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# Bike Shop

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Age 14 to 16

ShortChallenge Level

- Problem
- 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.

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.