Question

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

$\frac{4+\frac{\hspace{0.17em}4}{b-5\hspace{0.17em}}}{\frac{1}{b-5}+\frac{b}{4}\hspace{0.17em}}$

Short answer

The simplified expression is$\frac{16}{b-1}$.

Step-by-step solution

Step 1. Given Information.

We have:

$\frac{4+\frac{\hspace{0.17em}4}{b-5\hspace{0.17em}}}{\frac{1}{b-5}+\frac{b}{4}\hspace{0.17em}}$

Step 2. Simplify the numerator. 

The numerator is $4+\frac{\hspace{0.17em}4}{b-5\hspace{0.17em}}$.

Apply fraction rule: $a+\frac{b}{c}=\frac{ac+b}{c}$

$$\begin{gathered} =\frac{4\left(b-5\right)+4}{b-5} \\ =\frac{4b-16}{b-5} \end{gathered}$$

Step 3. Simplify the denominator. 

The denominator is $\frac{1}{b-5}+\frac{b}{4}$.

The least common divisor is $4\left(b-5\right)$.

Adjust the fractions based on the LCM:

$$\begin{gathered} =\frac{4}{4\left(b-5\right)}+\frac{b\left(b-5\right)}{4\left(b-5\right)} \\ =\frac{4+b\left(b-5\right)}{4\left(b-5\right)} \\ =\frac{4+b^2-5b}{4\left(b-5\right)} \end{gathered}$$

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

We have:

$\frac{4+\frac{\hspace{0.17em}4}{b-5\hspace{0.17em}}}{\frac{1}{b-5}+\frac{b}{4}\hspace{0.17em}}=\frac{\frac{4b-16}{b-5}}{\frac{4+b^2-5b}{4\left(b-5\right)}}$

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

localid="1662393230322" $$\begin{gathered} =\frac{\left(4b-16\right)·\hspace{0.17em}4\left(b-5\right)}{\left(b-5\right)\left(4+b^2-5b\right)} \\ =\frac{4\left(b-4\right)·\hspace{0.17em}4\left(b-5\right)}{\left(b-5\right)\left(b-1\right)\left(b-4\right)} \\ =\frac{16}{b-1} \end{gathered}$$