Question

Simplify the complex rational expression by writing it as a division:

$\frac{{\displaystyle \frac{1}{2}}+{\displaystyle \frac{2}{3}}}{{\displaystyle \frac{5}{6}}+{\displaystyle \frac{1}{12}}}$

Short answer

$\frac{14}{11}$

Step-by-step solution

Step 1. Given information.

The following rational expression is given:

$\frac{{\displaystyle \frac{1}{2}+\frac{2}{3}}}{{\displaystyle \frac{5}{6}+\frac{1}{12}}}$

Step 2. Simplification of the rational expression.

Find the LCD and add the numerator and the denominator terms:

$\frac{{\displaystyle \frac{1.3}{2.3}}+{\displaystyle \frac{2.2}{3.2}}}{{\displaystyle \frac{5.2}{6.2}}+{\displaystyle \frac{1}{12}}}$

Simplify the numerator and the denominator:

$$\begin{gathered} \frac{{\displaystyle \frac{3}{6}}+{\displaystyle \frac{4}{6}}}{{\displaystyle \frac{10}{12}}+{\displaystyle \frac{1}{12}}} \\ \frac{{\displaystyle \frac{7}{6}}}{{\displaystyle \frac{11}{12}}} \end{gathered}$$

Rewrite as a division problem:

$\frac{7}{6}÷\frac{11}{12}$

Rewrite as the product of the first with the reciprocal of the second:

$\frac{7}{6}.\frac{12}{11}$

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

$$\begin{gathered} \frac{7}{6}.\frac{2.6}{11} \\ \frac{14}{11} \end{gathered}$$