Solve 3/b+5+6/5b-2/frac{2{b-2}+4/5b-2}

$$\frac { \frac { 3 } { b + 5 } + \frac { 6 } { 5 b - 2 } } { \frac { 2 } { b - 2 } + \frac { 4 } { 5 b - 2 } }$$

Evaluate

$$\frac{3\left(b-2\right)\left(7b+8\right)}{2\left(7b-6\right)\left(b+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b+5 and 5b-2 is \left(5b-2\right)\left(b+5\right). Multiply \frac{3}{b+5} times \frac{5b-2}{5b-2}. Multiply \frac{6}{5b-2} times \frac{b+5}{b+5}.

$$\frac{\frac{3\left(5b-2\right)}{\left(5b-2\right)\left(b+5\right)}+\frac{6\left(b+5\right)}{\left(5b-2\right)\left(b+5\right)}}{\frac{2}{b-2}+\frac{4}{5b-2}}$$

Since \frac{3\left(5b-2\right)}{\left(5b-2\right)\left(b+5\right)} and \frac{6\left(b+5\right)}{\left(5b-2\right)\left(b+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3\left(5b-2\right)+6\left(b+5\right)}{\left(5b-2\right)\left(b+5\right)}}{\frac{2}{b-2}+\frac{4}{5b-2}}$$

Do the multiplications in 3\left(5b-2\right)+6\left(b+5\right).

$$\frac{\frac{15b-6+6b+30}{\left(5b-2\right)\left(b+5\right)}}{\frac{2}{b-2}+\frac{4}{5b-2}}$$

Combine like terms in 15b-6+6b+30.

$$\frac{\frac{21b+24}{\left(5b-2\right)\left(b+5\right)}}{\frac{2}{b-2}+\frac{4}{5b-2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b-2 and 5b-2 is \left(b-2\right)\left(5b-2\right). Multiply \frac{2}{b-2} times \frac{5b-2}{5b-2}. Multiply \frac{4}{5b-2} times \frac{b-2}{b-2}.

$$\frac{\frac{21b+24}{\left(5b-2\right)\left(b+5\right)}}{\frac{2\left(5b-2\right)}{\left(b-2\right)\left(5b-2\right)}+\frac{4\left(b-2\right)}{\left(b-2\right)\left(5b-2\right)}}$$

Since \frac{2\left(5b-2\right)}{\left(b-2\right)\left(5b-2\right)} and \frac{4\left(b-2\right)}{\left(b-2\right)\left(5b-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{21b+24}{\left(5b-2\right)\left(b+5\right)}}{\frac{2\left(5b-2\right)+4\left(b-2\right)}{\left(b-2\right)\left(5b-2\right)}}$$

Do the multiplications in 2\left(5b-2\right)+4\left(b-2\right).

$$\frac{\frac{21b+24}{\left(5b-2\right)\left(b+5\right)}}{\frac{10b-4+4b-8}{\left(b-2\right)\left(5b-2\right)}}$$

Combine like terms in 10b-4+4b-8.

$$\frac{\frac{21b+24}{\left(5b-2\right)\left(b+5\right)}}{\frac{14b-12}{\left(b-2\right)\left(5b-2\right)}}$$

Divide \frac{21b+24}{\left(5b-2\right)\left(b+5\right)} by \frac{14b-12}{\left(b-2\right)\left(5b-2\right)} by multiplying \frac{21b+24}{\left(5b-2\right)\left(b+5\right)} by the reciprocal of \frac{14b-12}{\left(b-2\right)\left(5b-2\right)}.

$$\frac{\left(21b+24\right)\left(b-2\right)\left(5b-2\right)}{\left(5b-2\right)\left(b+5\right)\left(14b-12\right)}$$

Cancel out 5b-2 in both numerator and denominator.

$$\frac{\left(b-2\right)\left(21b+24\right)}{\left(14b-12\right)\left(b+5\right)}$$

Apply the distributive property by multiplying each term of b-2 by each term of 21b+24.

$$\frac{21b^{2}+24b-42b-48}{\left(14b-12\right)\left(b+5\right)}$$

Combine 24b and -42b to get -18b.

$$\frac{21b^{2}-18b-48}{\left(14b-12\right)\left(b+5\right)}$$

Apply the distributive property by multiplying each term of 14b-12 by each term of b+5.

$$\frac{21b^{2}-18b-48}{14b^{2}+70b-12b-60}$$

Combine 70b and -12b to get 58b.

$$\frac{21b^{2}-18b-48}{14b^{2}+58b-60}$$

Expand

$$\frac{3\left(7b^{2}-6b-16\right)}{2\left(7b-6\right)\left(b+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b+5 and 5b-2 is \left(5b-2\right)\left(b+5\right). Multiply \frac{3}{b+5} times \frac{5b-2}{5b-2}. Multiply \frac{6}{5b-2} times \frac{b+5}{b+5}.

$$\frac{\frac{3\left(5b-2\right)}{\left(5b-2\right)\left(b+5\right)}+\frac{6\left(b+5\right)}{\left(5b-2\right)\left(b+5\right)}}{\frac{2}{b-2}+\frac{4}{5b-2}}$$

Since \frac{3\left(5b-2\right)}{\left(5b-2\right)\left(b+5\right)} and \frac{6\left(b+5\right)}{\left(5b-2\right)\left(b+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3\left(5b-2\right)+6\left(b+5\right)}{\left(5b-2\right)\left(b+5\right)}}{\frac{2}{b-2}+\frac{4}{5b-2}}$$

Do the multiplications in 3\left(5b-2\right)+6\left(b+5\right).

$$\frac{\frac{15b-6+6b+30}{\left(5b-2\right)\left(b+5\right)}}{\frac{2}{b-2}+\frac{4}{5b-2}}$$

Combine like terms in 15b-6+6b+30.

$$\frac{\frac{21b+24}{\left(5b-2\right)\left(b+5\right)}}{\frac{2}{b-2}+\frac{4}{5b-2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b-2 and 5b-2 is \left(b-2\right)\left(5b-2\right). Multiply \frac{2}{b-2} times \frac{5b-2}{5b-2}. Multiply \frac{4}{5b-2} times \frac{b-2}{b-2}.

$$\frac{\frac{21b+24}{\left(5b-2\right)\left(b+5\right)}}{\frac{2\left(5b-2\right)}{\left(b-2\right)\left(5b-2\right)}+\frac{4\left(b-2\right)}{\left(b-2\right)\left(5b-2\right)}}$$

Since \frac{2\left(5b-2\right)}{\left(b-2\right)\left(5b-2\right)} and \frac{4\left(b-2\right)}{\left(b-2\right)\left(5b-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{21b+24}{\left(5b-2\right)\left(b+5\right)}}{\frac{2\left(5b-2\right)+4\left(b-2\right)}{\left(b-2\right)\left(5b-2\right)}}$$

Do the multiplications in 2\left(5b-2\right)+4\left(b-2\right).

$$\frac{\frac{21b+24}{\left(5b-2\right)\left(b+5\right)}}{\frac{10b-4+4b-8}{\left(b-2\right)\left(5b-2\right)}}$$

Combine like terms in 10b-4+4b-8.

$$\frac{\frac{21b+24}{\left(5b-2\right)\left(b+5\right)}}{\frac{14b-12}{\left(b-2\right)\left(5b-2\right)}}$$

Divide \frac{21b+24}{\left(5b-2\right)\left(b+5\right)} by \frac{14b-12}{\left(b-2\right)\left(5b-2\right)} by multiplying \frac{21b+24}{\left(5b-2\right)\left(b+5\right)} by the reciprocal of \frac{14b-12}{\left(b-2\right)\left(5b-2\right)}.

$$\frac{\left(21b+24\right)\left(b-2\right)\left(5b-2\right)}{\left(5b-2\right)\left(b+5\right)\left(14b-12\right)}$$

Cancel out 5b-2 in both numerator and denominator.

$$\frac{\left(b-2\right)\left(21b+24\right)}{\left(14b-12\right)\left(b+5\right)}$$

Apply the distributive property by multiplying each term of b-2 by each term of 21b+24.

$$\frac{21b^{2}+24b-42b-48}{\left(14b-12\right)\left(b+5\right)}$$

Combine 24b and -42b to get -18b.

$$\frac{21b^{2}-18b-48}{\left(14b-12\right)\left(b+5\right)}$$

Apply the distributive property by multiplying each term of 14b-12 by each term of b+5.

$$\frac{21b^{2}-18b-48}{14b^{2}+70b-12b-60}$$

Combine 70b and -12b to get 58b.

$$\frac{21b^{2}-18b-48}{14b^{2}+58b-60}$$