Overtake
Tom and Tim are travelling towards Glasgow, but leave at different times. If Tim overtakes Tom, how fast is he travelling?
Age
14 to 16
Challenge level
Problem
Tom left a motorway service station and travelled towards Glasgow at a steady speed of 60 mph. Tim left the same service station 10 minutes after Tom and travelled in the same direction at a steady speed, overtaking Tom after a further 1 hour 40 minutes.
At what speed did Tim travel?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 66 mph
| Tom starts | 60 miles per hour | 1 hour 50 minutes = 110 minutes |
|---|---|---|
| Tim starts 10 minutes later | speed unknown | 1 hour 40 minutes = 100 minutes |
Finding the distance
Tom: 60 miles per hour $\Rightarrow$ 110 miles in 110 minutes
Tim travels 110 miles in 100 minutes
11 miles in 10 minutes
66 miles in 1 hour
Using ratio
Both travel the same distance.
Times in ratio 110 : 100
$\therefore$ speeds in ratio 100 : 110 = 10 : 11 = 60 : 66
Using algebra
Say Tim's speed is $v$.
Tom and Tim travel the same distance, so $$\begin{align}
60\times1\frac56 &= v\times 1\frac46\\
\Rightarrow 110&=\frac53v\\
\Rightarrow 22 &= \frac13v\\
\Rightarrow 66 & =v \end{align}$$