Solve (4+3/2/5+5-5/2/3)div(frac{frac{3}{4}}{frac{2}{3}}-frac{frac{2}{3}}{frac{5}{4}})

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

Evaluate

$$\frac{232}{71}\approx 3.267605634$$

Solution Steps

Convert 4 to fraction \frac{8}{2}.

$$\frac{\frac{\frac{8}{2}+\frac{3}{2}}{5}+\frac{5-\frac{5}{2}}{3}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Since \frac{8}{2} and \frac{3}{2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\frac{8+3}{2}}{5}+\frac{5-\frac{5}{2}}{3}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Add 8 and 3 to get 11.

$$\frac{\frac{\frac{11}{2}}{5}+\frac{5-\frac{5}{2}}{3}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Express \frac{\frac{11}{2}}{5} as a single fraction.

$$\frac{\frac{11}{2\times 5}+\frac{5-\frac{5}{2}}{3}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Multiply 2 and 5 to get 10.

$$\frac{\frac{11}{10}+\frac{5-\frac{5}{2}}{3}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

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

$$\frac{\frac{11}{10}+\frac{\frac{10}{2}-\frac{5}{2}}{3}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Since \frac{10}{2} and \frac{5}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{11}{10}+\frac{\frac{10-5}{2}}{3}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Subtract 5 from 10 to get 5.

$$\frac{\frac{11}{10}+\frac{\frac{5}{2}}{3}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

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

$$\frac{\frac{11}{10}+\frac{5}{2\times 3}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Multiply 2 and 3 to get 6.

$$\frac{\frac{11}{10}+\frac{5}{6}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Least common multiple of 10 and 6 is 30. Convert \frac{11}{10} and \frac{5}{6} to fractions with denominator 30.

$$\frac{\frac{33}{30}+\frac{25}{30}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Since \frac{33}{30} and \frac{25}{30} have the same denominator, add them by adding their numerators.

$$\frac{\frac{33+25}{30}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Add 33 and 25 to get 58.

$$\frac{\frac{58}{30}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Reduce the fraction \frac{58}{30} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{29}{15}}{\frac{\frac{3}{4}}{\frac{2}{3}}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Divide \frac{3}{4} by \frac{2}{3} by multiplying \frac{3}{4} by the reciprocal of \frac{2}{3}.

$$\frac{\frac{29}{15}}{\frac{3}{4}\times \frac{3}{2}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Multiply \frac{3}{4} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{29}{15}}{\frac{3\times 3}{4\times 2}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Do the multiplications in the fraction \frac{3\times 3}{4\times 2}.

$$\frac{\frac{29}{15}}{\frac{9}{8}-\frac{\frac{2}{3}}{\frac{5}{4}}}$$

Divide \frac{2}{3} by \frac{5}{4} by multiplying \frac{2}{3} by the reciprocal of \frac{5}{4}.

$$\frac{\frac{29}{15}}{\frac{9}{8}-\frac{2}{3}\times \frac{4}{5}}$$

Multiply \frac{2}{3} times \frac{4}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{29}{15}}{\frac{9}{8}-\frac{2\times 4}{3\times 5}}$$

Do the multiplications in the fraction \frac{2\times 4}{3\times 5}.

$$\frac{\frac{29}{15}}{\frac{9}{8}-\frac{8}{15}}$$

Least common multiple of 8 and 15 is 120. Convert \frac{9}{8} and \frac{8}{15} to fractions with denominator 120.

$$\frac{\frac{29}{15}}{\frac{135}{120}-\frac{64}{120}}$$

Since \frac{135}{120} and \frac{64}{120} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{29}{15}}{\frac{135-64}{120}}$$

Subtract 64 from 135 to get 71.

$$\frac{\frac{29}{15}}{\frac{71}{120}}$$

Divide \frac{29}{15} by \frac{71}{120} by multiplying \frac{29}{15} by the reciprocal of \frac{71}{120}.

$$\frac{29}{15}\times \frac{120}{71}$$

Multiply \frac{29}{15} times \frac{120}{71} by multiplying numerator times numerator and denominator times denominator.

$$\frac{29\times 120}{15\times 71}$$

Do the multiplications in the fraction \frac{29\times 120}{15\times 71}.

$$\frac{3480}{1065}$$

Reduce the fraction \frac{3480}{1065} to lowest terms by extracting and canceling out 15.

$$\frac{232}{71}$$

Factor

$$\frac{2 ^ {3} \cdot 29}{71} = 3\frac{19}{71} = 3.267605633802817$$