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

Meeting: olympiad question 9

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

Ka-lok and Tsz-ching set out from Tai Po for Sha Tin, and at the same moment Ho-yin sets out from Sha Tin for Tai Po; all three cycle along the same cycle track at steady speeds.

Ka-lok covers 65 m a minute, Tsz-ching 75 m and Ho-yin 85 m.

Ho-yin meets Ka-lok 2 minutes after meeting Tsz-ching.

How far apart are Tai Po and Sha Tin, in metres?

Show the hints (3)
  1. Hint 1: At the moment Ho-yin meets Tsz-ching, how far is Ho-yin from Ka-lok? How do they move in the next 2 minutes?
  2. Hint 2: At that moment Ho-yin and Tsz-ching are at the same spot, so the gap from Ka-lok to Ho-yin is how far Ka-lok is behind Tsz-ching. How much more does Tsz-ching cover each minute?
  3. Hint 3: At that moment Ka-lok is (65 + 85) × 2 = 300 m behind Tsz-ching; Tsz-ching gains 10 m a minute.

Worked solution

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

  1. 1.After Ho-yin meets Tsz-ching, Ka-lok and Ho-yin head towards each other and meet 2 minutes later, so at that moment they were (65 + 85) × 2 = 300 m apart.
  2. 2.At that moment Ho-yin and Tsz-ching are together, so 300 m is also Tsz-ching's lead over Ka-lok. Tsz-ching gains 75 − 65 = 10 m a minute, so they have been going 300 ÷ 10 = 30 minutes.
  3. 3.Tsz-ching and Ho-yin covered the whole way in 30 minutes: (75 + 85) × 30 = 4800 m.
  4. 4.Check: Ka-lok and Ho-yin meet at minute 32: (65 + 85) × 32 = 4800 = 4800 ✓
  5. ⚠️Common mistake: taking (65 + 85) × 2 as the whole way, or 2 minutes as the time all three walked. The key is the moment Ho-yin meets Tsz-ching: the gap to Ka-lok is exactly what Ka-lok and Tsz-ching have drifted apart.

Answer: 4800 m

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