Simplify

$-\frac12$

Explanation

To write the numerator as a single fraction, we build \(-\frac14\) and \(\frac25\) to have an LCD of 20, and then add.  To write the denominator as a single fraction, we build \(\frac12\) and \(\frac45\) to have an LCD of 10, and subtract.

The LCD for the numerator is 20.  Build each fraction so that each has a denominator of 20.  The LCD for the denominator is 10.  Build each fraction so that each has a denominator of 10.

$\large{\frac{-\frac14 + \frac25}{\frac12 - \frac45} = \frac{-\frac14 \cdot \color{red}{\frac55} + \frac25 \cdot \color{blue}{\frac44}}{\frac12 \cdot \color{green}{\frac55} - \frac45 \cdot \color{brown}{\frac22}}}$

Multiply in the numerator.  Multiply in the denominator.

$\large{=\frac{-\frac{5}{20} + \frac{8}{20}}{\frac{5}{10} - \frac{8}{10}}}$

In the numerator of the complex fraction, add the fractions.

$\large{= \frac{\frac{3}{20}}{-\frac{3}{10}}}$

Write the division indicated by the main fraction by using a ÷ symbol.

$= \frac{3}{20} ÷ \left(-\frac{3}{10}\right)$

Multiply the first fraction by the reciprocal of \(-\frac{3}{10}\), which is \(-\frac{10}{3}\)

$= \frac{3}{20} \left(-\frac{10}{3}\right)$

The product of two fractions with unlike signs is negative.  Multiply the numerators.  Multiply the denominators.

$= - \frac{3 \cdot 10}{20 \cdot 3}$

To simplify, factor 20 as 2 • 10.  Then remove the common factors of 3 and 10 from the numerator and denominator.

$\require{cancel} = - \frac{\cancel{3}^1 \cdot \cancel{10}^1}{2 \cdot \cancel{10}_1 \cdot \cancel{3}_1}$

Multiply the remaining factors in the numerator.  Multiply the remaining factors in the denominator.

$= -\frac12$