Solve [6/5*(1/2-1/3)+3/2:(frac{3}{2}+frac{9}{4})]*frac{1}{3}-(frac{1}{5}+frac{3}{2})

$$[ \frac { 6 } { 5 } \cdot ( \frac { 1 } { 2 } - \frac { 1 } { 3 } ) + \frac { 3 } { 2 } : ( \frac { 3 } { 2 } + \frac { 9 } { 4 } ) ] \cdot \frac { 1 } { 3 } - ( \frac { 1 } { 5 } + \frac { 3 } { 2 } )$$

Evaluate

$$-\frac{3}{2}=-1.5$$

Solution Steps

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

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

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

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

Subtract 2 from 3 to get 1.

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

Multiply \frac{6}{5} times \frac{1}{6} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 6 in both numerator and denominator.

$$\left(\frac{1}{5}+\frac{\frac{3}{2}}{\frac{3}{2}+\frac{9}{4}}\right)\times \frac{1}{3}-\left(\frac{1}{5}+\frac{3}{2}\right)$$

Least common multiple of 2 and 4 is 4. Convert \frac{3}{2} and \frac{9}{4} to fractions with denominator 4.

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

Since \frac{6}{4} and \frac{9}{4} have the same denominator, add them by adding their numerators.

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

Add 6 and 9 to get 15.

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

Divide \frac{3}{2} by \frac{15}{4} by multiplying \frac{3}{2} by the reciprocal of \frac{15}{4}.

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

Multiply \frac{3}{2} times \frac{4}{15} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{3\times 4}{2\times 15}.

$$\left(\frac{1}{5}+\frac{12}{30}\right)\times \frac{1}{3}-\left(\frac{1}{5}+\frac{3}{2}\right)$$

Reduce the fraction \frac{12}{30} to lowest terms by extracting and canceling out 6.

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

Since \frac{1}{5} and \frac{2}{5} have the same denominator, add them by adding their numerators.

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

Add 1 and 2 to get 3.

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

Multiply \frac{3}{5} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 3 in both numerator and denominator.

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

Least common multiple of 5 and 2 is 10. Convert \frac{1}{5} and \frac{3}{2} to fractions with denominator 10.

$$\frac{1}{5}-\left(\frac{2}{10}+\frac{15}{10}\right)$$

Since \frac{2}{10} and \frac{15}{10} have the same denominator, add them by adding their numerators.

$$\frac{1}{5}-\frac{2+15}{10}$$

Add 2 and 15 to get 17.

$$\frac{1}{5}-\frac{17}{10}$$

Least common multiple of 5 and 10 is 10. Convert \frac{1}{5} and \frac{17}{10} to fractions with denominator 10.

$$\frac{2}{10}-\frac{17}{10}$$

Since \frac{2}{10} and \frac{17}{10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2-17}{10}$$

Subtract 17 from 2 to get -15.

$$\frac{-15}{10}$$

Reduce the fraction \frac{-15}{10} to lowest terms by extracting and canceling out 5.

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

Factor

$$-\frac{3}{2} = -1\frac{1}{2} = -1.5$$