Maths olympiad question · P6★★★★ Expert
Boats and currents: olympiad question 10
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 stand 2080 m apart on the Shing Mun River in Sha Tin, A upstream and B downstream.
A rowing boat sets off downstream from A towards B, and at the same moment a speedboat sets off upstream from B towards A.
In still water the rowing boat does 50 m a minute and the speedboat 130; the river flows at 20 m a minute.
All three speeds stay the same.
Just as they set off, a float drops from the rowing boat and drifts downstream.
The boats pass and carry on; when the speedboat reaches the float, it turns straight round (now going downstream) to chase the rowing boat.
When the speedboat catches the rowing boat, how far is it from pier B?
Show the hints (3)
- Hint 1: The river carries both boats and the float alike. Seen from the float, does the current matter at all?
- Hint 2: The speedboat closes on the float at 130 m a minute; the rowing boat pulls 50 m a minute further from it. After turning, the speedboat chases the rowing boat the same way: how fast does it gain?
- Hint 3: The speedboat reaches the float after 16 minutes, when the rowing boat is 800 m beyond it; it catches up 10 minutes later. 26 minutes in all, the rowing boat doing 70 m a minute downstream.
Worked solution
Have a go first, then read the solution step by step.
- 1.The float and both boats are carried alike, so the current never changes the distances between them: the speedboat closes on the float at 130 m a minute (its still-water speed), and the rowing boat moves away from it at 50.
- 2.The speedboat starts 2080 m from the float: 2080 ÷ 130 = 16 minutes to reach it. By then the rowing boat is 50 × 16 = 800 m from the float — and from the speedboat — downstream.
- 3.The speedboat turns and chases downstream, the same way: it gains 130 − 50 = 80 m a minute (the current cancels again), taking 800 ÷ 80 = 10 minutes.
- 4.From the start to the catch is 16 + 10 = 26 minutes. The rowing boat does 50 + 20 = 70 m a minute downstream: 70 × 26 = 1820 m, leaving 2080 − 1820 = 260 m to B.
- 5.Check: the speedboat goes 1760 m upstream in 16 minutes, then 1500 m downstream in 10: 260 m from B ✓
- ⚠️Common mistake: chasing with the speedboat's downstream speed against the rowing boat's still-water speed — cancelling only half the current. Both boats ride the water, so the current adds to both; only at the end, measuring from the pier, switch back to speeds over the bank.
Answer: 260 m
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