Solve m+n/12m+m-n/8n

$$\frac { m + n } { 12 m } + \frac { m - n } { 8 n }$$

Evaluate

$$\frac{3m^{2}-mn+2n^{2}}{24mn}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12m and 8n is 24mn. Multiply \frac{m+n}{12m} times \frac{2n}{2n}. Multiply \frac{m-n}{8n} times \frac{3m}{3m}.

$$\frac{\left(m+n\right)\times 2n}{24mn}+\frac{\left(m-n\right)\times 3m}{24mn}$$

Since \frac{\left(m+n\right)\times 2n}{24mn} and \frac{\left(m-n\right)\times 3m}{24mn} have the same denominator, add them by adding their numerators.

$$\frac{\left(m+n\right)\times 2n+\left(m-n\right)\times 3m}{24mn}$$

Do the multiplications in \left(m+n\right)\times 2n+\left(m-n\right)\times 3m.

$$\frac{2mn+2n^{2}+3m^{2}-3nm}{24mn}$$

Combine like terms in 2mn+2n^{2}+3m^{2}-3nm.

$$\frac{2n^{2}-mn+3m^{2}}{24mn}$$

Expand

$$\frac{3m^{2}-mn+2n^{2}}{24mn}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12m and 8n is 24mn. Multiply \frac{m+n}{12m} times \frac{2n}{2n}. Multiply \frac{m-n}{8n} times \frac{3m}{3m}.

$$\frac{\left(m+n\right)\times 2n}{24mn}+\frac{\left(m-n\right)\times 3m}{24mn}$$

Since \frac{\left(m+n\right)\times 2n}{24mn} and \frac{\left(m-n\right)\times 3m}{24mn} have the same denominator, add them by adding their numerators.

$$\frac{\left(m+n\right)\times 2n+\left(m-n\right)\times 3m}{24mn}$$

Do the multiplications in \left(m+n\right)\times 2n+\left(m-n\right)\times 3m.

$$\frac{2mn+2n^{2}+3m^{2}-3nm}{24mn}$$

Combine like terms in 2mn+2n^{2}+3m^{2}-3nm.

$$\frac{2n^{2}-mn+3m^{2}}{24mn}$$