# How do you evaluate (5/9 - 3)/(5/6) -:3/2 -: 3/4?

Feb 27, 2017

$\frac{\frac{5}{9} - 3}{\frac{5}{6}} \div \frac{3}{2} \div \frac{3}{4} = - \frac{352}{135}$

#### Explanation:

Dividing a fraction by a fraction, whether as in $\frac{\frac{a}{b}}{\frac{c}{d}}$ or $\frac{a}{b} \div \frac{c}{d}$, is equivalent to multiplying by the reciprocal or multiplicative inverse of denominator (or the fraction written after sign ''$\div$ and hence

$\frac{\frac{a}{b}}{\textcolor{red}{\frac{c}{d}}} = \frac{a}{b} \textcolor{red}{\div \frac{c}{d}} = \frac{a}{b} \textcolor{red}{\times \left(\frac{d}{c}\right)}$

Now $\frac{5}{9} - 3 = \frac{5}{9} - \frac{3 \times 9}{1 \times 9} = \frac{5}{9} - \frac{27}{9} = \frac{5 - 27}{9} = - \frac{22}{9}$

Hence $\frac{\frac{5}{9} - 3}{\frac{5}{6}} \div \frac{3}{2} \div \frac{3}{4}$

= $\frac{- \frac{22}{9}}{\frac{5}{6}} \div \frac{3}{2} \div \frac{3}{4}$

= $- \frac{22}{9} \times \frac{6}{5} \times \frac{2}{3} \times \frac{4}{3}$

= $- \frac{22}{9} \times \frac{2 \cancel{6}}{5} \times \frac{2}{1 \cancel{3}} \times \frac{4}{3}$

= $- \frac{22 \times 2 \times 2 \times 4}{9 \times 5 \times 3}$

= $- \frac{352}{135}$