Question

Chester rode his bike uphill $24\mathrm{miles}$ and then back downhill at $2\mathrm{mph}$ faster than his uphill. If it took him $2$ hours longer to ride uphill than downhill, what was his uphill rate?

Answer 1

Step 1: Find the time taken to ride uphill and downhill in terms of rate.

Let the rate of downhill be $r$ and rate of uphill be $r-2$.
The table below shows the rate and time:

	rate$\left(r\right)$ mph	time$\left(t\right)$ hour	distance$=\frac{r}{t}$ (miles)
Downhill	$r$	$\frac{24}{r}$	$24\mathrm{miles}$
Uphill	$r-2$	$\frac{24}{r-2}$	$24\mathrm{miles}$

Step 2: Finding the rate of uphill

it took him $2$ hours longer to ride uphill than downhill, the equation is
$$\begin{gathered} \Rightarrow \frac{24}{r-2}=\frac{24}{r}+2 \\ \Rightarrow \frac{24}{r-2}=\frac{24+2r}{r} \\ \Rightarrow 24r-48+2r^2-4r=24r \\ \Rightarrow r^2-2r-24=0 \\ \Rightarrow (r-6)(r+4)=0 \\ \Rightarrow r=6,-4 \end{gathered}$$
Checking $r=6\mathrm{mph}$ is a reasonable speed
Time take for downhill $\frac{24}{6}=4\mathrm{hr}$
Time take for uphill $\frac{24}{4}=6\mathrm{hr}$
Rate of uphill is $6-2=4\mathrm{mph}$

Hence, his uphill rate is $4\mathrm{mph}$.