EDGE Math 8

Example 2

Simplify $\frac{\frac{1}{2} + \frac{3}{x}}{\frac{2}{3} + \frac{5}{x}} .$

Solution

Simplify both the numerator and the denominator. The LCD of the fractions in the numerator is 2x while that in the denominator is 3x.

\frac{\frac{1}{2} + \frac{3}{x}}{\frac{2}{3} + \frac{5}{x}}

= \frac{\frac{1}{2} \cdot \frac{x}{x} + \frac{3}{x} \cdot \frac{2}{2}}{\frac{2}{3} \cdot \frac{x}{x} + \frac{5}{x} \cdot \frac{3}{3}}

= \frac{\frac{x}{2 x} + \frac{6}{2 x}}{\frac{2 x}{3 x} + \frac{15}{3 x}}

= \frac{\frac{x + 6}{2 x}}{\frac{2 x + 15}{3 x}}

= \frac{x + 6}{2 x} \div \frac{2 x + 15}{3 x}

= \frac{x + 6}{2 x} \cdot \frac{3 x}{2 x + 15}

= \frac{3 x (x + 6)}{2 x (2 x + 15)}

= \frac{3 (x + 6)}{2 (2 x + 15)}