Solve 3/d+5+6/5d-6/frac{3{d-6}+5/5d-6}=

$$\frac { \frac { 3 } { d + 5 } + \frac { 6 } { 5 d - 6 } } { \frac { 3 } { d - 6 } + \frac { 5 } { 5 d - 6 } } =$$

Evaluate

$$\frac{3\left(d-6\right)\left(7d+4\right)}{4\left(5d-12\right)\left(d+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of d+5 and 5d-6 is \left(5d-6\right)\left(d+5\right). Multiply \frac{3}{d+5} times \frac{5d-6}{5d-6}. Multiply \frac{6}{5d-6} times \frac{d+5}{d+5}.

$$\frac{\frac{3\left(5d-6\right)}{\left(5d-6\right)\left(d+5\right)}+\frac{6\left(d+5\right)}{\left(5d-6\right)\left(d+5\right)}}{\frac{3}{d-6}+\frac{5}{5d-6}}$$

Since \frac{3\left(5d-6\right)}{\left(5d-6\right)\left(d+5\right)} and \frac{6\left(d+5\right)}{\left(5d-6\right)\left(d+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3\left(5d-6\right)+6\left(d+5\right)}{\left(5d-6\right)\left(d+5\right)}}{\frac{3}{d-6}+\frac{5}{5d-6}}$$

Do the multiplications in 3\left(5d-6\right)+6\left(d+5\right).

$$\frac{\frac{15d-18+6d+30}{\left(5d-6\right)\left(d+5\right)}}{\frac{3}{d-6}+\frac{5}{5d-6}}$$

Combine like terms in 15d-18+6d+30.

$$\frac{\frac{21d+12}{\left(5d-6\right)\left(d+5\right)}}{\frac{3}{d-6}+\frac{5}{5d-6}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of d-6 and 5d-6 is \left(d-6\right)\left(5d-6\right). Multiply \frac{3}{d-6} times \frac{5d-6}{5d-6}. Multiply \frac{5}{5d-6} times \frac{d-6}{d-6}.

$$\frac{\frac{21d+12}{\left(5d-6\right)\left(d+5\right)}}{\frac{3\left(5d-6\right)}{\left(d-6\right)\left(5d-6\right)}+\frac{5\left(d-6\right)}{\left(d-6\right)\left(5d-6\right)}}$$

Since \frac{3\left(5d-6\right)}{\left(d-6\right)\left(5d-6\right)} and \frac{5\left(d-6\right)}{\left(d-6\right)\left(5d-6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{21d+12}{\left(5d-6\right)\left(d+5\right)}}{\frac{3\left(5d-6\right)+5\left(d-6\right)}{\left(d-6\right)\left(5d-6\right)}}$$

Do the multiplications in 3\left(5d-6\right)+5\left(d-6\right).

$$\frac{\frac{21d+12}{\left(5d-6\right)\left(d+5\right)}}{\frac{15d-18+5d-30}{\left(d-6\right)\left(5d-6\right)}}$$

Combine like terms in 15d-18+5d-30.

$$\frac{\frac{21d+12}{\left(5d-6\right)\left(d+5\right)}}{\frac{20d-48}{\left(d-6\right)\left(5d-6\right)}}$$

Divide \frac{21d+12}{\left(5d-6\right)\left(d+5\right)} by \frac{20d-48}{\left(d-6\right)\left(5d-6\right)} by multiplying \frac{21d+12}{\left(5d-6\right)\left(d+5\right)} by the reciprocal of \frac{20d-48}{\left(d-6\right)\left(5d-6\right)}.

$$\frac{\left(21d+12\right)\left(d-6\right)\left(5d-6\right)}{\left(5d-6\right)\left(d+5\right)\left(20d-48\right)}$$

Cancel out 5d-6 in both numerator and denominator.

$$\frac{\left(d-6\right)\left(21d+12\right)}{\left(20d-48\right)\left(d+5\right)}$$

Apply the distributive property by multiplying each term of d-6 by each term of 21d+12.

$$\frac{21d^{2}+12d-126d-72}{\left(20d-48\right)\left(d+5\right)}$$

Combine 12d and -126d to get -114d.

$$\frac{21d^{2}-114d-72}{\left(20d-48\right)\left(d+5\right)}$$

Apply the distributive property by multiplying each term of 20d-48 by each term of d+5.

$$\frac{21d^{2}-114d-72}{20d^{2}+100d-48d-240}$$

Combine 100d and -48d to get 52d.

$$\frac{21d^{2}-114d-72}{20d^{2}+52d-240}$$

Expand

$$\frac{3\left(7d^{2}-38d-24\right)}{4\left(5d-12\right)\left(d+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of d+5 and 5d-6 is \left(5d-6\right)\left(d+5\right). Multiply \frac{3}{d+5} times \frac{5d-6}{5d-6}. Multiply \frac{6}{5d-6} times \frac{d+5}{d+5}.

$$\frac{\frac{3\left(5d-6\right)}{\left(5d-6\right)\left(d+5\right)}+\frac{6\left(d+5\right)}{\left(5d-6\right)\left(d+5\right)}}{\frac{3}{d-6}+\frac{5}{5d-6}}$$

Since \frac{3\left(5d-6\right)}{\left(5d-6\right)\left(d+5\right)} and \frac{6\left(d+5\right)}{\left(5d-6\right)\left(d+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3\left(5d-6\right)+6\left(d+5\right)}{\left(5d-6\right)\left(d+5\right)}}{\frac{3}{d-6}+\frac{5}{5d-6}}$$

Do the multiplications in 3\left(5d-6\right)+6\left(d+5\right).

$$\frac{\frac{15d-18+6d+30}{\left(5d-6\right)\left(d+5\right)}}{\frac{3}{d-6}+\frac{5}{5d-6}}$$

Combine like terms in 15d-18+6d+30.

$$\frac{\frac{21d+12}{\left(5d-6\right)\left(d+5\right)}}{\frac{3}{d-6}+\frac{5}{5d-6}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of d-6 and 5d-6 is \left(d-6\right)\left(5d-6\right). Multiply \frac{3}{d-6} times \frac{5d-6}{5d-6}. Multiply \frac{5}{5d-6} times \frac{d-6}{d-6}.

$$\frac{\frac{21d+12}{\left(5d-6\right)\left(d+5\right)}}{\frac{3\left(5d-6\right)}{\left(d-6\right)\left(5d-6\right)}+\frac{5\left(d-6\right)}{\left(d-6\right)\left(5d-6\right)}}$$

Since \frac{3\left(5d-6\right)}{\left(d-6\right)\left(5d-6\right)} and \frac{5\left(d-6\right)}{\left(d-6\right)\left(5d-6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{21d+12}{\left(5d-6\right)\left(d+5\right)}}{\frac{3\left(5d-6\right)+5\left(d-6\right)}{\left(d-6\right)\left(5d-6\right)}}$$

Do the multiplications in 3\left(5d-6\right)+5\left(d-6\right).

$$\frac{\frac{21d+12}{\left(5d-6\right)\left(d+5\right)}}{\frac{15d-18+5d-30}{\left(d-6\right)\left(5d-6\right)}}$$

Combine like terms in 15d-18+5d-30.

$$\frac{\frac{21d+12}{\left(5d-6\right)\left(d+5\right)}}{\frac{20d-48}{\left(d-6\right)\left(5d-6\right)}}$$

Divide \frac{21d+12}{\left(5d-6\right)\left(d+5\right)} by \frac{20d-48}{\left(d-6\right)\left(5d-6\right)} by multiplying \frac{21d+12}{\left(5d-6\right)\left(d+5\right)} by the reciprocal of \frac{20d-48}{\left(d-6\right)\left(5d-6\right)}.

$$\frac{\left(21d+12\right)\left(d-6\right)\left(5d-6\right)}{\left(5d-6\right)\left(d+5\right)\left(20d-48\right)}$$

Cancel out 5d-6 in both numerator and denominator.

$$\frac{\left(d-6\right)\left(21d+12\right)}{\left(20d-48\right)\left(d+5\right)}$$

Apply the distributive property by multiplying each term of d-6 by each term of 21d+12.

$$\frac{21d^{2}+12d-126d-72}{\left(20d-48\right)\left(d+5\right)}$$

Combine 12d and -126d to get -114d.

$$\frac{21d^{2}-114d-72}{\left(20d-48\right)\left(d+5\right)}$$

Apply the distributive property by multiplying each term of 20d-48 by each term of d+5.

$$\frac{21d^{2}-114d-72}{20d^{2}+100d-48d-240}$$

Combine 100d and -48d to get 52d.

$$\frac{21d^{2}-114d-72}{20d^{2}+52d-240}$$