Question

In the following exercises, simplify each complex rational expression by writing it as division.

$\frac{\frac{1}{2}+\frac{3}{4}}{\frac{3}{5}+\frac{7}{10}}$

Short answer

The required solution is$\frac{25}{26}$.

Step-by-step solution

Step 1. Given Information.  

We have:

$\frac{\frac{1}{2}+\frac{3}{4}}{\frac{3}{5}+\frac{7}{10}}$

Step 2. Simplify the numerator: 

The numerator is $\frac{1}{2}+\frac{3}{4}$.

The least common multiplier of $2,4$is $4.$

$$\begin{gathered} =\frac{2}{4}+\frac{3}{4} \\ =\frac{2+3}{4} \\ =\frac{5}{4} \end{gathered}$$

Step 3. Simplify the denominator. 

Here, the denominator is $\frac{3}{5}+\frac{7}{10}$.

The least common multiplier of $5,10$is $50.$

$$\begin{gathered} =\frac{6}{10}+\frac{7}{10} \\ =\frac{6+7}{10} \\ =\frac{13}{10} \end{gathered}$$

Step 4. Substitute the simplified numerator and denominator into the original expression. 

We have:

$\frac{\frac{1}{2}+\frac{3}{4}}{\frac{3}{5}+\frac{7}{10}}=\frac{\frac{5}{4}}{\frac{13}{10}}$

Apply fraction rule: $\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a·\hspace{0.17em}d}{b·\hspace{0.17em}c}$

$$\begin{gathered} =\frac{5·\hspace{0.17em}10}{4·\hspace{0.17em}13} \\ =\frac{25}{26} \end{gathered}$$