Solve (3/4-1/3)2/3/(1-frac{1{6})div5}*3+frac{frac{frac{4}{3}+frac{1}{2}}{frac{1}{2}-frac{2}{5}}}{frac{1}{2}-frac{2}{5}}

$$\frac { ( \frac { 3 } { 4 } - \frac { 1 } { 3 } ) \frac { 2 } { 3 } } { ( 1 - \frac { 1 } { 6 } ) \div 5 } \times 3 + \frac { \frac { \frac { 4 } { 3 } + \frac { 1 } { 2 } } { \frac { 1 } { 2 } - \frac { 2 } { 5 } } } { \frac { 1 } { 2 } - \frac { 2 } { 5 } }$$

Evaluate

$$\frac{565}{3}\approx 188.333333333$$

Solution Steps

Least common multiple of 4 and 3 is 12. Convert \frac{3}{4} and \frac{1}{3} to fractions with denominator 12.

$$\frac{\left(\frac{9}{12}-\frac{4}{12}\right)\times \frac{2}{3}}{\frac{1-\frac{1}{6}}{5}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Since \frac{9}{12} and \frac{4}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{9-4}{12}\times \frac{2}{3}}{\frac{1-\frac{1}{6}}{5}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Subtract 4 from 9 to get 5.

$$\frac{\frac{5}{12}\times \frac{2}{3}}{\frac{1-\frac{1}{6}}{5}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Multiply \frac{5}{12} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{5\times 2}{12\times 3}}{\frac{1-\frac{1}{6}}{5}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Do the multiplications in the fraction \frac{5\times 2}{12\times 3}.

$$\frac{\frac{10}{36}}{\frac{1-\frac{1}{6}}{5}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Reduce the fraction \frac{10}{36} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{5}{18}}{\frac{1-\frac{1}{6}}{5}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Convert 1 to fraction \frac{6}{6}.

$$\frac{\frac{5}{18}}{\frac{\frac{6}{6}-\frac{1}{6}}{5}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Since \frac{6}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{5}{18}}{\frac{\frac{6-1}{6}}{5}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Subtract 1 from 6 to get 5.

$$\frac{\frac{5}{18}}{\frac{\frac{5}{6}}{5}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Express \frac{\frac{5}{6}}{5} as a single fraction.

$$\frac{\frac{5}{18}}{\frac{5}{6\times 5}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{\frac{5}{18}}{\frac{1}{6}}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Divide \frac{5}{18} by \frac{1}{6} by multiplying \frac{5}{18} by the reciprocal of \frac{1}{6}.

$$\frac{5}{18}\times 6\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Express \frac{5}{18}\times 6 as a single fraction.

$$\frac{5\times 6}{18}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Multiply 5 and 6 to get 30.

$$\frac{30}{18}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Reduce the fraction \frac{30}{18} to lowest terms by extracting and canceling out 6.

$$\frac{5}{3}\times 3+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Cancel out 3 and 3.

$$5+\frac{\frac{\frac{4}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Least common multiple of 3 and 2 is 6. Convert \frac{4}{3} and \frac{1}{2} to fractions with denominator 6.

$$5+\frac{\frac{\frac{8}{6}+\frac{3}{6}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Since \frac{8}{6} and \frac{3}{6} have the same denominator, add them by adding their numerators.

$$5+\frac{\frac{\frac{8+3}{6}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Add 8 and 3 to get 11.

$$5+\frac{\frac{\frac{11}{6}}{\frac{1}{2}-\frac{2}{5}}}{\frac{1}{2}-\frac{2}{5}}$$

Least common multiple of 2 and 5 is 10. Convert \frac{1}{2} and \frac{2}{5} to fractions with denominator 10.

$$5+\frac{\frac{\frac{11}{6}}{\frac{5}{10}-\frac{4}{10}}}{\frac{1}{2}-\frac{2}{5}}$$

Since \frac{5}{10} and \frac{4}{10} have the same denominator, subtract them by subtracting their numerators.

$$5+\frac{\frac{\frac{11}{6}}{\frac{5-4}{10}}}{\frac{1}{2}-\frac{2}{5}}$$

Subtract 4 from 5 to get 1.

$$5+\frac{\frac{\frac{11}{6}}{\frac{1}{10}}}{\frac{1}{2}-\frac{2}{5}}$$

Divide \frac{11}{6} by \frac{1}{10} by multiplying \frac{11}{6} by the reciprocal of \frac{1}{10}.

$$5+\frac{\frac{11}{6}\times 10}{\frac{1}{2}-\frac{2}{5}}$$

Express \frac{11}{6}\times 10 as a single fraction.

$$5+\frac{\frac{11\times 10}{6}}{\frac{1}{2}-\frac{2}{5}}$$

Multiply 11 and 10 to get 110.

$$5+\frac{\frac{110}{6}}{\frac{1}{2}-\frac{2}{5}}$$

Reduce the fraction \frac{110}{6} to lowest terms by extracting and canceling out 2.

$$5+\frac{\frac{55}{3}}{\frac{1}{2}-\frac{2}{5}}$$

Least common multiple of 2 and 5 is 10. Convert \frac{1}{2} and \frac{2}{5} to fractions with denominator 10.

$$5+\frac{\frac{55}{3}}{\frac{5}{10}-\frac{4}{10}}$$

Since \frac{5}{10} and \frac{4}{10} have the same denominator, subtract them by subtracting their numerators.

$$5+\frac{\frac{55}{3}}{\frac{5-4}{10}}$$

Subtract 4 from 5 to get 1.

$$5+\frac{\frac{55}{3}}{\frac{1}{10}}$$

Divide \frac{55}{3} by \frac{1}{10} by multiplying \frac{55}{3} by the reciprocal of \frac{1}{10}.

$$5+\frac{55}{3}\times 10$$

Express \frac{55}{3}\times 10 as a single fraction.

$$5+\frac{55\times 10}{3}$$

Multiply 55 and 10 to get 550.

$$5+\frac{550}{3}$$

Convert 5 to fraction \frac{15}{3}.

$$\frac{15}{3}+\frac{550}{3}$$

Since \frac{15}{3} and \frac{550}{3} have the same denominator, add them by adding their numerators.

$$\frac{15+550}{3}$$

Add 15 and 550 to get 565.

$$\frac{565}{3}$$

Factor

$$\frac{5 \cdot 113}{3} = 188\frac{1}{3} = 188.33333333333334$$