(58.) DIVISION OF FRACTIONS. You have already seen, that when you multiply by a proper fraction, the product is always less than the multiplicand; you will be prepared to expect, therefore, that when you divide by a proper fraction, the quotient will always be greater than the dividend. If your divisor is a whole unit, or 1, the quotient is only equal to the dividend; so that when your divisor is less than 1, that is, a proper fraction, the quotient must be greater than the dividend, for the less your divisor the greater the quotient, and if one divisor be a third part, a fourth part, &c., of another, the quotient due to that part must be three times, four times, &c., the quotient due to that other. All this is plain: suppose you have to divide by, that is, the fourth part of 3, the quotient must be four times the quotient you would get by dividing by 3 itself; so that you would get the true quotient by first dividing by 3, and then multiplying what you would get, by 4. Division by a fraction, therefore, would thus imply both division and multiplication, for what is here said of the divisor, you must see would equally apply to any other fractional divisor: the true quotient would always be got by dividing the dividend by the numerator of the divisor, and then multiplying the result by the denominator. For instance, if you had to divide by, you would first divide the by 5; this would give you; for 2, first divided by 3, and the result by 5, is, in fact, 2 divided by 15: but the true quotient must be 7 times the quotient 1, because your true divisor, namely,, is only a seventh part of the divisor 5; therefore, the true quotient is 2×7, or 1: you thus see, that the quotient of one fraction by another, is got by multiplying the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor; or, which is the same thing, the quotient is got by turning the divisor upside down, and then proceeding as if the operation were multiplication instead of division. Hence the rule. (59.) RULE. Invert the divisor, or make numerator and denominator change places, and then proceed as if it were multiplication. Mixed numbers are, of course, to be reduced to improper fractions, as before. NOTE. Before inverting the divisor, see whether the numerators of both fractions have a common factor; if they have, expunge it: see if the denominators have a common factor; if they have, expunge it. Remember to do this before applying the rule. Ex. 1. Divide by Here ÷ ÷ 3 = §×÷= 11, the factor 6, common to both denominators, being cancelled before applying the rule, because if allowed to remain, it would enter numerator and denominator of the result. 3. Divide of 81 by of. 22 First, 2×33 = 21. 22 166 X 13: 65 Second, x= }. Then, ÷ 2x3 = = 77. = 63 In inverting the divisor here, the denominator 1 is omitted. as useless. If the divisor had been 3 instead of, the divisor inverted would have been, since 3 is; and would have been called the reciprocal of 3; the reciprocal of any number being 1 divided by that number: the reciprocal of a fraction is merely that fraction inverted; thus, 1÷2 = 1 × 3 = { ; 1÷÷=1× 3 = 4, and so on; so that the rule for division of fractions may be stated thus: multiply the dividend by the reciprocal of the divisor, and you will get the quotient. NOTE. A fraction like either of the fractions in Ex. 12 is 1 called a complex fraction; the word complex implying that one of the terms of the fraction, at least, does itself contain a fraction. Each of the preceding exercises might have been written as a complex fraction, so as to dispense with the sign; and this way of indicating division is often the more convenient, as will be seen in the next article: before proceeding to which, however, it may be well to state the two following obvious inferences from the rules for multiplication and division in a distinct form, on account of the frequent application of them. 1. When a fraction is to be multiplied by a whole number, the correct product is obtained, whether we multiply the numerator by the number and leave the denominator untouched, or divide the denominator and leave the numerator untouched; the latter way, when the denominator is divisible by the number, is generally to be preferred. If the multiplier be equal to the denominator, the product is simply the nu merator. 2. When a fraction is to be divided by a whole number, the correct quotient is obtained whether we divide the numerator by the number, leaving the denominator untouched, or multiply the denominator and leave the numerator untouched; the former when practicable is generally to be preferred. (60.) To find what Fraction one Quantity is of another Quantity. The fraction that one quantity is of another is expressed by writing the former as numerator, and the latter as denominator; both quantities, if concrete, being first reduced to the same denomination. Ex. 1. What fraction of is? ៖ Here we have only to divide by ; therefore &=}; so that is of 3. 2. What fraction of 123 is 7? 7 35 the fraction required. 123 63' 3. What fraction of £1 is 7s. 8d.? Reducing these to the same denomination, we have 78. 20s. 92 23 that is, 78. 8d. is the 8ths of £1. If 240 60 8 had been replaced by, the result would have come out in the lowest terms at once. 4. What fraction of 8s. 4d. is 3s. 9d.? As 4d.s., and 9d.s., we have 3÷83-÷Y=} ; so that 3s. 9d. is ths of 88. 4d. Exercises. 1. What fraction of 5s. is 1s. 10d.? 2. What fraction of 2s. 6d. is 8d.? 3. Reduce 38. 11d. to the fraction of a guinea. 6. What fraction of a ton is 3 cwt. 1 qr.? 7. Reduce of 10 minutes to the fraction of an hour. 8. What fraction of 23 is of 3? 9. What fraction of £14 is 72s.? 10. What fraction of 90° is 12° 23'? 11. What fraction of a week is 2 d. 17 h. ? 15. Reduce 3 roo. 21 po. 3 yds. to the fraction of 11 ac. 2 roo. 6 po. (61.) Examples of Multiplication and Division of concrete Quantities. Though the rules given for multiplying and dividing by a fraction or mixed number apply generally, whatever the things multiplied or divided may be, yet as these rules have been actually applied hitherto only to abstract numbers, it may be proper to give a few examples in which concrete quantities are concerned. Ex. 1. Multiply 7s. 14d. by 115, or 127. S. d. 7 1×7 9)49 101/ 5 6 Or, since 78. 14d.=748.58., we have s. x 115=18. X 115=21858.=91s. £4 11s. 04d. The best way to multiply by 19, is to multiply by 20 and subtract the multiplicand. 3 24 = |