Solve 4/9*[11/2-(3/5+2/15)*(-frac{3}{22})+frac{1}{4}]+[2-(frac{1}{6}+frac{1}{3})]*(-frac{5}{2})

$$\frac { 4 } { 9 } \cdot [ \frac { 11 } { 2 } - ( \frac { 3 } { 5 } + \frac { 2 } { 15 } ) \cdot ( - \frac { 3 } { 22 } ) + \frac { 1 } { 4 } ] + [ 2 - ( \frac { 1 } { 6 } + \frac { 1 } { 3 } ) ] \cdot ( - \frac { 5 } { 2 } )$$

Evaluate

$$-\frac{23}{20}=-1.15$$

Solution Steps

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

$$\frac{4}{9}\left(\frac{11}{2}-\left(\frac{9}{15}+\frac{2}{15}\right)\left(-\frac{3}{22}\right)+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Since \frac{9}{15} and \frac{2}{15} have the same denominator, add them by adding their numerators.

$$\frac{4}{9}\left(\frac{11}{2}-\frac{9+2}{15}\left(-\frac{3}{22}\right)+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Add 9 and 2 to get 11.

$$\frac{4}{9}\left(\frac{11}{2}-\frac{11}{15}\left(-\frac{3}{22}\right)+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Multiply \frac{11}{15} times -\frac{3}{22} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4}{9}\left(\frac{11}{2}-\frac{11\left(-3\right)}{15\times 22}+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Do the multiplications in the fraction \frac{11\left(-3\right)}{15\times 22}.

$$\frac{4}{9}\left(\frac{11}{2}-\frac{-33}{330}+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Reduce the fraction \frac{-33}{330} to lowest terms by extracting and canceling out 33.

$$\frac{4}{9}\left(\frac{11}{2}-\left(-\frac{1}{10}\right)+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

The opposite of -\frac{1}{10} is \frac{1}{10}.

$$\frac{4}{9}\left(\frac{11}{2}+\frac{1}{10}+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Least common multiple of 2 and 10 is 10. Convert \frac{11}{2} and \frac{1}{10} to fractions with denominator 10.

$$\frac{4}{9}\left(\frac{55}{10}+\frac{1}{10}+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Since \frac{55}{10} and \frac{1}{10} have the same denominator, add them by adding their numerators.

$$\frac{4}{9}\left(\frac{55+1}{10}+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Add 55 and 1 to get 56.

$$\frac{4}{9}\left(\frac{56}{10}+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Reduce the fraction \frac{56}{10} to lowest terms by extracting and canceling out 2.

$$\frac{4}{9}\left(\frac{28}{5}+\frac{1}{4}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Least common multiple of 5 and 4 is 20. Convert \frac{28}{5} and \frac{1}{4} to fractions with denominator 20.

$$\frac{4}{9}\left(\frac{112}{20}+\frac{5}{20}\right)+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Since \frac{112}{20} and \frac{5}{20} have the same denominator, add them by adding their numerators.

$$\frac{4}{9}\times \frac{112+5}{20}+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Add 112 and 5 to get 117.

$$\frac{4}{9}\times \frac{117}{20}+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Multiply \frac{4}{9} times \frac{117}{20} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\times 117}{9\times 20}+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Do the multiplications in the fraction \frac{4\times 117}{9\times 20}.

$$\frac{468}{180}+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

Reduce the fraction \frac{468}{180} to lowest terms by extracting and canceling out 36.

$$\frac{13}{5}+\left(2-\left(\frac{1}{6}+\frac{1}{3}\right)\right)\left(-\frac{5}{2}\right)$$

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

$$\frac{13}{5}+\left(2-\left(\frac{1}{6}+\frac{2}{6}\right)\right)\left(-\frac{5}{2}\right)$$

Since \frac{1}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$\frac{13}{5}+\left(2-\frac{1+2}{6}\right)\left(-\frac{5}{2}\right)$$

Add 1 and 2 to get 3.

$$\frac{13}{5}+\left(2-\frac{3}{6}\right)\left(-\frac{5}{2}\right)$$

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

$$\frac{13}{5}+\left(2-\frac{1}{2}\right)\left(-\frac{5}{2}\right)$$

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

$$\frac{13}{5}+\left(\frac{4}{2}-\frac{1}{2}\right)\left(-\frac{5}{2}\right)$$

Since \frac{4}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{13}{5}+\frac{4-1}{2}\left(-\frac{5}{2}\right)$$

Subtract 1 from 4 to get 3.

$$\frac{13}{5}+\frac{3}{2}\left(-\frac{5}{2}\right)$$

Multiply \frac{3}{2} times -\frac{5}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{13}{5}+\frac{3\left(-5\right)}{2\times 2}$$

Do the multiplications in the fraction \frac{3\left(-5\right)}{2\times 2}.

$$\frac{13}{5}+\frac{-15}{4}$$

Fraction \frac{-15}{4} can be rewritten as -\frac{15}{4} by extracting the negative sign.

$$\frac{13}{5}-\frac{15}{4}$$

Least common multiple of 5 and 4 is 20. Convert \frac{13}{5} and \frac{15}{4} to fractions with denominator 20.

$$\frac{52}{20}-\frac{75}{20}$$

Since \frac{52}{20} and \frac{75}{20} have the same denominator, subtract them by subtracting their numerators.

$$\frac{52-75}{20}$$

Subtract 75 from 52 to get -23.

$$-\frac{23}{20}$$

Factor

$$-\frac{23}{20} = -1\frac{3}{20} = -1.15$$