Solve {[(5/6-2/3)*6/5:(1/15+frac{1}{3}+frac{3}{5})]*(frac{8}{15}:frac{4}{5}-frac{1}{9})}*frac{5}{2}*frac{32}{5}

$$\{ [ ( \frac { 5 } { 6 } - \frac { 2 } { 3 } ) \times \frac { 6 } { 5 } : ( \frac { 1 } { 15 } + \frac { 1 } { 3 } + \frac { 3 } { 5 } ) ] \times ( \frac { 8 } { 15 } : \frac { 4 } { 5 } - \frac { 1 } { 9 } ) \} \times \frac { 5 } { 2 } \times \frac { 32 } { 5 }$$

Evaluate

$$\frac{16}{9}\approx 1.777777778$$

Solution Steps

Least common multiple of 6 and 3 is 6. Convert \frac{5}{6} and \frac{2}{3} to fractions with denominator 6.

$$\frac{\left(\frac{5}{6}-\frac{4}{6}\right)\times \frac{6}{5}}{\frac{1}{15}+\frac{1}{3}+\frac{3}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Since \frac{5}{6} and \frac{4}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{5-4}{6}\times \frac{6}{5}}{\frac{1}{15}+\frac{1}{3}+\frac{3}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Subtract 4 from 5 to get 1.

$$\frac{\frac{1}{6}\times \frac{6}{5}}{\frac{1}{15}+\frac{1}{3}+\frac{3}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Multiply \frac{1}{6} times \frac{6}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{1\times 6}{6\times 5}}{\frac{1}{15}+\frac{1}{3}+\frac{3}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Cancel out 6 in both numerator and denominator.

$$\frac{\frac{1}{5}}{\frac{1}{15}+\frac{1}{3}+\frac{3}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

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

$$\frac{\frac{1}{5}}{\frac{1}{15}+\frac{5}{15}+\frac{3}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Since \frac{1}{15} and \frac{5}{15} have the same denominator, add them by adding their numerators.

$$\frac{\frac{1}{5}}{\frac{1+5}{15}+\frac{3}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Add 1 and 5 to get 6.

$$\frac{\frac{1}{5}}{\frac{6}{15}+\frac{3}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Reduce the fraction \frac{6}{15} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{1}{5}}{\frac{2}{5}+\frac{3}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Since \frac{2}{5} and \frac{3}{5} have the same denominator, add them by adding their numerators.

$$\frac{\frac{1}{5}}{\frac{2+3}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Add 2 and 3 to get 5.

$$\frac{\frac{1}{5}}{\frac{5}{5}}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Divide 5 by 5 to get 1.

$$\frac{\frac{1}{5}}{1}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Anything divided by one gives itself.

$$\frac{1}{5}\left(\frac{\frac{8}{15}}{\frac{4}{5}}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Divide \frac{8}{15} by \frac{4}{5} by multiplying \frac{8}{15} by the reciprocal of \frac{4}{5}.

$$\frac{1}{5}\left(\frac{8}{15}\times \frac{5}{4}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Multiply \frac{8}{15} times \frac{5}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{5}\left(\frac{8\times 5}{15\times 4}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Do the multiplications in the fraction \frac{8\times 5}{15\times 4}.

$$\frac{1}{5}\left(\frac{40}{60}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Reduce the fraction \frac{40}{60} to lowest terms by extracting and canceling out 20.

$$\frac{1}{5}\left(\frac{2}{3}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Least common multiple of 3 and 9 is 9. Convert \frac{2}{3} and \frac{1}{9} to fractions with denominator 9.

$$\frac{1}{5}\left(\frac{6}{9}-\frac{1}{9}\right)\times \frac{5}{2}\times \frac{32}{5}$$

Since \frac{6}{9} and \frac{1}{9} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{5}\times \frac{6-1}{9}\times \frac{5}{2}\times \frac{32}{5}$$

Subtract 1 from 6 to get 5.

$$\frac{1}{5}\times \frac{5}{9}\times \frac{5}{2}\times \frac{32}{5}$$

Multiply \frac{1}{5} times \frac{5}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 5}{5\times 9}\times \frac{5}{2}\times \frac{32}{5}$$

Cancel out 5 in both numerator and denominator.

$$\frac{1}{9}\times \frac{5}{2}\times \frac{32}{5}$$

Multiply \frac{1}{9} times \frac{5}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 5}{9\times 2}\times \frac{32}{5}$$

Do the multiplications in the fraction \frac{1\times 5}{9\times 2}.

$$\frac{5}{18}\times \frac{32}{5}$$

Multiply \frac{5}{18} times \frac{32}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\times 32}{18\times 5}$$

Cancel out 5 in both numerator and denominator.

$$\frac{32}{18}$$

Reduce the fraction \frac{32}{18} to lowest terms by extracting and canceling out 2.

$$\frac{16}{9}$$

Factor

$$\frac{2 ^ {4}}{3 ^ {2}} = 1\frac{7}{9} = 1.7777777777777777$$