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Maths olympiad question · P6★★★★ Expert

Trains and bridges: olympiad question 10

An expert question on trains and bridges for P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P5 and P6 qualifying rounds and finals; harder ones have two trains passing, or a train wholly inside a tunnel.

The question

A railway tunnel is 560 m long.

A worker is in it, 210 m from the entrance, when a train heads for the entrance.

He runs for it at once: running to the entrance, he gets out just as the train's front reaches the entrance; running to the exit (the far end), he gets out just as the train's front reaches the exit.

Both keep steady speeds.

When the worker starts running, how far is the train's front from the entrance?

Show the hints (3)
  1. Hint 1: Compare the two ways: how much further is it to run to the exit than to the entrance?
  2. Hint 2: Picture him running for the exit: once he has run as far as the entrance is, the train is just at the entrance. While he runs the extra stretch, how far does the train go?
  3. Hint 3: While the worker runs the extra 560 − 2 × 210 = 140 m, the train goes the whole tunnel, 560 m: the train is 560 ÷ 140 times as fast.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.To the entrance is 210 m; to the exit, 560 − 210 = 350 m: 350 − 210 = 140 m further.
  2. 2.Running for the exit, after his first 210 m the train is just at the entrance (as when he runs the other way). Then he runs the last 140 m while the train goes from entrance to exit, 560 m.
  3. 3.In the same time the train goes 560 m and the worker 140 m, so the train is 560 ÷ 140 = 4 times as fast.
  4. 4.In the time the worker runs 210 m to the entrance, the train reaches it, going 210 × 4 = 840 m: the front was 840 m from the entrance.
  5. 5.Check: the train has 840 m to the entrance and 1400 m to the exit; 840 : 1400 = 210 : 350, the same ratio as the worker's two runs ✓
  6. ⚠️Common mistake: taking the extra as 350 m, or dividing the tunnel by the worker's distance. Compare the same stretch of time: his extra 140 m against the train's whole tunnel.

Answer: 840 m

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