Chapter A.3, Problem 66E

Solution

Simplify the compound fraction.

  $\frac{x-3}{x+5}$ .

Given:

The given compound fraction is $\frac{2-\frac{13}{x+5}}{2+\frac{3}{x-3}}$ .

Concept Used:

• Compound fraction is a fraction with fractions in the numerator or denominator.
• First multiply the denominator of the fraction by the whole number. 
• Then, add that number to the numerator of the fraction, and keep the original denominator the same.

Calculation:

The given compound fraction is $\frac{2-\frac{13}{x+5}}{2+\frac{3}{x-3}}$ .

  $\begin{array}{c}\frac{2-\frac{13}{x+5}}{2+\frac{3}{x-3}}=\frac{\frac{2(x+5)-13}{x+5}}{\frac{2(x-3)+3}{x-3}}\\ =\frac{\frac{2x+10-13}{x+5}}{\frac{2x-6+3}{x-3}}\\ =\frac{\frac{2x-3}{x+5}}{\frac{2x-3}{x-3}}\\ =\frac{(2x-3)(x-3)}{(x+5)(2x-3)}\\ =\frac{x-3}{x+5}\end{array}$

Thus by simplifying the compound fraction $\frac{2-\frac{13}{x+5}}{2+\frac{3}{x-3}}$ the solution is $\frac{x-3}{x+5}$ .