Elements of Algebra

30 ELEMENTS SECT. I. 2. Reduce $\frac{a}{b}$ and $\frac{a+b}{c}$ to a common denominator. Ans. $\frac{ac}{bc}$ and $\frac{ab+b^2}{bc}$. 3. Reduce $\frac{3x}{2a}$, $\frac{2b}{3c}$, and $d$ to fractions having a common denominator. Ans. $\frac{9cx}{6ac}$, $\frac{4ab}{6ac}$, and $\frac{6acd}{6ac}$. 4. Reduce $\frac{3}{4}$, $\frac{2x}{3}$, and $a+\frac{2x}{a}$ to a common denominator. Ans. $\frac{9a}{12a}$, $\frac{8ax}{12a}$, and $\frac{12a^2+24x}{12a}$. 5. Reduce $\frac{1}{2}$, $\frac{a^2}{3}$, and $\frac{x^2+a^2}{x+a}$ to a common denominator. Ans. $\frac{3x+3a}{6x+6a}$, $\frac{2a^2x+2a^3}{6x+6a}$, $\frac{6x^2+6a^2}{6x+6a}$. 6. Reduce $\frac{b}{2a^2}$, $\frac{c}{2a^3}$, and $\frac{d}{a}$ to a common denominator. Ans. $\frac{2a^2b}{4a^4}$, $\frac{2a^3c}{4a^4}$, and $\frac{4a^3d}{4a^4}$; or $\frac{b}{2a^2}$, $\frac{ac}{2a^2}$, and $\frac{2ad}{2a^2}$. CHAP. X. Of the Multiplication and Division of Fractions. 101. The rule for the multiplication of a fraction by an integer, or whole number, is to multiply the numerator only by the given number, and not to change the denominator: thus, 2 times, or twice $\frac{1}{2}$ makes $\frac{2}{2}$, or 1 integer; 2 times, or twice $\frac{1}{4}$ makes $\frac{2}{4}$; and 3 times, or thrice $\frac{1}{6}$ makes $\frac{3}{6}$, or $\frac{1}{2}$; 4 times $\frac{5}{12}$ makes $\frac{20}{12}$, or $1\frac{8}{12}$, or $1\frac{2}{3}$. But, instead of this rule, we may use that of dividing the denominator by the given integer, which is preferable, when it can be done, because it shortens the operation. Let it be required, for example, to multiply $\frac{8}{9}$ by 3; if we multiply the numerator by the given integer we obtain $\frac{24}{9}$, which