Solve [(1/7-2/4)*(3/2-1/3)]:frac{5}{6}+frac{1}{4}-frac{3}{2}-[frac{1}{3}*(2+frac{1}{4})-frac{2}{3}]-frac{1}{6}

$$[ ( \frac { 1 } { 7 } - \frac { 2 } { 4 } ) \cdot ( \frac { 3 } { 2 } - \frac { 1 } { 3 } ) ] : \frac { 5 } { 6 } + \frac { 1 } { 4 } - \frac { 3 } { 2 } - [ \frac { 1 } { 3 } \cdot ( 2 + \frac { 1 } { 4 } ) - \frac { 2 } { 3 } ] - \frac { 1 } { 6 }$$

Evaluate

$$-2$$

Solution Steps

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

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

Least common multiple of 7 and 2 is 14. Convert \frac{1}{7} and \frac{1}{2} to fractions with denominator 14.

$$\frac{\left(\frac{2}{14}-\frac{7}{14}\right)\left(\frac{3}{2}-\frac{1}{3}\right)}{\frac{5}{6}}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Since \frac{2}{14} and \frac{7}{14} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2-7}{14}\left(\frac{3}{2}-\frac{1}{3}\right)}{\frac{5}{6}}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Subtract 7 from 2 to get -5.

$$\frac{-\frac{5}{14}\left(\frac{3}{2}-\frac{1}{3}\right)}{\frac{5}{6}}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

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

$$\frac{-\frac{5}{14}\left(\frac{9}{6}-\frac{2}{6}\right)}{\frac{5}{6}}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Since \frac{9}{6} and \frac{2}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\frac{5}{14}\times \frac{9-2}{6}}{\frac{5}{6}}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Subtract 2 from 9 to get 7.

$$\frac{-\frac{5}{14}\times \frac{7}{6}}{\frac{5}{6}}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Multiply -\frac{5}{14} times \frac{7}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{-5\times 7}{14\times 6}}{\frac{5}{6}}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Do the multiplications in the fraction \frac{-5\times 7}{14\times 6}.

$$\frac{\frac{-35}{84}}{\frac{5}{6}}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Reduce the fraction \frac{-35}{84} to lowest terms by extracting and canceling out 7.

$$\frac{-\frac{5}{12}}{\frac{5}{6}}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Divide -\frac{5}{12} by \frac{5}{6} by multiplying -\frac{5}{12} by the reciprocal of \frac{5}{6}.

$$-\frac{5}{12}\times \frac{6}{5}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Multiply -\frac{5}{12} times \frac{6}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-5\times 6}{12\times 5}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Do the multiplications in the fraction \frac{-5\times 6}{12\times 5}.

$$\frac{-30}{60}+\frac{1}{4}-\frac{3}{2}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Reduce the fraction \frac{-30}{60} to lowest terms by extracting and canceling out 30.

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

Least common multiple of 2 and 4 is 4. Convert -\frac{1}{2} and \frac{1}{4} to fractions with denominator 4.

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

Since -\frac{2}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

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

Add -2 and 1 to get -1.

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

Least common multiple of 4 and 2 is 4. Convert -\frac{1}{4} and \frac{3}{2} to fractions with denominator 4.

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

Since -\frac{1}{4} and \frac{6}{4} have the same denominator, subtract them by subtracting their numerators.

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

Subtract 6 from -1 to get -7.

$$-\frac{7}{4}-\left(\frac{1}{3}\left(2+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

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

$$-\frac{7}{4}-\left(\frac{1}{3}\left(\frac{8}{4}+\frac{1}{4}\right)-\frac{2}{3}\right)-\frac{1}{6}$$

Since \frac{8}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$-\frac{7}{4}-\left(\frac{1}{3}\times \frac{8+1}{4}-\frac{2}{3}\right)-\frac{1}{6}$$

Add 8 and 1 to get 9.

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

Multiply \frac{1}{3} times \frac{9}{4} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\times 9}{3\times 4}.

$$-\frac{7}{4}-\left(\frac{9}{12}-\frac{2}{3}\right)-\frac{1}{6}$$

Reduce the fraction \frac{9}{12} to lowest terms by extracting and canceling out 3.

$$-\frac{7}{4}-\left(\frac{3}{4}-\frac{2}{3}\right)-\frac{1}{6}$$

Least common multiple of 4 and 3 is 12. Convert \frac{3}{4} and \frac{2}{3} to fractions with denominator 12.

$$-\frac{7}{4}-\left(\frac{9}{12}-\frac{8}{12}\right)-\frac{1}{6}$$

Since \frac{9}{12} and \frac{8}{12} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{7}{4}-\frac{9-8}{12}-\frac{1}{6}$$

Subtract 8 from 9 to get 1.

$$-\frac{7}{4}-\frac{1}{12}-\frac{1}{6}$$

Least common multiple of 4 and 12 is 12. Convert -\frac{7}{4} and \frac{1}{12} to fractions with denominator 12.

$$-\frac{21}{12}-\frac{1}{12}-\frac{1}{6}$$

Since -\frac{21}{12} and \frac{1}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-21-1}{12}-\frac{1}{6}$$

Subtract 1 from -21 to get -22.

$$\frac{-22}{12}-\frac{1}{6}$$

Reduce the fraction \frac{-22}{12} to lowest terms by extracting and canceling out 2.

$$-\frac{11}{6}-\frac{1}{6}$$

Since -\frac{11}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-11-1}{6}$$

Subtract 1 from -11 to get -12.

$$\frac{-12}{6}$$

Divide -12 by 6 to get -2.

$$-2$$

Factor

$$-2$$