Solve (5/2-1/3-7/4)5/9-(frac{1}{2}*frac{5}{4}-frac{2}{3}:10):(frac{2}{5}+frac{1}{2}:frac{3}{4})

$$( \frac { 5 } { 2 } - \frac { 1 } { 3 } - \frac { 7 } { 4 } ) \frac { 5 } { 9 } - ( \frac { 1 } { 2 } \cdot \frac { 5 } { 4 } - \frac { 2 } { 3 } : 10 ) : ( \frac { 2 } { 5 } + \frac { 1 } { 2 } : \frac { 3 } { 4 } )$$

Evaluate

$$-\frac{1009}{3456}\approx -0.291956019$$

Solution Steps

Least common multiple of 2 and 3 is 6. Convert \frac{5}{2} and \frac{1}{3} to fractions with denominator 6.

$$\left(\frac{15}{6}-\frac{2}{6}-\frac{7}{4}\right)\times \frac{5}{9}-\frac{\frac{1}{2}\times \frac{5}{4}-\frac{\frac{2}{3}}{10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Since \frac{15}{6} and \frac{2}{6} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{15-2}{6}-\frac{7}{4}\right)\times \frac{5}{9}-\frac{\frac{1}{2}\times \frac{5}{4}-\frac{\frac{2}{3}}{10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Subtract 2 from 15 to get 13.

$$\left(\frac{13}{6}-\frac{7}{4}\right)\times \frac{5}{9}-\frac{\frac{1}{2}\times \frac{5}{4}-\frac{\frac{2}{3}}{10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Least common multiple of 6 and 4 is 12. Convert \frac{13}{6} and \frac{7}{4} to fractions with denominator 12.

$$\left(\frac{26}{12}-\frac{21}{12}\right)\times \frac{5}{9}-\frac{\frac{1}{2}\times \frac{5}{4}-\frac{\frac{2}{3}}{10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Since \frac{26}{12} and \frac{21}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{26-21}{12}\times \frac{5}{9}-\frac{\frac{1}{2}\times \frac{5}{4}-\frac{\frac{2}{3}}{10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Subtract 21 from 26 to get 5.

$$\frac{5}{12}\times \frac{5}{9}-\frac{\frac{1}{2}\times \frac{5}{4}-\frac{\frac{2}{3}}{10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Multiply \frac{5}{12} times \frac{5}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\times 5}{12\times 9}-\frac{\frac{1}{2}\times \frac{5}{4}-\frac{\frac{2}{3}}{10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Do the multiplications in the fraction \frac{5\times 5}{12\times 9}.

$$\frac{25}{108}-\frac{\frac{1}{2}\times \frac{5}{4}-\frac{\frac{2}{3}}{10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Multiply \frac{1}{2} times \frac{5}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25}{108}-\frac{\frac{1\times 5}{2\times 4}-\frac{\frac{2}{3}}{10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Do the multiplications in the fraction \frac{1\times 5}{2\times 4}.

$$\frac{25}{108}-\frac{\frac{5}{8}-\frac{\frac{2}{3}}{10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Express \frac{\frac{2}{3}}{10} as a single fraction.

$$\frac{25}{108}-\frac{\frac{5}{8}-\frac{2}{3\times 10}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Multiply 3 and 10 to get 30.

$$\frac{25}{108}-\frac{\frac{5}{8}-\frac{2}{30}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

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

$$\frac{25}{108}-\frac{\frac{5}{8}-\frac{1}{15}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Least common multiple of 8 and 15 is 120. Convert \frac{5}{8} and \frac{1}{15} to fractions with denominator 120.

$$\frac{25}{108}-\frac{\frac{75}{120}-\frac{8}{120}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Since \frac{75}{120} and \frac{8}{120} have the same denominator, subtract them by subtracting their numerators.

$$\frac{25}{108}-\frac{\frac{75-8}{120}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Subtract 8 from 75 to get 67.

$$\frac{25}{108}-\frac{\frac{67}{120}}{\frac{2}{5}+\frac{\frac{1}{2}}{\frac{3}{4}}}$$

Divide \frac{1}{2} by \frac{3}{4} by multiplying \frac{1}{2} by the reciprocal of \frac{3}{4}.

$$\frac{25}{108}-\frac{\frac{67}{120}}{\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}}$$

Multiply \frac{1}{2} times \frac{4}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25}{108}-\frac{\frac{67}{120}}{\frac{2}{5}+\frac{1\times 4}{2\times 3}}$$

Do the multiplications in the fraction \frac{1\times 4}{2\times 3}.

$$\frac{25}{108}-\frac{\frac{67}{120}}{\frac{2}{5}+\frac{4}{6}}$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$\frac{25}{108}-\frac{\frac{67}{120}}{\frac{2}{5}+\frac{2}{3}}$$

Least common multiple of 5 and 3 is 15. Convert \frac{2}{5} and \frac{2}{3} to fractions with denominator 15.

$$\frac{25}{108}-\frac{\frac{67}{120}}{\frac{6}{15}+\frac{10}{15}}$$

Since \frac{6}{15} and \frac{10}{15} have the same denominator, add them by adding their numerators.

$$\frac{25}{108}-\frac{\frac{67}{120}}{\frac{6+10}{15}}$$

Add 6 and 10 to get 16.

$$\frac{25}{108}-\frac{\frac{67}{120}}{\frac{16}{15}}$$

Divide \frac{67}{120} by \frac{16}{15} by multiplying \frac{67}{120} by the reciprocal of \frac{16}{15}.

$$\frac{25}{108}-\frac{67}{120}\times \frac{15}{16}$$

Multiply \frac{67}{120} times \frac{15}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25}{108}-\frac{67\times 15}{120\times 16}$$

Do the multiplications in the fraction \frac{67\times 15}{120\times 16}.

$$\frac{25}{108}-\frac{1005}{1920}$$

Reduce the fraction \frac{1005}{1920} to lowest terms by extracting and canceling out 15.

$$\frac{25}{108}-\frac{67}{128}$$

Least common multiple of 108 and 128 is 3456. Convert \frac{25}{108} and \frac{67}{128} to fractions with denominator 3456.

$$\frac{800}{3456}-\frac{1809}{3456}$$

Since \frac{800}{3456} and \frac{1809}{3456} have the same denominator, subtract them by subtracting their numerators.

$$\frac{800-1809}{3456}$$

Subtract 1809 from 800 to get -1009.

$$-\frac{1009}{3456}$$

Factor

$$-\frac{1009}{3456} = -0.29195601851851855$$