Question

Perform the indicated operation and simplify.

$\frac{{\displaystyle \frac{1}{m}}-{\displaystyle \frac{1}{n}}}{{\displaystyle \frac{1}{n}}+{\displaystyle \frac{1}{m}}}$

Short answer

The solution is$\frac{n-m}{n+m}$.

Step-by-step solution

Step 1. Given information.

The given expression is$\frac{{\displaystyle \frac{1}{m}-\frac{1}{n}}}{{\displaystyle \frac{1}{n}+\frac{1}{m}}}$.

Step 2. Simplify the given expression.

$$\begin{gathered} \frac{{\displaystyle \frac{1}{m}-\frac{1}{n}}}{{\displaystyle \frac{1}{n}+\frac{1}{m}}}=\frac{{\displaystyle \frac{n-m}{mn}}}{{\displaystyle \frac{m+n}{mn}}} \\ \frac{{\displaystyle \frac{1}{m}-\frac{1}{n}}}{{\displaystyle \frac{1}{n}+\frac{1}{m}}}=\frac{n-m}{mn}\times \frac{mn}{n+m} \\ \frac{{\displaystyle \frac{1}{m}-\frac{1}{n}}}{{\displaystyle \frac{1}{n}+\frac{1}{m}}}=\frac{n-m}{n+m} \end{gathered}$$

Step 3. Simplified answer.

Hence, the simplified expression is$\frac{n-m}{n+m}$.