A hard question on trains and bridges for P5–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
An express and a slow train run on parallel tracks at steady speeds.
The express is 130 m long and the slow train 74 m.
Going the same way, from the express's front reaching the slow train's back to the express's back passing the slow train's front takes 17 seconds.
Going towards each other, from their fronts meeting to their backs parting takes 6 seconds.
How many metres a second does the express travel?
Show the hints (3)
💡Hint 1: In both cases, how far do the trains have to travel past each other?
💡Hint 2: Head-on, they cover 204 ÷ 6 m a second between them; same way, the express gains 204 ÷ 17 m a second.
💡Hint 3: The two speeds add up to 34 and differ by 12.
Worked solution
Have a go first, then read the solution step by step.
1.Head-on: from fronts meeting to backs parting, the fronts of the express and the slow train travel both lengths between them: 130 + 74 = 204 m.
2.So the sum of speeds = 204 ÷ 6 = 34 m/s.
3.Same way: from the express's front reaching the slow train's back to its back passing the slow train's front, the express travels both lengths more than the slow train: again 204 m.
4.So the difference = 204 ÷ 17 = 12 m/s.
5.Give the slow train the missing 12 and they are equal: express = (34 + 12) ÷ 2 = 23 m/s.
⚠️Common mistake: using only the express's length when overtaking, or mixing up which time gives the sum and which the difference.
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