Solve (a/b+2a/3b)div(3x/8divx/9+frac{1}{4})

$$( \frac { a } { b } + \frac { 2 a } { 3 b } ) \div ( \frac { 3 x } { 8 } \div \frac { x } { 9 } + \frac { 1 } { 4 } )$$

Evaluate

$$\frac{40a}{87b}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b and 3b is 3b. Multiply \frac{a}{b} times \frac{3}{3}.

$$\frac{\frac{3a}{3b}+\frac{2a}{3b}}{\frac{\frac{3x}{8}}{\frac{x}{9}}+\frac{1}{4}}$$

Since \frac{3a}{3b} and \frac{2a}{3b} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3a+2a}{3b}}{\frac{\frac{3x}{8}}{\frac{x}{9}}+\frac{1}{4}}$$

Combine like terms in 3a+2a.

$$\frac{\frac{5a}{3b}}{\frac{\frac{3x}{8}}{\frac{x}{9}}+\frac{1}{4}}$$

Divide \frac{3x}{8} by \frac{x}{9} by multiplying \frac{3x}{8} by the reciprocal of \frac{x}{9}.

$$\frac{\frac{5a}{3b}}{\frac{3x\times 9}{8x}+\frac{1}{4}}$$

Cancel out x in both numerator and denominator.

$$\frac{\frac{5a}{3b}}{\frac{3\times 9}{8}+\frac{1}{4}}$$

Multiply 3 and 9 to get 27.

$$\frac{\frac{5a}{3b}}{\frac{27}{8}+\frac{1}{4}}$$

Least common multiple of 8 and 4 is 8. Convert \frac{27}{8} and \frac{1}{4} to fractions with denominator 8.

$$\frac{\frac{5a}{3b}}{\frac{27}{8}+\frac{2}{8}}$$

Since \frac{27}{8} and \frac{2}{8} have the same denominator, add them by adding their numerators.

$$\frac{\frac{5a}{3b}}{\frac{27+2}{8}}$$

Add 27 and 2 to get 29.

$$\frac{\frac{5a}{3b}}{\frac{29}{8}}$$

Divide \frac{5a}{3b} by \frac{29}{8} by multiplying \frac{5a}{3b} by the reciprocal of \frac{29}{8}.

$$\frac{5a\times 8}{3b\times 29}$$

Multiply 5 and 8 to get 40.

$$\frac{40a}{3b\times 29}$$

Multiply 3 and 29 to get 87.

$$\frac{40a}{87b}$$

Expand

$$\frac{40a}{87b}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b and 3b is 3b. Multiply \frac{a}{b} times \frac{3}{3}.

$$\frac{\frac{3a}{3b}+\frac{2a}{3b}}{\frac{\frac{3x}{8}}{\frac{x}{9}}+\frac{1}{4}}$$

Since \frac{3a}{3b} and \frac{2a}{3b} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3a+2a}{3b}}{\frac{\frac{3x}{8}}{\frac{x}{9}}+\frac{1}{4}}$$

Combine like terms in 3a+2a.

$$\frac{\frac{5a}{3b}}{\frac{\frac{3x}{8}}{\frac{x}{9}}+\frac{1}{4}}$$

Divide \frac{3x}{8} by \frac{x}{9} by multiplying \frac{3x}{8} by the reciprocal of \frac{x}{9}.

$$\frac{\frac{5a}{3b}}{\frac{3x\times 9}{8x}+\frac{1}{4}}$$

Cancel out x in both numerator and denominator.

$$\frac{\frac{5a}{3b}}{\frac{3\times 9}{8}+\frac{1}{4}}$$

Multiply 3 and 9 to get 27.

$$\frac{\frac{5a}{3b}}{\frac{27}{8}+\frac{1}{4}}$$

Least common multiple of 8 and 4 is 8. Convert \frac{27}{8} and \frac{1}{4} to fractions with denominator 8.

$$\frac{\frac{5a}{3b}}{\frac{27}{8}+\frac{2}{8}}$$

Since \frac{27}{8} and \frac{2}{8} have the same denominator, add them by adding their numerators.

$$\frac{\frac{5a}{3b}}{\frac{27+2}{8}}$$

Add 27 and 2 to get 29.

$$\frac{\frac{5a}{3b}}{\frac{29}{8}}$$

Divide \frac{5a}{3b} by \frac{29}{8} by multiplying \frac{5a}{3b} by the reciprocal of \frac{29}{8}.

$$\frac{5a\times 8}{3b\times 29}$$

Multiply 5 and 8 to get 40.

$$\frac{40a}{3b\times 29}$$

Multiply 3 and 29 to get 87.

$$\frac{40a}{87b}$$