Chapter A.3, Problem 64E

Solution

Simplify the compound fraction.

  $\frac{xy}{(y-x)}$ .

Given:

The given compound fraction is $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}}$ .

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 $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}}$ .

  $\begin{array}{c}\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}}=\frac{\frac{y+x}{xy}}{\frac{y^2-x^2}{x^2{y}^2}}\\ =\frac{(y+x)x^2{y}^2}{(y^2-x^2)xy}\\ =\frac{(y+x)x^2{y}^2}{(y+x)(y-x)xy}\\ =\frac{x^2{y}^2}{(y-x)xy}\\ =\frac{xy}{(y-x)}\end{array}$

Thus by simplifying the compound fraction $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}}$ the solution is $\frac{xy}{(y-x)}$ .