Maths olympiad question · P6★★★★ Expert
Boats and currents: olympiad question 9
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 on a river, A upstream and B downstream.
A speedboat and a pleasure boat leave A together, heading downstream for B.
In still water the speedboat does 160 m a minute and the pleasure boat 130; the river flows 10 m a minute.
All three speeds stay the same.
The speedboat turns straight back at B and meets the pleasure boat 450 m from B.
How far apart are piers A and B?
Show the hints (3)
- Hint 1: Work back from the meeting: after turning at B the speedboat went 450 m upstream. How many minutes did that take?
- Hint 2: After the turn the boats head for each other and the current's effect cancels: how fast do they close? On the way down, how much faster is the speedboat each minute?
- Hint 3: They meet 3 minutes after the turn; when the speedboat reached B, the boats were (160 + 130) × 3 = 870 m apart.
Worked solution
Have a go first, then read the solution step by step.
- 1.Upstream the speedboat does 160 − 10 = 150 m a minute, so 450 m from B took 450 ÷ 150 = 3 minutes.
- 2.Meanwhile the pleasure boat does 130 + 10 = 140 m a minute downstream; heading for each other they close 150 + 140 = 290 m a minute (the current cancels). So when the speedboat reached B they were 290 × 3 = 870 m apart.
- 3.On the way down the speedboat gains (160 + 10) − (130 + 10) = 30 m a minute; 870 m takes 870 ÷ 30 = 29 minutes — the speedboat's time from A to B.
- 4.A to B = (160 + 10) × 29 = 4930 m.
- 5.Check: in 32 minutes the pleasure boat went 140 × 32 = 4480 m = 4930 − 450 ✓
- ⚠️Common mistake: subtracting speeds for the meeting, or taking 450 m as the gap built up on the way down. Towards each other, speeds add; same way, they subtract — and the gap at B comes from working back from the meeting.
Answer: 4930 m
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