Solve 1/2+(1/2-1/3)-(-1/2-frac{1}{3}-frac{1}{6})-(frac{1}{3}-frac{1}{2})

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

Evaluate

$$\frac{11}{6}\approx 1.833333333$$

Solution Steps

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

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

Add 1 and 1 to get 2.

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

Divide 2 by 2 to get 1.

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

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

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

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

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

Subtract 1 from 3 to get 2.

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

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

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

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

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

Since -\frac{3}{6} and \frac{2}{6} have the same denominator, subtract them by subtracting their numerators.

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

Subtract 2 from -3 to get -5.

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

Since -\frac{5}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

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

Subtract 1 from -5 to get -6.

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

Divide -6 by 6 to get -1.

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

The opposite of -1 is 1.

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

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

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

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

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

Add 2 and 3 to get 5.

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

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

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

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

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

Subtract 3 from 2 to get -1.

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

The opposite of -\frac{1}{6} is \frac{1}{6}.

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

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

$$\frac{10}{6}+\frac{1}{6}$$

Since \frac{10}{6} and \frac{1}{6} have the same denominator, add them by adding their numerators.

$$\frac{10+1}{6}$$

Add 10 and 1 to get 11.

$$\frac{11}{6}$$

Factor

$$\frac{11}{2 \cdot 3} = 1\frac{5}{6} = 1.8333333333333333$$