 b2. Therefore, if these three terins be divided by 4 a' b2, the quotient will be the multiplicand. Again, the three terms 4 a' b3 c-6 a3 b3c2 + 2.a2 bac of the product were formed by multiplying each term of the multiplicand by 2 a b c. Therefore, if these three terms be divided by 2 abc, the quotient will be the multiplicand. |