Hillwalking
Andrew walks along a flat path, then up and down a hill, then back along the path. Is it possible to work out how far he has walked?
Age
14 to 16
Challenge level
Problem
Andrew starts his walk along a flat path at an average speed of $4\text{km/h}$. Then he climbs a hill at an average speed of $3\text{km/h}$.
On reaching the top, he comes straight back down at a speed of $6\text{km/h}$, then goes back on the flat path at a speed of $4\text{km/h}$ again.
The walk takes him $2$ hours altogether. What is the total distance he walks?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: 8 km
Using ratio
Up the hill and down the hill was the same distance each way
Uphill: 6 km would take 2 hours Downhill: 6 km would take 1 hour
Total: 12 km would take 3 hours = 4 km per hour
Average speed for whole journey = 4 km per hour
4 km per hour $\times$ 2 hours = 8 km
Using algebra
If the distance along the flat was $f$ km and the distance up the hill was $h$ km, then since $Speed = \frac{Distance}{Time} \Rightarrow Time = \frac{Distance}{Speed}$ then the total time taken was: $\frac{f}{4} + \frac{h}{3} + \frac{h}{6} + \frac{f}{4} = \frac{f}{2} + \frac{h}{2} = 2$.
Then, doubling this gives $f+h = 4$.
Therefore the total distance walked is $2(f+h) = 2 \times 4 = 8\text{km}$