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Maths olympiad question · P6★★★★ Expert

Trains and bridges: olympiad question 8

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 passenger train runs at a steady speed.

It crosses a bridge 700 m long: from its front reaching the bridge to its back leaving it takes 45 s.

Then it goes through a tunnel 2156 m long: from its back entering to its front reaching the far end — all that time wholly inside — takes 74 s.

Out of the tunnel it meets a goods train 402 m long coming the other way: from fronts meeting to backs parting takes 23 s.

Both trains keep steady speeds.

How many metres a second does the goods train go?

Show the hints (3)
  1. Hint 1: Crossing the bridge, the train goes bridge + its own length. Wholly inside the tunnel, how far does it go?
  2. Hint 2: Add the two distances and the train's length cancels: find the passenger train's speed first, then its length.
  3. Hint 3: Once you have the passenger train's speed and length, look at the meeting: from fronts meeting to backs parting, how far do the two go together — and how far a second?

Worked solution

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

  1. 1.The bridge: front on to back off, the train goes bridge + length = 700 + length, in 45 s.
  2. 2.Wholly inside the tunnel: back in to front out, it goes tunnel − length = 2156 − length, in 74 s.
  3. 3.Added, the length cancels: in 45 + 74 = 119 s it went 700 + 2156 = 2856 m, so 2856 ÷ 119 = 24 m a second; its length = 24 × 45 − 700 = 380 m.
  4. 4.The meeting: fronts meeting to backs parting, the two together go both lengths, 380 + 402 = 782 m, in 23 s, so their speeds add to 782 ÷ 23 = 34 m a second.
  5. 5.The goods train does 34 − 24 = 10 m a second.
  6. 6.Check: 24 × 45 = 1080 = 700 + 380 ✓; 24 × 74 = 1776 = 2156 − 380 ✓; (24 + 10) × 23 = 782 = 380 + 402 ✓
  7. ⚠️Common mistake: treating "wholly inside" as tunnel + length; or counting only the goods train's length when they pass. Head-on, together they cover both lengths.

Answer: 10 m a second

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