Solve n/m-m/n/frac{n+m{m}}

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

Evaluate

$$\frac{n-m}{n}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m and n is mn. Multiply \frac{n}{m} times \frac{n}{n}. Multiply \frac{m}{n} times \frac{m}{m}.

$$\frac{\frac{nn}{mn}-\frac{mm}{mn}}{\frac{n+m}{m}}$$

Since \frac{nn}{mn} and \frac{mm}{mn} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{nn-mm}{mn}}{\frac{n+m}{m}}$$

Do the multiplications in nn-mm.

$$\frac{\frac{n^{2}-m^{2}}{mn}}{\frac{n+m}{m}}$$

Divide \frac{n^{2}-m^{2}}{mn} by \frac{n+m}{m} by multiplying \frac{n^{2}-m^{2}}{mn} by the reciprocal of \frac{n+m}{m}.

$$\frac{\left(n^{2}-m^{2}\right)m}{mn\left(n+m\right)}$$

Cancel out m in both numerator and denominator.

$$\frac{-m^{2}+n^{2}}{n\left(m+n\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(m+n\right)\left(-m+n\right)}{n\left(m+n\right)}$$

Cancel out m+n in both numerator and denominator.

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

Expand

$$\frac{n-m}{n}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m and n is mn. Multiply \frac{n}{m} times \frac{n}{n}. Multiply \frac{m}{n} times \frac{m}{m}.

$$\frac{\frac{nn}{mn}-\frac{mm}{mn}}{\frac{n+m}{m}}$$

Since \frac{nn}{mn} and \frac{mm}{mn} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{nn-mm}{mn}}{\frac{n+m}{m}}$$

Do the multiplications in nn-mm.

$$\frac{\frac{n^{2}-m^{2}}{mn}}{\frac{n+m}{m}}$$

Divide \frac{n^{2}-m^{2}}{mn} by \frac{n+m}{m} by multiplying \frac{n^{2}-m^{2}}{mn} by the reciprocal of \frac{n+m}{m}.

$$\frac{\left(n^{2}-m^{2}\right)m}{mn\left(n+m\right)}$$

Cancel out m in both numerator and denominator.

$$\frac{-m^{2}+n^{2}}{n\left(m+n\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(m+n\right)\left(-m+n\right)}{n\left(m+n\right)}$$

Cancel out m+n in both numerator and denominator.

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