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Maths olympiad question · P6★★★ Hard

Boats and currents: olympiad question 8

A hard 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

On a school study tour, a cruise boat sails downstream from pier A to pier B and comes straight back upstream.

The round trip takes 13 hours in all, and the way back takes 3 hours longer than the way out.

The river flows at 3 km an hour, and the boat's still-water speed does not change.

How far apart are piers A and B, in km?

Show the hints (3)
  1. Hint 1: How many hours did the way out and the way back each take?
  2. Hint 2: Over the same stretch, what is the ratio of the downstream and upstream speeds? How many km an hour apart are they?
  3. Hint 3: Out 5 hours, back 8, so downstream : upstream = 8 : 5; the speeds are 6 km/h apart.

Worked solution

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

  1. 1.Way out: (13 − 3) ÷ 2 = 5 hours; way back: 5 + 3 = 8 hours.
  2. 2.Same stretch, less time, more speed: downstream : upstream = 8 : 5.
  3. 3.Downstream gains one current and upstream loses one: the speeds differ by 3 × 2 = 6 km/h.
  4. 4.The speeds differ by 8 − 5 = 3 parts, which is 6 km/h, so one part is 6 ÷ 3 = 2: downstream = 2 × 8 = 16 km/h.
  5. 5.The piers are 16 × 5 = 80 km apart.
  6. ⚠️Common mistake: thinking downstream and upstream differ by only one current, 3 km/h; that gives exactly half the right distance.

Answer: 80 km

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