Solve 2/3{-1/6+5/9-[-2/3-frac{1}{6}-(frac{1}{3}-frac{5}{6})]-frac{2}{9}}

$$\frac { 2 } { 3 } \{ - \frac { 1 } { 6 } + \frac { 5 } { 9 } - [ - \frac { 2 } { 3 } - \frac { 1 } { 6 } - ( \frac { 1 } { 3 } - \frac { 5 } { 6 } ) ] - \frac { 2 } { 9 } \}$$

Evaluate

$$\frac{1}{3}\approx 0.333333333$$

Solution Steps

Least common multiple of 6 and 9 is 18. Convert -\frac{1}{6} and \frac{5}{9} to fractions with denominator 18.

$$\frac{2}{3}\left(-\frac{3}{18}+\frac{10}{18}-\left(-\frac{2}{3}-\frac{1}{6}-\left(\frac{1}{3}-\frac{5}{6}\right)\right)-\frac{2}{9}\right)$$

Since -\frac{3}{18} and \frac{10}{18} have the same denominator, add them by adding their numerators.

$$\frac{2}{3}\left(\frac{-3+10}{18}-\left(-\frac{2}{3}-\frac{1}{6}-\left(\frac{1}{3}-\frac{5}{6}\right)\right)-\frac{2}{9}\right)$$

Add -3 and 10 to get 7.

$$\frac{2}{3}\left(\frac{7}{18}-\left(-\frac{2}{3}-\frac{1}{6}-\left(\frac{1}{3}-\frac{5}{6}\right)\right)-\frac{2}{9}\right)$$

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

$$\frac{2}{3}\left(\frac{7}{18}-\left(-\frac{4}{6}-\frac{1}{6}-\left(\frac{1}{3}-\frac{5}{6}\right)\right)-\frac{2}{9}\right)$$

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

$$\frac{2}{3}\left(\frac{7}{18}-\left(\frac{-4-1}{6}-\left(\frac{1}{3}-\frac{5}{6}\right)\right)-\frac{2}{9}\right)$$

Subtract 1 from -4 to get -5.

$$\frac{2}{3}\left(\frac{7}{18}-\left(-\frac{5}{6}-\left(\frac{1}{3}-\frac{5}{6}\right)\right)-\frac{2}{9}\right)$$

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

$$\frac{2}{3}\left(\frac{7}{18}-\left(-\frac{5}{6}-\left(\frac{2}{6}-\frac{5}{6}\right)\right)-\frac{2}{9}\right)$$

Since \frac{2}{6} and \frac{5}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2}{3}\left(\frac{7}{18}-\left(-\frac{5}{6}-\frac{2-5}{6}\right)-\frac{2}{9}\right)$$

Subtract 5 from 2 to get -3.

$$\frac{2}{3}\left(\frac{7}{18}-\left(-\frac{5}{6}-\frac{-3}{6}\right)-\frac{2}{9}\right)$$

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

$$\frac{2}{3}\left(\frac{7}{18}-\left(-\frac{5}{6}-\left(-\frac{1}{2}\right)\right)-\frac{2}{9}\right)$$

The opposite of -\frac{1}{2} is \frac{1}{2}.

$$\frac{2}{3}\left(\frac{7}{18}-\left(-\frac{5}{6}+\frac{1}{2}\right)-\frac{2}{9}\right)$$

Least common multiple of 6 and 2 is 6. Convert -\frac{5}{6} and \frac{1}{2} to fractions with denominator 6.

$$\frac{2}{3}\left(\frac{7}{18}-\left(-\frac{5}{6}+\frac{3}{6}\right)-\frac{2}{9}\right)$$

Since -\frac{5}{6} and \frac{3}{6} have the same denominator, add them by adding their numerators.

$$\frac{2}{3}\left(\frac{7}{18}-\frac{-5+3}{6}-\frac{2}{9}\right)$$

Add -5 and 3 to get -2.

$$\frac{2}{3}\left(\frac{7}{18}-\frac{-2}{6}-\frac{2}{9}\right)$$

Reduce the fraction \frac{-2}{6} to lowest terms by extracting and canceling out 2.

$$\frac{2}{3}\left(\frac{7}{18}-\left(-\frac{1}{3}\right)-\frac{2}{9}\right)$$

The opposite of -\frac{1}{3} is \frac{1}{3}.

$$\frac{2}{3}\left(\frac{7}{18}+\frac{1}{3}-\frac{2}{9}\right)$$

Least common multiple of 18 and 3 is 18. Convert \frac{7}{18} and \frac{1}{3} to fractions with denominator 18.

$$\frac{2}{3}\left(\frac{7}{18}+\frac{6}{18}-\frac{2}{9}\right)$$

Since \frac{7}{18} and \frac{6}{18} have the same denominator, add them by adding their numerators.

$$\frac{2}{3}\left(\frac{7+6}{18}-\frac{2}{9}\right)$$

Add 7 and 6 to get 13.

$$\frac{2}{3}\left(\frac{13}{18}-\frac{2}{9}\right)$$

Least common multiple of 18 and 9 is 18. Convert \frac{13}{18} and \frac{2}{9} to fractions with denominator 18.

$$\frac{2}{3}\left(\frac{13}{18}-\frac{4}{18}\right)$$

Since \frac{13}{18} and \frac{4}{18} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2}{3}\times \frac{13-4}{18}$$

Subtract 4 from 13 to get 9.

$$\frac{2}{3}\times \frac{9}{18}$$

Reduce the fraction \frac{9}{18} to lowest terms by extracting and canceling out 9.

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

Multiply \frac{2}{3} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2 in both numerator and denominator.

$$\frac{1}{3}$$

Factor

$$\frac{1}{3} = 0.3333333333333333$$