Question

James and Lucas competed in climbing a mountain and coming back. James climbed at an average rate of $3$ mph and ran back at $4$ mph.

Lucas climbed at an average rate of $2$ mph and ran back at $5$ mph.

If the path from the bottom to the top of the mountain was $2$ miles, how much time did each take to finish and who was faster $?$

Answer 1

Step-1: Find the time taken by James in climbing up the mountain:

Let time taken by James in climbing up the mountain be ${\mathrm{t}}_1$:

The climbing rate of James is $3$ mph and the distance covered by him is $2$ miles:

${\mathrm{t}}_1=\frac{2}{3}\mathrm{hour}\because [\mathrm{Time}=\frac{\mathrm{Distance}}{\mathrm{Speed}}]$

Step-2: Find the time taken by James in coming back the mountain.

Let time taken by James in coming back the mountain be ${\mathrm{t}}_2$.

The coming back rate of James is $4$ mph and the distance covered by him is $2$ miles:

$\begin{array}{rcl}{\mathrm{t}}_2& =& \frac{2}{4}\\ & =& \frac{1}{2}\mathrm{hour}\end{array}$

Step-3: Find the total time taken by the James:

Since the total time taken by him is the sum of the time taken in climbing up and the time taken in coming back the mountain:

$\begin{array}{rcl}{\mathrm{t}}_1+{\mathrm{t}}_2& =& \frac{2}{3}+\frac{1}{2}\\ & =& \frac{7}{6}\mathrm{hour}\\ & =& \frac{7}{6}\times 60\\ & =& 70\mathrm{minutes}\end{array}$

Therefore the total time taken by him is $70\mathrm{minutes}.$

Step-4: Find the time taken by Lucas in climbing up the mountain

Let time taken by Lucas in climbing up the mountain be ${\mathrm{t}}_3$.

The climbing rate of Lucas is $2$ mph and the distance covered by him is $2$ miles:

${\mathrm{t}}_3=\frac{2}{2}=1\mathrm{hour}$ ,

Step-4: Find the time taken by Lucas in coming back the mountain

Let time taken by Lucas in coming back the mountain be ${\mathrm{t}}_4$

The coming back rate of Lucas is $5$ mph and the distance covered by him is $2$ miles:

$\begin{array}{rcl}{\mathrm{t}}_4& =& \frac{2}{5}\mathrm{hour}\end{array}$

Step-3: Find the total time taken by the Lucas:

Since the total time taken by him is the sum of the time taken in climbing up and the time taken in coming back the mountain:

$\begin{array}{rcl}{\mathrm{t}}_3+{\mathrm{t}}_4& =& 1+\frac{2}{5}\\ & =& \frac{7}{5}\mathrm{hours}\\ & =& \frac{7}{5}\times 60\\ & =& 84\mathrm{minutes}\end{array}$

Therefore the total time taken by him is $84\mathrm{minutes}.$

Hence, it took $\mathbf{70}\mathbf{}\mathbf{minutes}$ for James to finish and $\mathbf{84}\mathbf{}\mathbf{minutes}$ for Lucas to finish and James was faster.