Solve (3/5+1/2*2/3):(frac{1/5+frac{4}{2})}{(1-frac{5}{6})*(1+frac{5}{6})+frac{1}{36}}*(1+frac{2}{21})

$$\frac { ( \frac { 3 } { 5 } + \frac { 1 } { 2 } \cdot \frac { 2 } { 3 } ) : ( \frac { 1 } { 5 } + \frac { 4 } { 2 } ) } { ( 1 - \frac { 5 } { 6 } ) \cdot ( 1 + \frac { 5 } { 6 } ) + \frac { 1 } { 36 } } \cdot ( 1 + \frac { 2 } { 21 } )$$

Evaluate

$$\frac{46}{33}\approx 1.393939394$$

Solution Steps

Multiply \frac{1}{2} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\frac{3}{5}+\frac{1\times 2}{2\times 3}}{\frac{1}{5}+\frac{4}{2}}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Cancel out 2 in both numerator and denominator.

$$\frac{\frac{\frac{3}{5}+\frac{1}{3}}{\frac{1}{5}+\frac{4}{2}}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Least common multiple of 5 and 3 is 15. Convert \frac{3}{5} and \frac{1}{3} to fractions with denominator 15.

$$\frac{\frac{\frac{9}{15}+\frac{5}{15}}{\frac{1}{5}+\frac{4}{2}}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Since \frac{9}{15} and \frac{5}{15} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\frac{9+5}{15}}{\frac{1}{5}+\frac{4}{2}}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Add 9 and 5 to get 14.

$$\frac{\frac{\frac{14}{15}}{\frac{1}{5}+\frac{4}{2}}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Divide 4 by 2 to get 2.

$$\frac{\frac{\frac{14}{15}}{\frac{1}{5}+2}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Convert 2 to fraction \frac{10}{5}.

$$\frac{\frac{\frac{14}{15}}{\frac{1}{5}+\frac{10}{5}}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Since \frac{1}{5} and \frac{10}{5} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\frac{14}{15}}{\frac{1+10}{5}}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Add 1 and 10 to get 11.

$$\frac{\frac{\frac{14}{15}}{\frac{11}{5}}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Divide \frac{14}{15} by \frac{11}{5} by multiplying \frac{14}{15} by the reciprocal of \frac{11}{5}.

$$\frac{\frac{14}{15}\times \frac{5}{11}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Multiply \frac{14}{15} times \frac{5}{11} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{14\times 5}{15\times 11}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Do the multiplications in the fraction \frac{14\times 5}{15\times 11}.

$$\frac{\frac{70}{165}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Reduce the fraction \frac{70}{165} to lowest terms by extracting and canceling out 5.

$$\frac{\frac{14}{33}}{\left(1-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

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

$$\frac{\frac{14}{33}}{\left(\frac{6}{6}-\frac{5}{6}\right)\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Since \frac{6}{6} and \frac{5}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{14}{33}}{\frac{6-5}{6}\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Subtract 5 from 6 to get 1.

$$\frac{\frac{14}{33}}{\frac{1}{6}\left(1+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

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

$$\frac{\frac{14}{33}}{\frac{1}{6}\left(\frac{6}{6}+\frac{5}{6}\right)+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Since \frac{6}{6} and \frac{5}{6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{14}{33}}{\frac{1}{6}\times \frac{6+5}{6}+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Add 6 and 5 to get 11.

$$\frac{\frac{14}{33}}{\frac{1}{6}\times \frac{11}{6}+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Multiply \frac{1}{6} times \frac{11}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{14}{33}}{\frac{1\times 11}{6\times 6}+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Do the multiplications in the fraction \frac{1\times 11}{6\times 6}.

$$\frac{\frac{14}{33}}{\frac{11}{36}+\frac{1}{36}}\left(1+\frac{2}{21}\right)$$

Since \frac{11}{36} and \frac{1}{36} have the same denominator, add them by adding their numerators.

$$\frac{\frac{14}{33}}{\frac{11+1}{36}}\left(1+\frac{2}{21}\right)$$

Add 11 and 1 to get 12.

$$\frac{\frac{14}{33}}{\frac{12}{36}}\left(1+\frac{2}{21}\right)$$

Reduce the fraction \frac{12}{36} to lowest terms by extracting and canceling out 12.

$$\frac{\frac{14}{33}}{\frac{1}{3}}\left(1+\frac{2}{21}\right)$$

Divide \frac{14}{33} by \frac{1}{3} by multiplying \frac{14}{33} by the reciprocal of \frac{1}{3}.

$$\frac{14}{33}\times 3\left(1+\frac{2}{21}\right)$$

Express \frac{14}{33}\times 3 as a single fraction.

$$\frac{14\times 3}{33}\left(1+\frac{2}{21}\right)$$

Multiply 14 and 3 to get 42.

$$\frac{42}{33}\left(1+\frac{2}{21}\right)$$

Reduce the fraction \frac{42}{33} to lowest terms by extracting and canceling out 3.

$$\frac{14}{11}\left(1+\frac{2}{21}\right)$$

Convert 1 to fraction \frac{21}{21}.

$$\frac{14}{11}\left(\frac{21}{21}+\frac{2}{21}\right)$$

Since \frac{21}{21} and \frac{2}{21} have the same denominator, add them by adding their numerators.

$$\frac{14}{11}\times \frac{21+2}{21}$$

Add 21 and 2 to get 23.

$$\frac{14}{11}\times \frac{23}{21}$$

Multiply \frac{14}{11} times \frac{23}{21} by multiplying numerator times numerator and denominator times denominator.

$$\frac{14\times 23}{11\times 21}$$

Do the multiplications in the fraction \frac{14\times 23}{11\times 21}.

$$\frac{322}{231}$$

Reduce the fraction \frac{322}{231} to lowest terms by extracting and canceling out 7.

$$\frac{46}{33}$$

Factor

$$\frac{2 \cdot 23}{3 \cdot 11} = 1\frac{13}{33} = 1.393939393939394$$