Solve [3/4:(3/4-5/2)-5/14:(frac{3}{2}+1)]:[-frac{2}{5}:(frac{1}{3}-frac{1}{5})+frac{5}{7}]

$$[ \frac { 3 } { 4 } : ( \frac { 3 } { 4 } - \frac { 5 } { 2 } ) - \frac { 5 } { 14 } : ( \frac { 3 } { 2 } + 1 ) ] : [ - \frac { 2 } { 5 } : ( \frac { 1 } { 3 } - \frac { 1 } { 5 } ) + \frac { 5 } { 7 } ]$$

Evaluate

$$\frac{1}{4}=0.25$$

Solution Steps

Least common multiple of 4 and 2 is 4. Convert \frac{3}{4} and \frac{5}{2} to fractions with denominator 4.

$$\frac{\frac{\frac{3}{4}}{\frac{3}{4}-\frac{10}{4}}-\frac{\frac{5}{14}}{\frac{3}{2}+1}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Since \frac{3}{4} and \frac{10}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\frac{3}{4}}{\frac{3-10}{4}}-\frac{\frac{5}{14}}{\frac{3}{2}+1}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Subtract 10 from 3 to get -7.

$$\frac{\frac{\frac{3}{4}}{-\frac{7}{4}}-\frac{\frac{5}{14}}{\frac{3}{2}+1}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Divide \frac{3}{4} by -\frac{7}{4} by multiplying \frac{3}{4} by the reciprocal of -\frac{7}{4}.

$$\frac{\frac{3}{4}\left(-\frac{4}{7}\right)-\frac{\frac{5}{14}}{\frac{3}{2}+1}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Multiply \frac{3}{4} times -\frac{4}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{3\left(-4\right)}{4\times 7}-\frac{\frac{5}{14}}{\frac{3}{2}+1}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Do the multiplications in the fraction \frac{3\left(-4\right)}{4\times 7}.

$$\frac{\frac{-12}{28}-\frac{\frac{5}{14}}{\frac{3}{2}+1}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Reduce the fraction \frac{-12}{28} to lowest terms by extracting and canceling out 4.

$$\frac{-\frac{3}{7}-\frac{\frac{5}{14}}{\frac{3}{2}+1}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Convert 1 to fraction \frac{2}{2}.

$$\frac{-\frac{3}{7}-\frac{\frac{5}{14}}{\frac{3}{2}+\frac{2}{2}}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

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

$$\frac{-\frac{3}{7}-\frac{\frac{5}{14}}{\frac{3+2}{2}}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Add 3 and 2 to get 5.

$$\frac{-\frac{3}{7}-\frac{\frac{5}{14}}{\frac{5}{2}}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Divide \frac{5}{14} by \frac{5}{2} by multiplying \frac{5}{14} by the reciprocal of \frac{5}{2}.

$$\frac{-\frac{3}{7}-\frac{5}{14}\times \frac{2}{5}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Multiply \frac{5}{14} times \frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\frac{3}{7}-\frac{5\times 2}{14\times 5}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{-\frac{3}{7}-\frac{2}{14}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Reduce the fraction \frac{2}{14} to lowest terms by extracting and canceling out 2.

$$\frac{-\frac{3}{7}-\frac{1}{7}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Since -\frac{3}{7} and \frac{1}{7} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{-3-1}{7}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

Subtract 1 from -3 to get -4.

$$\frac{-\frac{4}{7}}{\frac{-\frac{2}{5}}{\frac{1}{3}-\frac{1}{5}}+\frac{5}{7}}$$

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

$$\frac{-\frac{4}{7}}{\frac{-\frac{2}{5}}{\frac{5}{15}-\frac{3}{15}}+\frac{5}{7}}$$

Since \frac{5}{15} and \frac{3}{15} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\frac{4}{7}}{\frac{-\frac{2}{5}}{\frac{5-3}{15}}+\frac{5}{7}}$$

Subtract 3 from 5 to get 2.

$$\frac{-\frac{4}{7}}{\frac{-\frac{2}{5}}{\frac{2}{15}}+\frac{5}{7}}$$

Divide -\frac{2}{5} by \frac{2}{15} by multiplying -\frac{2}{5} by the reciprocal of \frac{2}{15}.

$$\frac{-\frac{4}{7}}{-\frac{2}{5}\times \frac{15}{2}+\frac{5}{7}}$$

Multiply -\frac{2}{5} times \frac{15}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\frac{4}{7}}{\frac{-2\times 15}{5\times 2}+\frac{5}{7}}$$

Do the multiplications in the fraction \frac{-2\times 15}{5\times 2}.

$$\frac{-\frac{4}{7}}{\frac{-30}{10}+\frac{5}{7}}$$

Divide -30 by 10 to get -3.

$$\frac{-\frac{4}{7}}{-3+\frac{5}{7}}$$

Convert -3 to fraction -\frac{21}{7}.

$$\frac{-\frac{4}{7}}{-\frac{21}{7}+\frac{5}{7}}$$

Since -\frac{21}{7} and \frac{5}{7} have the same denominator, add them by adding their numerators.

$$\frac{-\frac{4}{7}}{\frac{-21+5}{7}}$$

Add -21 and 5 to get -16.

$$\frac{-\frac{4}{7}}{-\frac{16}{7}}$$

Divide -\frac{4}{7} by -\frac{16}{7} by multiplying -\frac{4}{7} by the reciprocal of -\frac{16}{7}.

$$-\frac{4}{7}\left(-\frac{7}{16}\right)$$

Multiply -\frac{4}{7} times -\frac{7}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-4\left(-7\right)}{7\times 16}$$

Do the multiplications in the fraction \frac{-4\left(-7\right)}{7\times 16}.

$$\frac{28}{112}$$

Reduce the fraction \frac{28}{112} to lowest terms by extracting and canceling out 28.

$$\frac{1}{4}$$

Factor

$$\frac{1}{2 ^ {2}} = 0.25$$