Multiplication and Division of Fractions.

a complex fraction is to multiply both terms of the fraction by the L. C. D. of the fractions contained in the numerator and denominator.

1. Simplify ${\displaystyle \frac{3x}{x-\frac{1}{4}}}$.

Multiply both terms by $4$, and we have $\frac{12x}{4x-1}.$

2. Simplify ${\displaystyle \frac{{\displaystyle \frac{a+x}{a-x}}-{\displaystyle \frac{a-x}{a+x}}}{{\displaystyle \frac{a+x}{a-x}}+{\displaystyle \frac{a-x}{a+x}}}}$.

The L. C. D. of the fractions in the numerator and denominator is $(a-x)(a+x).$

Multiply by $(a-x)(a+x)$, and the result is

$$\begin{gathered} \&\frac{(a+x{)}^2-(a-x{)}^2}{(a+x{)}^2+(a-x{)}^2} \\ \&=\frac{(a^2+2ax+x^2)-(a^2-2ax+x^2)}{(a^2+2ax+x^2)+(a^2-2ax+x^2)} \\ \&=\frac{a^2+2ax+x^2-a^2+2ax-x^2}{a^2+2ax+x^2+a^2-2ax+x^2} \\ \&=\frac{4ax}{2a^2+2x^2} \\ \&=\frac{2ax}{a^2+x^2}. \end{gathered}$$

Exercise 50.

Reduce to the simplest form:

1. ${\displaystyle \frac{{\displaystyle \frac{x}{b}}+{\displaystyle \frac{y}{b}}}{{\displaystyle \frac{z}{b}}}}$.