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# Bike Shop

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Age 14 to 16

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**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.

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. ... How many steps are there on the escalator?