Bike shop
Problem
I walked to the bike shop at a speed of 3 miles per hour, and cycled back along the same route at a speed of 12 miles per hour.
What was my average speed for the round trip?
Student Solutions
Answer: $4.8$ miles per hour
Using ratio
At 3 mph, the journey takes 4 times longer than at 12 mph.
Return journey: one unit of time, 12 mph
Outward journey: 4 units of time, 3 mph, 3 mph, 3 mph, 3mph
Average speed = (12 + 3 + 3 + 3 + 3)$\div$5 = 4.8
Using algebra
Suppose the distance to and from the bike shop is $x$ miles.
Then the time taken on the journey there is $\frac{x}{3}$ hours, and the time taken on the journey back is $\frac{x}{12}$ hours.
So altogether a distance of $2x$ miles is travelled in $\frac{x}{3} + \frac{x}{12} = \frac{5x}{12}$ hours.
So the average speed is $2x \div \frac{5x}{12} = 4.8$ miles per hour.