Solve (1/m-n-1/m+n):2/3m-3n=

$$( \frac { 1 } { m - n } - \frac { 1 } { m + n } ) : \frac { 2 } { 3 m - 3 n } =$$

Evaluate

$$\frac{3n}{m+n}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m-n and m+n is \left(m+n\right)\left(m-n\right). Multiply \frac{1}{m-n} times \frac{m+n}{m+n}. Multiply \frac{1}{m+n} times \frac{m-n}{m-n}.

$$\frac{\frac{m+n}{\left(m+n\right)\left(m-n\right)}-\frac{m-n}{\left(m+n\right)\left(m-n\right)}}{\frac{2}{3m-3n}}$$

Since \frac{m+n}{\left(m+n\right)\left(m-n\right)} and \frac{m-n}{\left(m+n\right)\left(m-n\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{m+n-\left(m-n\right)}{\left(m+n\right)\left(m-n\right)}}{\frac{2}{3m-3n}}$$

Do the multiplications in m+n-\left(m-n\right).

$$\frac{\frac{m+n-m+n}{\left(m+n\right)\left(m-n\right)}}{\frac{2}{3m-3n}}$$

Combine like terms in m+n-m+n.

$$\frac{\frac{2n}{\left(m+n\right)\left(m-n\right)}}{\frac{2}{3m-3n}}$$

Divide \frac{2n}{\left(m+n\right)\left(m-n\right)} by \frac{2}{3m-3n} by multiplying \frac{2n}{\left(m+n\right)\left(m-n\right)} by the reciprocal of \frac{2}{3m-3n}.

$$\frac{2n\left(3m-3n\right)}{\left(m+n\right)\left(m-n\right)\times 2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{n\left(3m-3n\right)}{\left(m+n\right)\left(m-n\right)}$$

Factor the expressions that are not already factored.

$$\frac{3n\left(m-n\right)}{\left(m+n\right)\left(m-n\right)}$$

Cancel out m-n in both numerator and denominator.

$$\frac{3n}{m+n}$$

Expand

$$\frac{3n}{m+n}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m-n and m+n is \left(m+n\right)\left(m-n\right). Multiply \frac{1}{m-n} times \frac{m+n}{m+n}. Multiply \frac{1}{m+n} times \frac{m-n}{m-n}.

$$\frac{\frac{m+n}{\left(m+n\right)\left(m-n\right)}-\frac{m-n}{\left(m+n\right)\left(m-n\right)}}{\frac{2}{3m-3n}}$$

Since \frac{m+n}{\left(m+n\right)\left(m-n\right)} and \frac{m-n}{\left(m+n\right)\left(m-n\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{m+n-\left(m-n\right)}{\left(m+n\right)\left(m-n\right)}}{\frac{2}{3m-3n}}$$

Do the multiplications in m+n-\left(m-n\right).

$$\frac{\frac{m+n-m+n}{\left(m+n\right)\left(m-n\right)}}{\frac{2}{3m-3n}}$$

Combine like terms in m+n-m+n.

$$\frac{\frac{2n}{\left(m+n\right)\left(m-n\right)}}{\frac{2}{3m-3n}}$$

Divide \frac{2n}{\left(m+n\right)\left(m-n\right)} by \frac{2}{3m-3n} by multiplying \frac{2n}{\left(m+n\right)\left(m-n\right)} by the reciprocal of \frac{2}{3m-3n}.

$$\frac{2n\left(3m-3n\right)}{\left(m+n\right)\left(m-n\right)\times 2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{n\left(3m-3n\right)}{\left(m+n\right)\left(m-n\right)}$$

Factor the expressions that are not already factored.

$$\frac{3n\left(m-n\right)}{\left(m+n\right)\left(m-n\right)}$$

Cancel out m-n in both numerator and denominator.

$$\frac{3n}{m+n}$$