Solve a^2/a+3+a^2/5a+15/frac{a{a+3}+1/5}

$$\frac { \frac { a ^ { 2 } } { a + 3 } + \frac { a ^ { 2 } } { 5 a + 15 } } { \frac { a } { a + 3 } + \frac { 1 } { 5 } }$$

Evaluate

$$\frac{2a^{2}}{2a+1}$$

Short Solution Steps

Factor 5a+15.

$$\frac{\frac{a^{2}}{a+3}+\frac{a^{2}}{5\left(a+3\right)}}{\frac{a}{a+3}+\frac{1}{5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+3 and 5\left(a+3\right) is 5\left(a+3\right). Multiply \frac{a^{2}}{a+3} times \frac{5}{5}.

$$\frac{\frac{5a^{2}}{5\left(a+3\right)}+\frac{a^{2}}{5\left(a+3\right)}}{\frac{a}{a+3}+\frac{1}{5}}$$

Since \frac{5a^{2}}{5\left(a+3\right)} and \frac{a^{2}}{5\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{5a^{2}+a^{2}}{5\left(a+3\right)}}{\frac{a}{a+3}+\frac{1}{5}}$$

Combine like terms in 5a^{2}+a^{2}.

$$\frac{\frac{6a^{2}}{5\left(a+3\right)}}{\frac{a}{a+3}+\frac{1}{5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+3 and 5 is 5\left(a+3\right). Multiply \frac{a}{a+3} times \frac{5}{5}. Multiply \frac{1}{5} times \frac{a+3}{a+3}.

$$\frac{\frac{6a^{2}}{5\left(a+3\right)}}{\frac{5a}{5\left(a+3\right)}+\frac{a+3}{5\left(a+3\right)}}$$

Since \frac{5a}{5\left(a+3\right)} and \frac{a+3}{5\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{6a^{2}}{5\left(a+3\right)}}{\frac{5a+a+3}{5\left(a+3\right)}}$$

Combine like terms in 5a+a+3.

$$\frac{\frac{6a^{2}}{5\left(a+3\right)}}{\frac{6a+3}{5\left(a+3\right)}}$$

Divide \frac{6a^{2}}{5\left(a+3\right)} by \frac{6a+3}{5\left(a+3\right)} by multiplying \frac{6a^{2}}{5\left(a+3\right)} by the reciprocal of \frac{6a+3}{5\left(a+3\right)}.

$$\frac{6a^{2}\times 5\left(a+3\right)}{5\left(a+3\right)\left(6a+3\right)}$$

Cancel out 5\left(a+3\right) in both numerator and denominator.

$$\frac{6a^{2}}{6a+3}$$

Factor the expressions that are not already factored.

$$\frac{6a^{2}}{3\left(2a+1\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{2a^{2}}{2a+1}$$

Expand

$$\frac{2a^{2}}{2a+1}$$

Short Solution Steps

Factor 5a+15.

$$\frac{\frac{a^{2}}{a+3}+\frac{a^{2}}{5\left(a+3\right)}}{\frac{a}{a+3}+\frac{1}{5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+3 and 5\left(a+3\right) is 5\left(a+3\right). Multiply \frac{a^{2}}{a+3} times \frac{5}{5}.

$$\frac{\frac{5a^{2}}{5\left(a+3\right)}+\frac{a^{2}}{5\left(a+3\right)}}{\frac{a}{a+3}+\frac{1}{5}}$$

Since \frac{5a^{2}}{5\left(a+3\right)} and \frac{a^{2}}{5\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{5a^{2}+a^{2}}{5\left(a+3\right)}}{\frac{a}{a+3}+\frac{1}{5}}$$

Combine like terms in 5a^{2}+a^{2}.

$$\frac{\frac{6a^{2}}{5\left(a+3\right)}}{\frac{a}{a+3}+\frac{1}{5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+3 and 5 is 5\left(a+3\right). Multiply \frac{a}{a+3} times \frac{5}{5}. Multiply \frac{1}{5} times \frac{a+3}{a+3}.

$$\frac{\frac{6a^{2}}{5\left(a+3\right)}}{\frac{5a}{5\left(a+3\right)}+\frac{a+3}{5\left(a+3\right)}}$$

Since \frac{5a}{5\left(a+3\right)} and \frac{a+3}{5\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{6a^{2}}{5\left(a+3\right)}}{\frac{5a+a+3}{5\left(a+3\right)}}$$

Combine like terms in 5a+a+3.

$$\frac{\frac{6a^{2}}{5\left(a+3\right)}}{\frac{6a+3}{5\left(a+3\right)}}$$

Divide \frac{6a^{2}}{5\left(a+3\right)} by \frac{6a+3}{5\left(a+3\right)} by multiplying \frac{6a^{2}}{5\left(a+3\right)} by the reciprocal of \frac{6a+3}{5\left(a+3\right)}.

$$\frac{6a^{2}\times 5\left(a+3\right)}{5\left(a+3\right)\left(6a+3\right)}$$

Cancel out 5\left(a+3\right) in both numerator and denominator.

$$\frac{6a^{2}}{6a+3}$$

Factor the expressions that are not already factored.

$$\frac{6a^{2}}{3\left(2a+1\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{2a^{2}}{2a+1}$$