Elements of Algebra

CHAP. X. OF ALGEBRA. 31 product we must reduce to $ \frac{8}{3} $. But if we do not change the numerator, and divide the denominator by the integer, we find immediately $ \frac{8}{3} $, or $ 2\frac{2}{3} $, for the given product; and, in the same manner, $ \frac{13}{24} $ multiplied by 6 gives $ \frac{13}{4} $, or $ 3\frac{1}{4} $. 102. In general, therefore, the product of the multiplication of a fraction $ \frac{a}{b} $ by $ c $ is $ \frac{ac}{b} $; and here it may be remarked, when the integer is exactly equal to the denominator, that the product must be equal to the numerator. So that $ \frac{1}{2} $ taken twice, gives 1; $ \frac{2}{3} $ taken thrice, gives 2; $ \frac{3}{4} $ taken four times, gives 3. And, in general, if we multiply the fraction $ \frac{a}{b} $ by the number $ b $, the product must be $ a $, as we have already shewn; for since $ \frac{a}{b} $ expresses the quotient resulting from the division of the dividend $ a $ by the divisor $ b $, and because it has been demonstrated that the quotient multiplied by the divisor will give the dividend, it is evident that $ \frac{a}{b} $ multiplied by $ b $ must produce $ a $. 103. Having thus shewn how a fraction is to be multiplied by an integer; let us now consider also how a fraction is to be divided by an integer. This inquiry is necessary, before we proceed to the multiplication of fractions by fractions. It is evident, if we have to divide the fraction $ \frac{2}{3} $ by 2, that the result must be $ \frac{1}{3} $; and that the quotient of $ \frac{6}{7} $ divided by 3 is $ \frac{2}{7} $. The rule therefore is, to divide the numerator by the integer without changing the denominator. Thus: $ \frac{12}{25} $ divided by 2 gives $ \frac{6}{25} $; $ \frac{12}{25} $ divided by 3 gives $ \frac{4}{25} $; and $ \frac{12}{25} $ divided by 4 gives $ \frac{3}{25} $; &c. 104. This rule may be easily practised, provided the numerator be divisible by the number proposed; but very often it is not: it must therefore be observed, that a fraction may be transformed into an infinite number of other expressions, and in that number there must be some, by which the numerator might be divided by the given integer. If it were required, for example, to divide $ \frac{3}{4} $ by 2, we should change the fraction into $ \frac{6}{8} $, and then dividing the numerator by 2, we should immediately have $ \frac{3}{8} $ for the quotient sought.