Solve (3/5div9/20*1/3)+8/5phi-frac{4}{3}

$$( \frac { 3 } { 5 } \div \frac { 9 } { 20 } \times \frac { 1 } { 3 } ) + \frac { 8 } { 5 } \phi - \frac { 4 } { 3 }$$

Evaluate

$$\frac{8ϕ}{5}-\frac{8}{9}$$

Solution Steps

Divide \frac{3}{5} by \frac{9}{20} by multiplying \frac{3}{5} by the reciprocal of \frac{9}{20}.

$$\frac{3}{5}\times \frac{20}{9}\times \frac{1}{3}+\frac{8}{5}ϕ-\frac{4}{3}$$

Multiply \frac{3}{5} times \frac{20}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\times 20}{5\times 9}\times \frac{1}{3}+\frac{8}{5}ϕ-\frac{4}{3}$$

Do the multiplications in the fraction \frac{3\times 20}{5\times 9}.

$$\frac{60}{45}\times \frac{1}{3}+\frac{8}{5}ϕ-\frac{4}{3}$$

Reduce the fraction \frac{60}{45} to lowest terms by extracting and canceling out 15.

$$\frac{4}{3}\times \frac{1}{3}+\frac{8}{5}ϕ-\frac{4}{3}$$

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

$$\frac{4\times 1}{3\times 3}+\frac{8}{5}ϕ-\frac{4}{3}$$

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

$$\frac{4}{9}+\frac{8}{5}ϕ-\frac{4}{3}$$

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

$$\frac{4}{9}+\frac{8}{5}ϕ-\frac{12}{9}$$

Since \frac{4}{9} and \frac{12}{9} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4-12}{9}+\frac{8}{5}ϕ$$

Subtract 12 from 4 to get -8.

$$-\frac{8}{9}+\frac{8}{5}ϕ$$

Factor

$$\frac{8\left(9ϕ-5\right)}{45}$$

Solution Steps

Factor out \frac{4}{45}.

$$\frac{4\left(-10+18ϕ\right)}{45}$$

Consider 5+18ϕ-15. Multiply and combine like terms.

$$18ϕ-10$$

Consider 18ϕ-10. Factor out 2.

$$2\left(9ϕ-5\right)$$

Rewrite the complete factored expression.

$$\frac{8\left(9ϕ-5\right)}{45}$$