Single correct option
Two runners start together from the same point on a circular track and run in opposite directions at 8 m/s and 5 m/s. They keep running until they meet at the starting point for the first time. At how many distinct points on the track do they meet (including the starting point)?
- A. 8
- B. 5
- C. 13
- D. 3
Show answer and solution
Answer: (C) 13
The speeds are 8 and 5 m/s, already in lowest terms (gcd(8, 5) = 1).
Shortcut: for speeds in lowest ratio a : b, running in opposite directions gives a + b distinct meeting points before they are both back at the start together (running the same direction would give a − b).
Here a + b = 8 + 5 = 13.
Why it works: relative speed = 13 units, so in the time both return to the start (which takes them a and b whole laps respectively), they close the relative gap of a + b track-lengths, creating a + b equally spaced meeting points.
Shortcut: for speeds in lowest ratio a : b, running in opposite directions gives a + b distinct meeting points before they are both back at the start together (running the same direction would give a − b).
Here a + b = 8 + 5 = 13.
Why it works: relative speed = 13 units, so in the time both return to the start (which takes them a and b whole laps respectively), they close the relative gap of a + b track-lengths, creating a + b equally spaced meeting points.
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