Question

Simplify

In the following exercises, use either method.

$\frac{\frac{3}{4}-\frac{2}{7}}{\frac{1}{2}+\frac{5}{14}}$

Short answer

$\frac{\frac{3}{4}-\frac{2}{7}}{\frac{1}{2}+\frac{5}{14}}=\frac{13}{24}$

Step-by-step solution

Step 1. Given information 

We have been given a complex rational expression $\frac{\frac{3}{4}-\frac{2}{7}}{\frac{1}{2}+\frac{5}{14}}$.

We have to simplify it.

Step 2. Find the LCD and add the fractions in the numerator. Find the LCD and add the fractions in the denominator. 

$$\begin{gathered} \frac{\frac{3}{4}-\frac{2}{7}}{\frac{1}{2}+\frac{5}{14}} \\ =\frac{\frac{3\mathit{\cdot }\mathit{7}}{4\mathit{\cdot }\mathit{7}}-\frac{2\mathit{\cdot }\mathit{4}}{7\mathit{\cdot }\mathit{4}}}{\frac{1\mathit{\cdot }\mathit{7}}{2\mathit{\cdot }\mathit{7}}+\frac{5}{14}} \end{gathered}$$

Step 3. Simplify the numerator and denominator. 

$$\begin{gathered} \frac{\frac{3\mathit{\cdot }\mathit{7}}{4\mathit{\cdot }\mathit{7}}-\frac{2\mathit{\cdot }\mathit{4}}{7\mathit{\cdot }\mathit{4}}}{\frac{1\mathit{\cdot }\mathit{7}}{2\mathit{\cdot }\mathit{7}}+\frac{5}{14}} \\ =\frac{\frac{21}{28}-\frac{8}{28}}{\frac{\mathit{7}}{14}+\frac{5}{14}} \end{gathered}$$

Step 4. Simplify the numerator and denominator, again 

$$\begin{gathered} \frac{\frac{21}{28}-\frac{8}{28}}{\frac{\mathit{7}}{14}+\frac{5}{14}} \\ =\frac{\frac{21-8}{28}}{\frac{\mathit{7}\mathit{+}\mathit{5}}{14}} \\ =\frac{\frac{13}{28}}{\frac{12}{14}} \end{gathered}$$

Step 5. Rewrite the complex rational expression as a division problem. Multiply the first times by the reciprocal of the second. 

$$\begin{gathered} \frac{\frac{13}{28}}{\frac{12}{14}} \\ =\frac{13}{28}\times \frac{14}{12} \\ =\frac{13}{24} \end{gathered}$$