Solve 1-1/2+3(1/3+frac{1/2)/3frac{1{2}}}{2+frac{3}{frac{2}{3}}}

$$\frac { 1 - \frac { \frac { 1 } { 2 } + 3 ( \frac { 1 } { 3 } + \frac { 1 } { 2 } ) } { 3 \frac { 1 } { 2 } } } { 2 + \frac { 3 } { \frac { 2 } { 3 } } }$$

Evaluate

$$\frac{2}{91}\approx 0.021978022$$

Solution Steps

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

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

Since \frac{2}{6} and \frac{3}{6} have the same denominator, add them by adding their numerators.

$$\frac{1-\frac{\frac{1}{2}+3\times \frac{2+3}{6}}{\frac{3\times 2+1}{2}}}{2+\frac{3}{\frac{2}{3}}}$$

Add 2 and 3 to get 5.

$$\frac{1-\frac{\frac{1}{2}+3\times \frac{5}{6}}{\frac{3\times 2+1}{2}}}{2+\frac{3}{\frac{2}{3}}}$$

Express 3\times \frac{5}{6} as a single fraction.

$$\frac{1-\frac{\frac{1}{2}+\frac{3\times 5}{6}}{\frac{3\times 2+1}{2}}}{2+\frac{3}{\frac{2}{3}}}$$

Multiply 3 and 5 to get 15.

$$\frac{1-\frac{\frac{1}{2}+\frac{15}{6}}{\frac{3\times 2+1}{2}}}{2+\frac{3}{\frac{2}{3}}}$$

Reduce the fraction \frac{15}{6} to lowest terms by extracting and canceling out 3.

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

Since \frac{1}{2} and \frac{5}{2} have the same denominator, add them by adding their numerators.

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

Add 1 and 5 to get 6.

$$\frac{1-\frac{\frac{6}{2}}{\frac{3\times 2+1}{2}}}{2+\frac{3}{\frac{2}{3}}}$$

Divide 6 by 2 to get 3.

$$\frac{1-\frac{3}{\frac{3\times 2+1}{2}}}{2+\frac{3}{\frac{2}{3}}}$$

Multiply 3 and 2 to get 6.

$$\frac{1-\frac{3}{\frac{6+1}{2}}}{2+\frac{3}{\frac{2}{3}}}$$

Add 6 and 1 to get 7.

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

Divide 3 by \frac{7}{2} by multiplying 3 by the reciprocal of \frac{7}{2}.

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

Express 3\times \frac{2}{7} as a single fraction.

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

Multiply 3 and 2 to get 6.

$$\frac{1-\frac{6}{7}}{2+\frac{3}{\frac{2}{3}}}$$

Convert 1 to fraction \frac{7}{7}.

$$\frac{\frac{7}{7}-\frac{6}{7}}{2+\frac{3}{\frac{2}{3}}}$$

Since \frac{7}{7} and \frac{6}{7} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{7-6}{7}}{2+\frac{3}{\frac{2}{3}}}$$

Subtract 6 from 7 to get 1.

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

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

$$\frac{\frac{1}{7}}{2+3\times \frac{3}{2}}$$

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

$$\frac{\frac{1}{7}}{2+\frac{3\times 3}{2}}$$

Multiply 3 and 3 to get 9.

$$\frac{\frac{1}{7}}{2+\frac{9}{2}}$$

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

$$\frac{\frac{1}{7}}{\frac{4}{2}+\frac{9}{2}}$$

Since \frac{4}{2} and \frac{9}{2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{1}{7}}{\frac{4+9}{2}}$$

Add 4 and 9 to get 13.

$$\frac{\frac{1}{7}}{\frac{13}{2}}$$

Divide \frac{1}{7} by \frac{13}{2} by multiplying \frac{1}{7} by the reciprocal of \frac{13}{2}.

$$\frac{1}{7}\times \frac{2}{13}$$

Multiply \frac{1}{7} times \frac{2}{13} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 2}{7\times 13}$$

Do the multiplications in the fraction \frac{1\times 2}{7\times 13}.

$$\frac{2}{91}$$

Factor

$$\frac{2}{7 \cdot 13} = 0.02197802197802198$$