Solve left(left(1/2+2/3right)left(3/4+2/7right)-left(frac{1}{6}+frac{5}{8}right)left(frac{1}{2}+frac{2}{3}-frac{1}{6}right)right)left(frac{2}{5}+frac{1}{2}frac{4}{3}right)

$$\left( \left( \frac{ 1 }{ 2 } + \frac{ 2 }{ 3 } \right) \left( \frac{ 3 }{ 4 } + \frac{ 2 }{ 7 } \right) - \left( \frac{ 1 }{ 6 } + \frac{ 5 }{ 8 } \right) \left( \frac{ 1 }{ 2 } + \frac{ 2 }{ 3 } - \frac{ 1 }{ 6 } \right) \right) \left( \frac{ 2 }{ 5 } + \frac{ 1 }{ 2 } \frac{ 4 }{ 3 } \right)$$

Evaluate

$$\frac{4}{9}\approx 0.444444444$$

Solution Steps

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

$$\left(\left(\frac{3}{6}+\frac{4}{6}\right)\left(\frac{3}{4}+\frac{2}{7}\right)-\left(\frac{1}{6}+\frac{5}{8}\right)\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Since \frac{3}{6} and \frac{4}{6} have the same denominator, add them by adding their numerators.

$$\left(\frac{3+4}{6}\left(\frac{3}{4}+\frac{2}{7}\right)-\left(\frac{1}{6}+\frac{5}{8}\right)\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Add 3 and 4 to get 7.

$$\left(\frac{7}{6}\left(\frac{3}{4}+\frac{2}{7}\right)-\left(\frac{1}{6}+\frac{5}{8}\right)\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Least common multiple of 4 and 7 is 28. Convert \frac{3}{4} and \frac{2}{7} to fractions with denominator 28.

$$\left(\frac{7}{6}\left(\frac{21}{28}+\frac{8}{28}\right)-\left(\frac{1}{6}+\frac{5}{8}\right)\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Since \frac{21}{28} and \frac{8}{28} have the same denominator, add them by adding their numerators.

$$\left(\frac{7}{6}\times \frac{21+8}{28}-\left(\frac{1}{6}+\frac{5}{8}\right)\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Add 21 and 8 to get 29.

$$\left(\frac{7}{6}\times \frac{29}{28}-\left(\frac{1}{6}+\frac{5}{8}\right)\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Multiply \frac{7}{6} times \frac{29}{28} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{7\times 29}{6\times 28}-\left(\frac{1}{6}+\frac{5}{8}\right)\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Do the multiplications in the fraction \frac{7\times 29}{6\times 28}.

$$\left(\frac{203}{168}-\left(\frac{1}{6}+\frac{5}{8}\right)\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Reduce the fraction \frac{203}{168} to lowest terms by extracting and canceling out 7.

$$\left(\frac{29}{24}-\left(\frac{1}{6}+\frac{5}{8}\right)\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Least common multiple of 6 and 8 is 24. Convert \frac{1}{6} and \frac{5}{8} to fractions with denominator 24.

$$\left(\frac{29}{24}-\left(\frac{4}{24}+\frac{15}{24}\right)\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Since \frac{4}{24} and \frac{15}{24} have the same denominator, add them by adding their numerators.

$$\left(\frac{29}{24}-\frac{4+15}{24}\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Add 4 and 15 to get 19.

$$\left(\frac{29}{24}-\frac{19}{24}\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

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

$$\left(\frac{29}{24}-\frac{19}{24}\left(\frac{3}{6}+\frac{4}{6}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Since \frac{3}{6} and \frac{4}{6} have the same denominator, add them by adding their numerators.

$$\left(\frac{29}{24}-\frac{19}{24}\left(\frac{3+4}{6}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Add 3 and 4 to get 7.

$$\left(\frac{29}{24}-\frac{19}{24}\left(\frac{7}{6}-\frac{1}{6}\right)\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Since \frac{7}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{29}{24}-\frac{19}{24}\times \frac{7-1}{6}\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Subtract 1 from 7 to get 6.

$$\left(\frac{29}{24}-\frac{19}{24}\times \frac{6}{6}\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Divide 6 by 6 to get 1.

$$\left(\frac{29}{24}-\frac{19}{24}\times 1\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Multiply \frac{19}{24} and 1 to get \frac{19}{24}.

$$\left(\frac{29}{24}-\frac{19}{24}\right)\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Since \frac{29}{24} and \frac{19}{24} have the same denominator, subtract them by subtracting their numerators.

$$\frac{29-19}{24}\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Subtract 19 from 29 to get 10.

$$\frac{10}{24}\left(\frac{2}{5}+\frac{1}{2}\times \frac{4}{3}\right)$$

Reduce the fraction \frac{10}{24} to lowest terms by extracting and canceling out 2.

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

Multiply \frac{1}{2} times \frac{4}{3} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\times 4}{2\times 3}.

$$\frac{5}{12}\left(\frac{2}{5}+\frac{4}{6}\right)$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$\frac{5}{12}\left(\frac{2}{5}+\frac{2}{3}\right)$$

Least common multiple of 5 and 3 is 15. Convert \frac{2}{5} and \frac{2}{3} to fractions with denominator 15.

$$\frac{5}{12}\left(\frac{6}{15}+\frac{10}{15}\right)$$

Since \frac{6}{15} and \frac{10}{15} have the same denominator, add them by adding their numerators.

$$\frac{5}{12}\times \frac{6+10}{15}$$

Add 6 and 10 to get 16.

$$\frac{5}{12}\times \frac{16}{15}$$

Multiply \frac{5}{12} times \frac{16}{15} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\times 16}{12\times 15}$$

Do the multiplications in the fraction \frac{5\times 16}{12\times 15}.

$$\frac{80}{180}$$

Reduce the fraction \frac{80}{180} to lowest terms by extracting and canceling out 20.

$$\frac{4}{9}$$

Factor

$$\frac{2 ^ {2}}{3 ^ {2}} = 0.4444444444444444$$