Solve [(31/4-41/3)-5/6]:{11/12-frac{1}{2}(2frac{1}{3}+1-1frac{1}{4})}

$$[ ( 3 \frac { 1 } { 4 } - 4 \frac { 1 } { 3 } ) - \frac { 5 } { 6 } ] : \{ \frac { 11 } { 12 } - \frac { 1 } { 2 } ( 2 \frac { 1 } { 3 } + 1 - 1 \frac { 1 } { 4 } ) \}$$

Evaluate

$$\frac{46}{3}\approx 15.333333333$$

Solution Steps

Multiply 3 and 4 to get 12.

$$\frac{\frac{12+1}{4}-\frac{4\times 3+1}{3}-\frac{5}{6}}{\frac{11}{12}-\frac{1}{2}\left(\frac{2\times 3+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Add 12 and 1 to get 13.

$$\frac{\frac{13}{4}-\frac{4\times 3+1}{3}-\frac{5}{6}}{\frac{11}{12}-\frac{1}{2}\left(\frac{2\times 3+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Multiply 4 and 3 to get 12.

$$\frac{\frac{13}{4}-\frac{12+1}{3}-\frac{5}{6}}{\frac{11}{12}-\frac{1}{2}\left(\frac{2\times 3+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Add 12 and 1 to get 13.

$$\frac{\frac{13}{4}-\frac{13}{3}-\frac{5}{6}}{\frac{11}{12}-\frac{1}{2}\left(\frac{2\times 3+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Least common multiple of 4 and 3 is 12. Convert \frac{13}{4} and \frac{13}{3} to fractions with denominator 12.

$$\frac{\frac{39}{12}-\frac{52}{12}-\frac{5}{6}}{\frac{11}{12}-\frac{1}{2}\left(\frac{2\times 3+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Since \frac{39}{12} and \frac{52}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{39-52}{12}-\frac{5}{6}}{\frac{11}{12}-\frac{1}{2}\left(\frac{2\times 3+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Subtract 52 from 39 to get -13.

$$\frac{-\frac{13}{12}-\frac{5}{6}}{\frac{11}{12}-\frac{1}{2}\left(\frac{2\times 3+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Least common multiple of 12 and 6 is 12. Convert -\frac{13}{12} and \frac{5}{6} to fractions with denominator 12.

$$\frac{-\frac{13}{12}-\frac{10}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{2\times 3+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Since -\frac{13}{12} and \frac{10}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{-13-10}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{2\times 3+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Subtract 10 from -13 to get -23.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{2\times 3+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Multiply 2 and 3 to get 6.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{6+1}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Add 6 and 1 to get 7.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{7}{3}+1-\frac{1\times 4+1}{4}\right)}$$

Convert 1 to fraction \frac{3}{3}.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{7}{3}+\frac{3}{3}-\frac{1\times 4+1}{4}\right)}$$

Since \frac{7}{3} and \frac{3}{3} have the same denominator, add them by adding their numerators.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{7+3}{3}-\frac{1\times 4+1}{4}\right)}$$

Add 7 and 3 to get 10.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{10}{3}-\frac{1\times 4+1}{4}\right)}$$

Multiply 1 and 4 to get 4.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{10}{3}-\frac{4+1}{4}\right)}$$

Add 4 and 1 to get 5.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{10}{3}-\frac{5}{4}\right)}$$

Least common multiple of 3 and 4 is 12. Convert \frac{10}{3} and \frac{5}{4} to fractions with denominator 12.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\left(\frac{40}{12}-\frac{15}{12}\right)}$$

Since \frac{40}{12} and \frac{15}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\times \frac{40-15}{12}}$$

Subtract 15 from 40 to get 25.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1}{2}\times \frac{25}{12}}$$

Multiply \frac{1}{2} times \frac{25}{12} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{1\times 25}{2\times 12}}$$

Do the multiplications in the fraction \frac{1\times 25}{2\times 12}.

$$\frac{-\frac{23}{12}}{\frac{11}{12}-\frac{25}{24}}$$

Least common multiple of 12 and 24 is 24. Convert \frac{11}{12} and \frac{25}{24} to fractions with denominator 24.

$$\frac{-\frac{23}{12}}{\frac{22}{24}-\frac{25}{24}}$$

Since \frac{22}{24} and \frac{25}{24} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\frac{23}{12}}{\frac{22-25}{24}}$$

Subtract 25 from 22 to get -3.

$$\frac{-\frac{23}{12}}{\frac{-3}{24}}$$

Reduce the fraction \frac{-3}{24} to lowest terms by extracting and canceling out 3.

$$\frac{-\frac{23}{12}}{-\frac{1}{8}}$$

Divide -\frac{23}{12} by -\frac{1}{8} by multiplying -\frac{23}{12} by the reciprocal of -\frac{1}{8}.

$$-\frac{23}{12}\left(-8\right)$$

Express -\frac{23}{12}\left(-8\right) as a single fraction.

$$\frac{-23\left(-8\right)}{12}$$

Multiply -23 and -8 to get 184.

$$\frac{184}{12}$$

Reduce the fraction \frac{184}{12} to lowest terms by extracting and canceling out 4.

$$\frac{46}{3}$$

Factor

$$\frac{2 \cdot 23}{3} = 15\frac{1}{3} \approx 15.333333333$$