Question

Sarah ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. If r represents Sarah’s speed when she ran, then her running time is modeled by the expression $\frac{8}{r}$and her biking time is modeled by the expression $\frac{24}{r+4}$. Add the rational expressions $\frac{8}{r}+\frac{24}{r+4}$to get an expression for the total amount of time Sarah ran and biked.

Short answer

The added expression is$\frac{32r+32}{r\left(r+4\right)}$.

Step-by-step solution

Step 1. Given information.

Consider the given expression,

$\frac{8}{r},\mathrm{and}\frac{24}{r+4}$

Step 2. Calculation.

Since the denominator are different in both expressions. Find the LCD of expressions.

$\frac{8}{r}\times \frac{r+4}{r+4}+\frac{24}{r+4}\times \frac{r}{r}=\frac{8\left(r+4\right)}{r\left(r+4\right)}+\frac{24r}{r\left(r+4\right)}$

Now, add the expressions.

$\begin{array}{rcl}\frac{8\left(r+4\right)}{r\left(r+4\right)}+\frac{24r}{r\left(r+4\right)}& =& \frac{8r+32+24r}{r\left(r+4\right)}\\ & =& \frac{32r+32}{r\left(r+4\right)}\end{array}$