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Maths olympiad question · P6★★★★ Expert

Trains and bridges: olympiad question 9

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

Beside a railway runs a straight cycle track.

Ka-lok and Tsz-ching cycle on it at 3 m a second each;

Ka-lok goes the train's way and Tsz-ching the other way, so the two are heading for each other.

A train passes at a steady speed: from its front catching Ka-lok to its back leaving Ka-lok takes 6 s;

278 s after its back leaves Ka-lok, its front meets Tsz-ching; from front meeting Tsz-ching to back leaving Tsz-ching takes 5 s.

After the train's back leaves Tsz-ching, how many seconds until Ka-lok and Tsz-ching meet?

Show the hints (3)
  1. Hint 1: Passing Ka-lok and passing Tsz-ching, the train's length is the same: overtaking uses the difference of speeds, passing head-on the sum.
  2. Hint 2: At the moment the front meets Tsz-ching, where is Ka-lok? From the moment the front caught Ka-lok, how much further has the train gone than Ka-lok?
  3. Hint 3: The train does 33 m a second. When its front meets Tsz-ching, the cyclists are (33 − 3) × (6 + 278) = 8520 m apart; for the next 5 s they keep closing.

Worked solution

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

  1. 1.Passing Ka-lok: length = (train − 3) × 6; passing Tsz-ching: length = (train + 3) × 5.
  2. 2.Equal lengths: train × 6 − 18 = train × 5 + 15, so train × 1 = 33: 33 m a second, length (33 − 3) × 6 = 180 m.
  3. 3.From the front catching Ka-lok to the front meeting Tsz-ching is 6 + 278 = 284 s, in which the train gains (33 − 3) × 284 = 8520 m on Ka-lok; the front is right beside Tsz-ching, so the cyclists are 8520 m apart.
  4. 4.In the next 5 s they close 3 × 2 × 5 = 30 m; when the back leaves Tsz-ching, they are 8520 − 30 = 8490 m apart.
  5. 5.They close 6 m a second, so they meet 8490 ÷ 6 = 1415 s later.
  6. 6.Check: (33 + 3) × 5 = 180 = the length, 180 ✓
  7. ⚠️Common mistake: taking the cyclists' gap as 33 × 278, forgetting to start from the front catching Ka-lok and that Ka-lok keeps moving; or missing how far they close during the 5 s pass.

Answer: 1415 s

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