Hiking the Hill
Problem
Sarah walks uphill at an average of 2 miles per hour, and downhill at an average of 4 miles per hour.
She goes for a 6 mile hike up a hill, and then returns downhill by the same route.
What is Sarah's average speed for the whole journey?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $2\frac23$ miles per hour
Total distance and total time
Uphill: $6$ miles, $2$ miles/hour takes $3$ hours
Downhill: $6$ miles, $4$ miles/hour takes $1\frac12$ hours
Average speed: $\dfrac{6+6\text{ miles}}{3 + 1\frac12\text{ hours}} = \dfrac{12}{4\frac12}$ miles per hour
$=\frac{24}{9}=\frac83=2\frac23$ miles per hour
Weighted average
Uphill: 2 miles per hour
Downhill: 4 miles per hour
Takes twice as long to go up as down
$\Rightarrow$ spends twice as long going up as down
$\Rightarrow$ spends twice as long travelling at 2 miles per hour as at 4 miles per hour
$\therefore$ average speed $=\frac{2+2+4}{3}=\frac83 = 2\frac23$ miles per hour