Solve n+5-12/n+1/frac{n+9{n+1}-5/n}

$$\frac { n + 5 - \frac { 12 } { n + 1 } } { \frac { n + 9 } { n + 1 } - \frac { 5 } { n } }$$

Evaluate

$$\frac{n\left(n+7\right)}{n+5}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply n+5 times \frac{n+1}{n+1}.

$$\frac{\frac{\left(n+5\right)\left(n+1\right)}{n+1}-\frac{12}{n+1}}{\frac{n+9}{n+1}-\frac{5}{n}}$$

Since \frac{\left(n+5\right)\left(n+1\right)}{n+1} and \frac{12}{n+1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(n+5\right)\left(n+1\right)-12}{n+1}}{\frac{n+9}{n+1}-\frac{5}{n}}$$

Do the multiplications in \left(n+5\right)\left(n+1\right)-12.

$$\frac{\frac{n^{2}+n+5n+5-12}{n+1}}{\frac{n+9}{n+1}-\frac{5}{n}}$$

Combine like terms in n^{2}+n+5n+5-12.

$$\frac{\frac{n^{2}+6n-7}{n+1}}{\frac{n+9}{n+1}-\frac{5}{n}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of n+1 and n is n\left(n+1\right). Multiply \frac{n+9}{n+1} times \frac{n}{n}. Multiply \frac{5}{n} times \frac{n+1}{n+1}.

$$\frac{\frac{n^{2}+6n-7}{n+1}}{\frac{\left(n+9\right)n}{n\left(n+1\right)}-\frac{5\left(n+1\right)}{n\left(n+1\right)}}$$

Since \frac{\left(n+9\right)n}{n\left(n+1\right)} and \frac{5\left(n+1\right)}{n\left(n+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{n^{2}+6n-7}{n+1}}{\frac{\left(n+9\right)n-5\left(n+1\right)}{n\left(n+1\right)}}$$

Do the multiplications in \left(n+9\right)n-5\left(n+1\right).

$$\frac{\frac{n^{2}+6n-7}{n+1}}{\frac{n^{2}+9n-5n-5}{n\left(n+1\right)}}$$

Combine like terms in n^{2}+9n-5n-5.

$$\frac{\frac{n^{2}+6n-7}{n+1}}{\frac{n^{2}+4n-5}{n\left(n+1\right)}}$$

Divide \frac{n^{2}+6n-7}{n+1} by \frac{n^{2}+4n-5}{n\left(n+1\right)} by multiplying \frac{n^{2}+6n-7}{n+1} by the reciprocal of \frac{n^{2}+4n-5}{n\left(n+1\right)}.

$$\frac{\left(n^{2}+6n-7\right)n\left(n+1\right)}{\left(n+1\right)\left(n^{2}+4n-5\right)}$$

Cancel out n+1 in both numerator and denominator.

$$\frac{n\left(n^{2}+6n-7\right)}{n^{2}+4n-5}$$

Factor the expressions that are not already factored.

$$\frac{n\left(n-1\right)\left(n+7\right)}{\left(n-1\right)\left(n+5\right)}$$

Cancel out n-1 in both numerator and denominator.

$$\frac{n\left(n+7\right)}{n+5}$$

Expand the expression.

$$\frac{n^{2}+7n}{n+5}$$

Expand

$$\frac{n^{2}+7n}{n+5}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply n+5 times \frac{n+1}{n+1}.

$$\frac{\frac{\left(n+5\right)\left(n+1\right)}{n+1}-\frac{12}{n+1}}{\frac{n+9}{n+1}-\frac{5}{n}}$$

Since \frac{\left(n+5\right)\left(n+1\right)}{n+1} and \frac{12}{n+1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(n+5\right)\left(n+1\right)-12}{n+1}}{\frac{n+9}{n+1}-\frac{5}{n}}$$

Do the multiplications in \left(n+5\right)\left(n+1\right)-12.

$$\frac{\frac{n^{2}+n+5n+5-12}{n+1}}{\frac{n+9}{n+1}-\frac{5}{n}}$$

Combine like terms in n^{2}+n+5n+5-12.

$$\frac{\frac{n^{2}+6n-7}{n+1}}{\frac{n+9}{n+1}-\frac{5}{n}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of n+1 and n is n\left(n+1\right). Multiply \frac{n+9}{n+1} times \frac{n}{n}. Multiply \frac{5}{n} times \frac{n+1}{n+1}.

$$\frac{\frac{n^{2}+6n-7}{n+1}}{\frac{\left(n+9\right)n}{n\left(n+1\right)}-\frac{5\left(n+1\right)}{n\left(n+1\right)}}$$

Since \frac{\left(n+9\right)n}{n\left(n+1\right)} and \frac{5\left(n+1\right)}{n\left(n+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{n^{2}+6n-7}{n+1}}{\frac{\left(n+9\right)n-5\left(n+1\right)}{n\left(n+1\right)}}$$

Do the multiplications in \left(n+9\right)n-5\left(n+1\right).

$$\frac{\frac{n^{2}+6n-7}{n+1}}{\frac{n^{2}+9n-5n-5}{n\left(n+1\right)}}$$

Combine like terms in n^{2}+9n-5n-5.

$$\frac{\frac{n^{2}+6n-7}{n+1}}{\frac{n^{2}+4n-5}{n\left(n+1\right)}}$$

Divide \frac{n^{2}+6n-7}{n+1} by \frac{n^{2}+4n-5}{n\left(n+1\right)} by multiplying \frac{n^{2}+6n-7}{n+1} by the reciprocal of \frac{n^{2}+4n-5}{n\left(n+1\right)}.

$$\frac{\left(n^{2}+6n-7\right)n\left(n+1\right)}{\left(n+1\right)\left(n^{2}+4n-5\right)}$$

Cancel out n+1 in both numerator and denominator.

$$\frac{n\left(n^{2}+6n-7\right)}{n^{2}+4n-5}$$

Factor the expressions that are not already factored.

$$\frac{n\left(n-1\right)\left(n+7\right)}{\left(n-1\right)\left(n+5\right)}$$

Cancel out n-1 in both numerator and denominator.

$$\frac{n\left(n+7\right)}{n+5}$$

Expand the expression.

$$\frac{n^{2}+7n}{n+5}$$