Maths olympiad question · P6★★★★ Expert
Boats and currents: olympiad question 11
An expert question on boats and currents for P6, with a step-by-step solution, diagrams and the answer. Boats going with and against the current appear in the Hua Xia Cup's P5 and P6 first rounds, qualifying rounds and finals.
The question
Piers A and B are 4180 m apart on a river, A upstream and B downstream.
Boat P leaves A downstream for B and boat Q leaves B upstream for A at the same moment.
In still water P does 90 m a minute and Q 80; the river flows 10 m a minute.
16 minutes after setting off, P's engine fails and it drifts with the current for 14 minutes before it is fixed; then it carries on as before.
Q never stops.
How far from pier A do the boats meet?
Show the hints (3)
- Hint 1: Heading for each other, the current's effects on the two boats cancel. So while P drifts with the current, what is it doing as far as the water is concerned?
- Hint 2: Seen from the water: together the boats must close 4180 m; during P's 14 drifting minutes, only Q is closing.
- Hint 3: They meet 32 minutes after setting off, P under its own power for 18 of them.
Worked solution
Have a go first, then read the solution step by step.
- 1.The current carries both boats alike and never changes the distance between them. Relative to the water, Q does 80 m a minute and P 90 when running and nothing while drifting.
- 2.Say they meet after T minutes: Q closes 80 × T and P 90 × (T − 14), 4180 m together. So 170 × T = 4180 + 90 × 14 = 5440, and T = 5440 ÷ 170 = 32 minutes (P was fixed by minute 30, so this fits).
- 3.Now on the bank: for its 18 running minutes P does 90 + 10 = 100 m a minute, and for the 14 drifting minutes 10: 100 × 18 + 10 × 14 = 1940 m.
- 4.Check: Q goes (80 − 10) × 32 = 2240 m upstream in 32 minutes, and 1940 + 2240 = 4180 ✓
- ⚠️Common mistake: thinking P stands still on the bank while it drifts; it still drifts 10 m a minute downstream. Another is timing the meeting with bank speeds, where the current doesn't cancel and the sums get tangled.
Answer: 1940 m
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