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

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

Evaluate

$$\frac{65}{6}\approx 10.833333333$$

Solution Steps

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

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

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

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

Subtract 4 from 9 to get 5.

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

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

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

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

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

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

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

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

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

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

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

Subtract 1 from 6 to get 5.

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

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

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

Cancel out 5 in both numerator and denominator.

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

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

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

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

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

Multiply 5 and 6 to get 30.

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

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

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

Cancel out 3 and 3.

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

Express \frac{\frac{\frac{2}{3}+\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}}}{2} as a single fraction.

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

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

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

Since \frac{4}{6} and \frac{3}{6} have the same denominator, add them by adding their numerators.

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

Add 4 and 3 to get 7.

$$5+\frac{\frac{7}{6}}{\left(\frac{1}{2}-\frac{2}{5}\right)\times 2}$$

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

$$5+\frac{\frac{7}{6}}{\left(\frac{5}{10}-\frac{4}{10}\right)\times 2}$$

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

$$5+\frac{\frac{7}{6}}{\frac{5-4}{10}\times 2}$$

Subtract 4 from 5 to get 1.

$$5+\frac{\frac{7}{6}}{\frac{1}{10}\times 2}$$

Multiply \frac{1}{10} and 2 to get \frac{2}{10}.

$$5+\frac{\frac{7}{6}}{\frac{2}{10}}$$

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

$$5+\frac{\frac{7}{6}}{\frac{1}{5}}$$

Divide \frac{7}{6} by \frac{1}{5} by multiplying \frac{7}{6} by the reciprocal of \frac{1}{5}.

$$5+\frac{7}{6}\times 5$$

Express \frac{7}{6}\times 5 as a single fraction.

$$5+\frac{7\times 5}{6}$$

Multiply 7 and 5 to get 35.

$$5+\frac{35}{6}$$

Convert 5 to fraction \frac{30}{6}.

$$\frac{30}{6}+\frac{35}{6}$$

Since \frac{30}{6} and \frac{35}{6} have the same denominator, add them by adding their numerators.

$$\frac{30+35}{6}$$

Add 30 and 35 to get 65.

$$\frac{65}{6}$$

Factor

$$\frac{5 \cdot 13}{2 \cdot 3} = 10\frac{5}{6} = 10.833333333333334$$