Chapter A.3, Problem 69E

Solution

Simplify the compound fraction.

  $(b+a)$ .

Given:

The given compound fraction is $\frac{\frac{b}{a}-\frac{a}{b}}{\frac{1}{a}-\frac{1}{b}}$ .

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 $\frac{\frac{b}{a}-\frac{a}{b}}{\frac{1}{a}-\frac{1}{b}}$ .

  $\begin{array}{c}\frac{\frac{b}{a}-\frac{a}{b}}{\frac{1}{a}-\frac{1}{b}}=\frac{\frac{b\cdot b-a\cdot a}{ab}}{\frac{b-a}{ab}}\\ =\frac{\frac{b^2-a^2}{ab}}{\frac{b-a}{ab}}\\ =\frac{\frac{(b+a)(b-a)}{ab}}{\frac{b-a}{ab}}\\ =\frac{(b+a)(b-a)ab}{ab(b-a)}\\ =\frac{ab(b+a)(b-a)}{ab(b-a)}\\ =\frac{(b+a)}{1}\\ =(b+a)\end{array}$

Thus by simplifying the compound fraction $\frac{\frac{b}{a}-\frac{a}{b}}{\frac{1}{a}-\frac{1}{b}}$ the solution is $(b+a)$ .