Solve 1+1/2/3+1-frac{1/3/2}{frac{2frac{1}{5}}{frac{5}{6}}-frac{frac{1}{3}}{frac{1}{6}}}

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

Evaluate

$$\frac{125}{96}\approx 1.302083333$$

Solution Steps

Convert 1 to fraction \frac{2}{2}.

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

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

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

Add 2 and 1 to get 3.

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

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

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

Cancel out 3 in both numerator and denominator.

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

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

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

Since \frac{3}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

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

Subtract 1 from 3 to get 2.

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

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

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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

Add 3 and 2 to get 5.

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

Divide \frac{2\times 5+1}{5} by \frac{5}{6} by multiplying \frac{2\times 5+1}{5} by the reciprocal of \frac{5}{6}.

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

Multiply 2 and 5 to get 10.

$$\frac{\frac{5}{6}}{\frac{\left(10+1\right)\times 6}{5\times 5}-\frac{\frac{1}{3}}{\frac{1}{6}}}$$

Add 10 and 1 to get 11.

$$\frac{\frac{5}{6}}{\frac{11\times 6}{5\times 5}-\frac{\frac{1}{3}}{\frac{1}{6}}}$$

Multiply 11 and 6 to get 66.

$$\frac{\frac{5}{6}}{\frac{66}{5\times 5}-\frac{\frac{1}{3}}{\frac{1}{6}}}$$

Multiply 5 and 5 to get 25.

$$\frac{\frac{5}{6}}{\frac{66}{25}-\frac{\frac{1}{3}}{\frac{1}{6}}}$$

Divide \frac{1}{3} by \frac{1}{6} by multiplying \frac{1}{3} by the reciprocal of \frac{1}{6}.

$$\frac{\frac{5}{6}}{\frac{66}{25}-\frac{1}{3}\times 6}$$

Multiply \frac{1}{3} and 6 to get \frac{6}{3}.

$$\frac{\frac{5}{6}}{\frac{66}{25}-\frac{6}{3}}$$

Divide 6 by 3 to get 2.

$$\frac{\frac{5}{6}}{\frac{66}{25}-2}$$

Convert 2 to fraction \frac{50}{25}.

$$\frac{\frac{5}{6}}{\frac{66}{25}-\frac{50}{25}}$$

Since \frac{66}{25} and \frac{50}{25} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{5}{6}}{\frac{66-50}{25}}$$

Subtract 50 from 66 to get 16.

$$\frac{\frac{5}{6}}{\frac{16}{25}}$$

Divide \frac{5}{6} by \frac{16}{25} by multiplying \frac{5}{6} by the reciprocal of \frac{16}{25}.

$$\frac{5}{6}\times \frac{25}{16}$$

Multiply \frac{5}{6} times \frac{25}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\times 25}{6\times 16}$$

Do the multiplications in the fraction \frac{5\times 25}{6\times 16}.

$$\frac{125}{96}$$

Factor

$$\frac{5 ^ {3}}{2 ^ {5} \cdot 3} = 1\frac{29}{96} = 1.3020833333333333$$