Question

A jogger runs $6\text{miles per hour}$ faster downhill than uphill.

If the jogger can run $5\text{miles}$ downhill in the same time that it takes to run $2\text{miles}$ uphill find the jogging rate in.

Answer 1

Find the jogging rate:

The relation between the time, speed, and distance are:

$\text{time}=\frac{\text{Distance}}{\text{speed}}$

The jogger can run $5\text{miles}$ downhill at the same time that it takes to run $2\text{miles}$ uphill.

It follows that the distance of the uphill is $2\text{miles}$.

And the distance of the downhill is $5\text{miles}$.

Consider that $t$ be the time taken to jogger run both uphill and downhill.

A jogger runs $6\text{miles per hour}$ faster downhill than uphill.

Consider that $x\text{miles per hour}$ be the speed of the jogger run uphill.

And $\left(x+6\right)\text{mile per hour}$ be the speed of the jogger run downhill.

Step 1. Calculate the time taken to run uphill.

Substitute $\mathrm{Distance}=2\mathrm{and}\mathrm{speed}=x$ in the above formula:

$\begin{array}{rcl}\text{time}& =& \frac{\text{Distance}}{\text{speed}}\\ & =& \frac{2}{x}\text{hours}\end{array}$

Step 2. Calculate the time taken to run downhill.

Substitute $\mathrm{Distance}=5\mathrm{and}\mathrm{speed}=x+6$ in the above formula:

$\begin{array}{rcl}\text{time}& =& \frac{\text{Distance}}{\text{speed}}\\ & =& \frac{5}{x+6}\text{hours}\end{array}$

Step 3. Calculate the value of speed $\mathit{x}$ when joggers run uphill(Since the time is same for the both cases).

Put both the time expression equal to find the speed:

$\begin{array}{rcl}\frac{2}{x}& =& \frac{5}{x+6}\\ & \Rightarrow & 2\left(x+6\right)=5x\\ & \Rightarrow & 2x+12=5x\\ & \Rightarrow & 5x-2x=12\\ & \Rightarrow & 3x=12\\ & \Rightarrow & x=\frac{12}{3}\\ & \Rightarrow & x=4\text{miles per hour}\end{array}$

Step 4. Calculate speed when joggers run downhill.

$\begin{array}{rcl}x+6& =& 4+6\text{Since}x=4\text{miles per hour}\\ & =& 10\text{miles per hour}\end{array}$

Therefore the jogging rate when uphill will be $\mathbf{4}\mathbf{}\mathbf{\text{miles per hour}}$ and when downhill will be $\mathbf{\text{10 miles per hour}}$.