# How do you evaluate \frac { \frac { 1} { 2} - \frac { 2} { 3} } { \frac { 5} { 8} - \frac { 2} { 3} }?

Feb 28, 2017

$\frac{\frac{1}{2} - \frac{2}{3}}{\frac{5}{8} - \frac{2}{3}} = 4$

#### Explanation:

First of all let us simplify numerator and denominator separately.

Numerator is $\frac{1}{2} - \frac{2}{3}$ and as LCD of denominators is $6$

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

Similarly in denominator $\frac{5}{8} - \frac{2}{3}$, as GCD of denominators is $24$

$\frac{5}{8} - \frac{2}{3} = \frac{5 \times 3}{8 \times 3} - \frac{2 \times 8}{3 \times 8} = \frac{15}{24} - \frac{16}{24} = \frac{15 - 16}{24} = - \frac{1}{24}$

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 and hence

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

$\frac{\frac{1}{2} - \frac{2}{3}}{\frac{5}{8} - \frac{2}{3}} = \frac{- \frac{1}{6}}{- \frac{1}{24}} = - \frac{1}{6} \times \frac{24}{- 1} = \frac{1}{{\cancel{6}}^{1}} \times \frac{{\cancel{24}}^{4}}{1} = 4$

Mar 1, 2017

$4$

#### Explanation:

$\frac{\frac{1}{2} - \frac{2}{3}}{\frac{5}{8} - \frac{2}{3}}$

L.C.D. of 2,3=6
L.C.D. of 8,3=24

$\therefore = \frac{\frac{3 - 4}{6}}{\frac{15 - 16}{24}}$

$\therefore = \frac{\frac{- 1}{6}}{\frac{- 1}{24}}$

$\therefore = - \frac{1}{\cancel{6}} ^ 1 \times - {\cancel{24}}^{4} / 1$

$\therefore = 4$