Solve [[1/2+3/4-2)-(1/4-3/8-frac{1}{2})]*(frac{1}{3}-frac{2}{5}+frac{5}{3})-frac{3}{2}*(frac{5}{2}-frac{19}{6})

$$[ [ \frac { 1 } { 2 } + \frac { 3 } { 4 } - 2 ) - ( \frac { 1 } { 4 } - \frac { 3 } { 8 } - \frac { 1 } { 2 } ) ] \cdot ( \frac { 1 } { 3 } - \frac { 2 } { 5 } + \frac { 5 } { 3 } ) - \frac { 3 } { 2 } \cdot ( \frac { 5 } { 2 } - \frac { 19 } { 6 } )$$

Evaluate

$$\frac{4}{5}=0.8$$

Solution Steps

Least common multiple of 2 and 4 is 4. Convert \frac{1}{2} and \frac{3}{4} to fractions with denominator 4.

$$\left(\frac{2}{4}+\frac{3}{4}-2-\left(\frac{1}{4}-\frac{3}{8}-\frac{1}{2}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Since \frac{2}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

$$\left(\frac{2+3}{4}-2-\left(\frac{1}{4}-\frac{3}{8}-\frac{1}{2}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Add 2 and 3 to get 5.

$$\left(\frac{5}{4}-2-\left(\frac{1}{4}-\frac{3}{8}-\frac{1}{2}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Convert 2 to fraction \frac{8}{4}.

$$\left(\frac{5}{4}-\frac{8}{4}-\left(\frac{1}{4}-\frac{3}{8}-\frac{1}{2}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Since \frac{5}{4} and \frac{8}{4} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{5-8}{4}-\left(\frac{1}{4}-\frac{3}{8}-\frac{1}{2}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Subtract 8 from 5 to get -3.

$$\left(-\frac{3}{4}-\left(\frac{1}{4}-\frac{3}{8}-\frac{1}{2}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Least common multiple of 4 and 8 is 8. Convert \frac{1}{4} and \frac{3}{8} to fractions with denominator 8.

$$\left(-\frac{3}{4}-\left(\frac{2}{8}-\frac{3}{8}-\frac{1}{2}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Since \frac{2}{8} and \frac{3}{8} have the same denominator, subtract them by subtracting their numerators.

$$\left(-\frac{3}{4}-\left(\frac{2-3}{8}-\frac{1}{2}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Subtract 3 from 2 to get -1.

$$\left(-\frac{3}{4}-\left(-\frac{1}{8}-\frac{1}{2}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Least common multiple of 8 and 2 is 8. Convert -\frac{1}{8} and \frac{1}{2} to fractions with denominator 8.

$$\left(-\frac{3}{4}-\left(-\frac{1}{8}-\frac{4}{8}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Since -\frac{1}{8} and \frac{4}{8} have the same denominator, subtract them by subtracting their numerators.

$$\left(-\frac{3}{4}-\frac{-1-4}{8}\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Subtract 4 from -1 to get -5.

$$\left(-\frac{3}{4}-\left(-\frac{5}{8}\right)\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

The opposite of -\frac{5}{8} is \frac{5}{8}.

$$\left(-\frac{3}{4}+\frac{5}{8}\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Least common multiple of 4 and 8 is 8. Convert -\frac{3}{4} and \frac{5}{8} to fractions with denominator 8.

$$\left(-\frac{6}{8}+\frac{5}{8}\right)\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Since -\frac{6}{8} and \frac{5}{8} have the same denominator, add them by adding their numerators.

$$\frac{-6+5}{8}\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Add -6 and 5 to get -1.

$$-\frac{1}{8}\left(\frac{1}{3}-\frac{2}{5}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Least common multiple of 3 and 5 is 15. Convert \frac{1}{3} and \frac{2}{5} to fractions with denominator 15.

$$-\frac{1}{8}\left(\frac{5}{15}-\frac{6}{15}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Since \frac{5}{15} and \frac{6}{15} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{1}{8}\left(\frac{5-6}{15}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Subtract 6 from 5 to get -1.

$$-\frac{1}{8}\left(-\frac{1}{15}+\frac{5}{3}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Least common multiple of 15 and 3 is 15. Convert -\frac{1}{15} and \frac{5}{3} to fractions with denominator 15.

$$-\frac{1}{8}\left(-\frac{1}{15}+\frac{25}{15}\right)-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Since -\frac{1}{15} and \frac{25}{15} have the same denominator, add them by adding their numerators.

$$-\frac{1}{8}\times \frac{-1+25}{15}-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Add -1 and 25 to get 24.

$$-\frac{1}{8}\times \frac{24}{15}-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Reduce the fraction \frac{24}{15} to lowest terms by extracting and canceling out 3.

$$-\frac{1}{8}\times \frac{8}{5}-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Multiply -\frac{1}{8} times \frac{8}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-8}{8\times 5}-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Cancel out 8 in both numerator and denominator.

$$\frac{-1}{5}-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Fraction \frac{-1}{5} can be rewritten as -\frac{1}{5} by extracting the negative sign.

$$-\frac{1}{5}-\frac{3}{2}\left(\frac{5}{2}-\frac{19}{6}\right)$$

Least common multiple of 2 and 6 is 6. Convert \frac{5}{2} and \frac{19}{6} to fractions with denominator 6.

$$-\frac{1}{5}-\frac{3}{2}\left(\frac{15}{6}-\frac{19}{6}\right)$$

Since \frac{15}{6} and \frac{19}{6} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{1}{5}-\frac{3}{2}\times \frac{15-19}{6}$$

Subtract 19 from 15 to get -4.

$$-\frac{1}{5}-\frac{3}{2}\times \frac{-4}{6}$$

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

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

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

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

Cancel out 3 in both numerator and denominator.

$$-\frac{1}{5}-\frac{-2}{2}$$

Divide -2 by 2 to get -1.

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

The opposite of -1 is 1.

$$-\frac{1}{5}+1$$

Convert 1 to fraction \frac{5}{5}.

$$-\frac{1}{5}+\frac{5}{5}$$

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

$$\frac{-1+5}{5}$$

Add -1 and 5 to get 4.

$$\frac{4}{5}$$

Factor

$$\frac{2 ^ {2}}{5} = 0.8$$