How far from end A is their face-to-face meeting number 5?

Question

The school field has a straight running track, 1200 m long, with ends A and B.

Ka-lok starts from A and Ka-yee from B at the same time, running back and forth: at each end they turn at once and keep going, at steady speeds.

Ka-lok runs 230 m a minute, Ka-yee 130 m.

Running into each other head on is a "face-to-face meeting"; one catching the other from behind doesn't count.

Accepted Answer

Answer: 300 m. At the first face-to-face meeting they have run one track length together, 1200 m. Then each runs on to an end and turns, and by the next face-to-face meeting they have run two more lengths together. Why don't overtakes upset this? Join the track and its return into a 2400 m loop: the two run opposite ways round it, and meeting on the loop is meeting face to face (an overtake only overlaps on the track; on the loop they are apart). Running opposite ways, they meet once every 2400 m run together. So at meeting 5 they have run 1200 + 2400 × 4 = 10800 m together, taking 10800 ÷ (230 + 130) minutes; Ka-lok has run 10800 × 230 ÷ 360 = 6900 m. Ka-lok is back at A after every 2400 m: 6900 = 2400 × 2 + 2100. The last 2100 m: 1200 m to B, then 900 m back, so 1200 − 900 = 300 m from A. Check: Ka-yee has run 3900 m from B, which also folds back to 300 m from A ✓ Common mistake: thinking they run one length together between meetings, or counting overtakes. Another is forgetting Ka-lok turns at B and giving 6900 − 2400 × 2 as the answer.

Meeting: olympiad question 10

The question

Worked solution