Elements of Algebra

32 ELEMENTS SECT. I. In general, if it be proposed to divide the fraction $\frac{a}{b}$ by $c$, we change it into $\frac{ac}{bc}$, and then dividing the numerator $ac$ by $c$, write $\frac{a}{bc}$ for the quotient sought. 105. When therefore a fraction $\frac{a}{b}$ is to be divided by an integer $c$, we have only to multiply the denominator by that number, and leave the numerator as it is. Thus $\frac{3}{4}$ divided by $3$ gives $\frac{3}{12}$, and $\frac{5}{8}$ divided by $5$ gives $\frac{5}{40}$. This operation becomes easier, when the numerator itself is divisible by the integer, as we have supposed in article 103. For example, $\frac{9}{16}$ divided by $3$ would give, according to our last rule, $\frac{9}{48}$; but by the first rule, which is applicable here, we obtain $\frac{3}{16}$, an expression equivalent to $\frac{9}{48}$, but more simple. 106. We shall now be able to understand how one fraction $\frac{a}{b}$ may be multiplied by another fraction $\frac{c}{d}$. For this purpose, we have only to consider that $\frac{c}{d}$ means that $c$ is divided by $d$; and on this principle we shall first multiply the fraction $\frac{a}{b}$ by $c$, which produces the result $\frac{ac}{b}$; after which we shall divide by $d$, which gives $\frac{ac}{bd}$. Hence the following rule for multiplying fractions. Multiply the numerators together for a numerator, and the denominators together for a denominator. Thus $\frac{1}{2}$ by $\frac{2}{3}$ gives the product $\frac{2}{6}$, or $\frac{1}{3}$; $\frac{2}{3}$ by $\frac{4}{5}$ makes $\frac{8}{15}$; $\frac{3}{4}$ by $\frac{5}{12}$ produces $\frac{15}{48}$, or $\frac{5}{16}$; &c. 107. It now remains to shew how one fraction may be divided by another. Here we remark first, that if the two fractions have the same number for a denominator, the division takes place only with respect to the numerators; for it is evident, that $\frac{3}{12}$ are contained as many times in $\frac{9}{12}$ as $3$ is contained in $9$, that is to say, three times; and, in the same manner, in order to divide $\frac{8}{12}$ by $\frac{9}{12}$, we have only to divide $8$ by $9$, which gives $\frac{8}{9}$. We shall also have $\frac{18}{100}$ in $\frac{20}{100}$, 3 times; $\frac{7}{100}$ in $\frac{49}{100}$, 7 times; $\frac{6}{25}$ in $\frac{7}{25}$, &c. 108. But when the fractions have not equal denominators,