Elements of Algebra

34 ELEMENTS SECT. I. which case we obtain $\frac{1 \times 2}{2 \times 1}$, or 1; and if it be required to divide $\frac{a}{b}$ by $\frac{a}{b}$, we multiply $\frac{a}{b}$ by $\frac{b}{a}$; where the product $\frac{ab}{ab}$ is also equal to 1. 113. We have still to explain an expression which is frequently used. It may be asked, for example, what is the half of $\frac{3}{4}$? This means, that we must multiply $\frac{3}{4}$ by $\frac{1}{2}$. So likewise, if the value of $\frac{2}{3}$ of $\frac{5}{8}$ were required, we should multiply $\frac{5}{8}$ by $\frac{2}{3}$, which produces $\frac{10}{24}$; and $\frac{3}{4}$ of $\frac{9}{16}$ is the same as $\frac{9}{16}$ multiplied by $\frac{3}{4}$, which produces $\frac{27}{64}$. 114. Lastly, we must here observe, with respect to the signs $+$ and $-$, the same rules that we before laid down for integers. Thus $+\frac{1}{2}$ multiplied by $-\frac{1}{3}$, makes $-\frac{1}{6}$; and $-\frac{2}{3}$ multiplied by $-\frac{4}{5}$, gives $+\frac{8}{15}$. Farther $-\frac{5}{8}$ divided by $+\frac{2}{3}$, gives $-\frac{15}{16}$; and $-\frac{3}{4}$ divided by $-\frac{3}{4}$, gives $+\frac{12}{12}$, or $+1$. QUESTIONS FOR PRACTICE. 1. Required the product of $\frac{x}{6}$ and $\frac{2x}{9}$. Ans. $\frac{x^2}{27}$. 2. Required the product of $\frac{x}{2}$, $\frac{4x}{5}$, and $\frac{10x}{21}$. Ans. $\frac{4x^3}{21}$. 3. Required the product of $\frac{x}{a}$ and $\frac{x+a}{a+c}$. Ans. $\frac{x^2+ax}{a^2+ac}$. 4. Required the product of $\frac{3x}{2}$ and $\frac{3a}{b}$. Ans. $\frac{9ax}{2b}$. 5. Required the product of $\frac{2x}{5}$ and $\frac{3x^2}{2a}$. Ans. $\frac{3x^3}{5a}$. 6. Required the product of $\frac{2x}{a}$, $\frac{3ab}{c}$, and $\frac{3ac}{2b}$. Ans. $9ax$. 7. Required the product of $b+\frac{bx}{a}$ and $\frac{a}{x}$. Ans. $\frac{ab+bx}{x}$. 8. Required the product of $\frac{x^2-b^2}{bc}$ and $\frac{x^2+b^2}{b+c}$. Ans. $\frac{x^4-b^4}{b^2c+bc^2}$.