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Maths olympiad question · P5–P6★★★★ Expert

Meeting: olympiad question 11

An expert question on meeting for P5–P6, with a step-by-step solution, diagrams and the answer. The basis of journey problems: meeting and catching up appear in the Hua Xia Cup's P4 to P6 qualifying rounds and finals.

The question

Dad sets out from the estate gate and his son from the beach at the same moment, walking towards each other along the same road; the two places are 15210 m apart.

The dog starts with his son and runs towards Dad; on reaching Dad it turns straight back to his son, then back to Dad, and so on.

Dad walks 90 m a minute, his son 60 m, and the dog runs 300 m.

When the dog reaches Dad for the second time, how far apart are Dad and his son?

Show the hints (3)
  1. Hint 1: Take it in stages: the dog runs from his son to Dad — how many minutes until it meets Dad? How far apart are the two walkers then?
  2. Hint 2: Each stage is a meeting: the dog and one walker close at the sum of their speeds; meanwhile the walkers' gap shrinks by 150 m a minute.
  3. Hint 3: Stage 1 takes 39 minutes, leaving the walkers 9360 m apart; in stage 2 the dog takes 26 minutes to get back to his son.

Worked solution

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

  1. 1.Stage 1: the dog and Dad close at 300 + 90: 15210 ÷ 390 = 39 minutes. The walkers cover (90 + 60) × 39 = 5850 m, leaving 15210 − 5850 = 9360 m.
  2. 2.Stage 2: the dog turns and closes on his son at 300 + 60: 9360 ÷ 360 = 26 minutes. The walkers cover 3900 m more, leaving 9360 − 3900 = 5460 m.
  3. 3.Stage 3: the dog turns again to Dad: 5460 ÷ 390 = 14 minutes — its second meeting with Dad. The walkers cover 2100 m more, leaving 5460 − 2100 = 3360 m.
  4. 4.Check: 39 + 26 + 14 = 79 minutes; the walkers cover 11850 m, and 15210 − 11850 = 3360 ✓
  5. ⚠️Common mistake: counting two stages for the second meeting with Dad; starting from his son, the dog meets Dad, then his son, then Dad: three. Others subtract speeds — heading towards each other, speeds add.

Answer: 3360 m

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