Single correct option
Two runners start together from the same point on a circular track of length 600 m and run in opposite directions at constant speeds of 5 m/s and 7 m/s. If they run indefinitely, at how many distinct points on the track will they meet?
- A. 6
- B. 12
- C. 2
- D. 35
Show answer and solution
Answer: (B) 12
Running in opposite directions, they close the gap at 5 + 7 = 12 m/s, so they meet every 600/12 = 50 seconds.
In 50 s the slower runner covers 250 m. So the meeting points are at multiples of 250 m (mod 600) from the start: 250, 500, 150, 400, 50, 300, 550, 200, 450, 100, 350, 0. Since gcd(250, 600) = 50, the positions repeat every 600/50 = 12 meetings.
Rule: For speeds in lowest ratio a : b, opposite directions give (a + b) distinct meeting points; the same direction gives |a − b|. Here 5 : 7 → 5 + 7 = 12.
In 50 s the slower runner covers 250 m. So the meeting points are at multiples of 250 m (mod 600) from the start: 250, 500, 150, 400, 50, 300, 550, 200, 450, 100, 350, 0. Since gcd(250, 600) = 50, the positions repeat every 600/50 = 12 meetings.
Rule: For speeds in lowest ratio a : b, opposite directions give (a + b) distinct meeting points; the same direction gives |a − b|. Here 5 : 7 → 5 + 7 = 12.
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