Question

Simplify the complex rational expression by writing it as division:

$\frac{{\displaystyle \frac{3}{4}}-{\displaystyle \frac{1}{3}}}{{\displaystyle \frac{1}{8}}+{\displaystyle \frac{5}{6}}}$

Short answer

$\frac{10}{23}$

Step-by-step solution

Step 1. Given information.

The following complex rational expression is given:

$\frac{{\displaystyle \frac{3}{4}-\frac{1}{3}}}{{\displaystyle \frac{1}{8}+\frac{5}{6}}}$

Step 2. Simplification of the complex rational expression.

Find out the LCD, subtract the numerator terms and add the denominator terms:

$\frac{{\displaystyle \frac{3.3}{4.3}}-{\displaystyle \frac{1.4}{3.4}}}{{\displaystyle \frac{1.3}{8.3}}+{\displaystyle \frac{5.4}{6.4}}}$

Simplify the numerator and the denominator:

$$\begin{gathered} \frac{{\displaystyle \frac{{\displaystyle 9}}{12}-\frac{4}{12}}}{{\displaystyle \frac{3}{24}+\frac{20}{24}}} \\ \frac{{\displaystyle \frac{5}{12}}}{{\displaystyle \frac{23}{24}}} \end{gathered}$$

Write as a division problem:

$\frac{5}{12}÷\frac{23}{24}$

Write as a product of the first with the reciprocal of the second:

$\frac{5}{12}.\frac{24}{23}$

Factor, cancel the common factors from the numerator and the denominator and then simplify:

$$\begin{gathered} \frac{5}{12}.\frac{2.12}{23} \\ \frac{10}{23} \end{gathered}$$