Question

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

$\frac{\frac{2}{3}-\frac{1}{9}}{\frac{3}{4}+\frac{5}{6}}$

Short answer

The required solutions is$\frac{20}{57}$.

Step-by-step solution

Step 1. Given Information.   

We have:

$\frac{\frac{2}{3}-\frac{1}{9}}{\frac{3}{4}+\frac{5}{6}}$

Step 2. Simplify the numerator:  

The numerator is $\frac{2}{3}-\frac{1}{9}$.

The least common multiplier of $3,9$is $9.$

Adjust the fractions based on the LCM:

$$\begin{gathered} =\frac{6}{9}-\frac{1}{9} \\ =\frac{6-1}{9} \\ =\frac{5}{9} \end{gathered}$$

Step 3. Simplify the denominator.  

The denominator is $\frac{3}{4}+\frac{5}{6}$.

The least common multiplier of $4,6$is $12.$

Adjust the fractions based on the LCM:

$$\begin{gathered} =\frac{9}{12}+\frac{10}{12} \\ =\frac{9+10}{12} \\ =\frac{19}{12} \end{gathered}$$

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

We have:

$\frac{\frac{2}{3}-\frac{1}{9}}{\frac{3}{4}+\frac{5}{6}}=\frac{\frac{5}{9}}{\frac{19}{12}}$

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

$$\begin{gathered} =\frac{5·\hspace{0.17em}12}{9·\hspace{0.17em}19} \\ =\frac{20}{57} \end{gathered}$$