To run or not to run?
Problem
An athlete covers three consecutive miles by walking the first mile, running the second mile and cycling the third. He runs twice as fast as he walks, and he cycles one and a half times as fast as he runs. He takes ten minutes longer than he would do if he cycled the three miles. How long does he take by walking, running and cycling?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Getting Started
Let the athlete take x minutes to cycle one mile.
Student Solutions
Answer: 22 minutes
In terms of the walking time
walk run cycle
$\rightarrow2\times$ as fast $\rightarrow\frac32\times$ as fast
$\rightarrow\frac12$ the time $\rightarrow\frac23$ the time
$w$ $\tfrac12w$ $\tfrac23\times\tfrac12w=\tfrac13w$
$\begin{align} w + \tfrac12w + \tfrac13w &= 3\times \tfrac13w+10\\
\Rightarrow \tfrac56w&=10\\
\Rightarrow w&=12\end{align}$
Total time: $12+\tfrac12 12 + \tfrac13 12 = 22$ minutes
In terms of the cycling time
Let the athlete take $x$ minutes to cycle one mile.
Therefore he takes $\frac{3}{2}x$ minutes to run one mile and $3x$ minutes to walk one mile.
So $3x+\frac{3}{2}x+x=3x+10$, i.e. $x=4$.
Therefore the cyclist takes $12$ minutes to walk the first mile, $6$ minutes to run the second mile and $4$ minutes to cycle the third mile. So the total time taken to walk, run and cycle the three miles is $22$ minutes.