Elements of Algebra

CHAP. X. OF ALGEBRA. 33 we must have recourse to the method already mentioned for reducing them to a common denominator. Let there be, for example, the fraction $\frac{a}{b}$ to be divided by the fraction $\frac{c}{d}$. We first reduce them to the same denominator, and there results $\frac{ad}{bd}$ to be divided by $\frac{cb}{db}$; it is now evident that the quotient must be represented simply by the division of $ad$ by $bc$; which gives $\frac{ad}{bc}$. Hence the following rule: Multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor; then the first product will be the numerator of the quotient, and the second will be its denominator. 109. Applying this rule to the division of $\frac{1}{2}$ by $\frac{2}{3}$, we shall have the quotient $\frac{3}{4}$: also the division of $\frac{3}{4}$ by $\frac{1}{2}$ will give $\frac{6}{4}$, or $\frac{3}{2}$, or $1\frac{1}{2}$; and $\frac{25}{48}$ by $\frac{5}{6}$ will give $\frac{150}{240}$, or $\frac{5}{8}$. 110. This rule for division is often expressed in a manner that is more easily remembered, as follows: Invert the terms of the divisor, so that the denominator may be in the place of the numerator, and the latter be written under the line; then multiply the fraction, which is the dividend by this inverted fraction, and the product will be the quotient sought. Thus, $\frac{3}{4}$ divided by $\frac{1}{2}$ is the same as multiplied by $\frac{2}{1}$, which makes $\frac{6}{4}$, or $1\frac{1}{2}$. Also $\frac{5}{8}$ divided by $\frac{2}{3}$ is the same as $\frac{5}{8}$ multiplied by $\frac{3}{2}$, which is $\frac{15}{16}$; or $\frac{25}{48}$ divided by $\frac{5}{6}$ gives the same as $\frac{25}{48}$ multiplied by $\frac{6}{5}$, the product of which is $\frac{150}{240}$, or $\frac{5}{8}$. We see then, in general, that to divide by the fraction $\frac{1}{2}$ is the same as to multiply by $\frac{2}{1}$, or $2$; and that dividing by $\frac{1}{3}$ amounts to multiplying by $\frac{3}{1}$, or by $3$, \ldots 111. The number $100$ divided by $\frac{1}{2}$ will give $200$; and $1000$ divided by $\frac{1}{3}$ will give $3000$. Farther, if it were required to divide $1$ by $\frac{1}{1000}$, the quotient would be $1000$; and dividing $1$ by $\frac{1}{100000}$, the quotient is $100000$. This enables us to conceive that, when any number is divided by $0$, the result must be a number indefinitely great; for even the division of $1$ by the small fraction $\frac{1}{1000000000}$ gives for the quotient the very great number $1000000000$. 112. Every number, when divided by itself, producing unity, it is evident that a fraction divided by itself must also give $1$ for the quotient; and the same follows from our rule: for, in order to divide $\frac{3}{4}$ by $\frac{3}{4}$, we must multiply $\frac{3}{4}$ by $\frac{4}{3}$, in D