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On the Road

Four vehicles travelled on a road with constant velocities. The car overtook the scooter at 12 o'clock, then met the bike at 14.00 and the motorcycle at 16.00. The motorcycle met the scooter at 17.00 then it overtook the bike at 18.00. At what time did the bike and the scooter meet?

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There and Back

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?

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Escalator

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?

Speeding Boats

Stage: 4 Challenge Level: Challenge Level:1

Why do this problem?

This problem offers a chance to think about the links between distance and speed. By solving parts of the problem mentally, there is an opportunity for students to use their visualisation skills and then discuss the different ways of thinking about the problem.

Possible approach

Explain the scenario to the class - the boats start from opposite ends and always turn when they get to the other end without slowing down. Ask them to consider whereabouts on the lake the boats will meet for the first time if they travel at the same speed, and where they will meet for the second time, third time and so on. Encourage them to picture it without writing anything down if they can. (It is important to highlight that we're interested in the boats being in the same position on the lake, regardless of which way they are travelling, so a meeting point could be when they cross each other or when one overtakes the other.)

Allow students to share in pairs or small groups their answer to this first question and their different ways of thinking about it. Then set them the task of investigating how the meeting points change if the ratios of the speeds of the boats are changed.

Encourage groups to share their ways of approaching the problem, and give them a chance to describe their way of seeing it. Sharing different strategies for recording can also be useful. Then present the problem of finding the ratio of speeds and length of the lake from the three statements given in the problem. The natural extension to this is to experiment with different distances and see how it affects the ratio of speeds.

Key questions

How do you picture what happens for different ratios of speeds?
How can you record what you are picturing?

Possible extension

Around and Back is a harder problem about ratio of speeds which can be solved using some of the same techniques.

Possible support

Circuit Training might be a simpler context to think about as an introduction to this problem.
It is well worth looking at the problem Bus Stop and its solutions to see a variety of approaches to this sort of problem.