Three runners, P, Q, and R start to run simultaneously and in the same…
2025
Three runners, P, Q, and R start to run simultaneously and in the same direction along a circular garden. P completes one round in 200 seconds, Q in 300 seconds, and R in 150 seconds when all of them start from the same point. How many minutes will they meet at the starting point?
- A.
11 minutes
- B.
12 minutes
- C.
23 minutes
- D.
10 minutes
Show answer & explanation
Correct answer: D
Concept: When several runners start together from the same point on a circular track and keep looping, all of them return to the starting point at the same instant after a time equal to the Least Common Multiple (LCM) of their individual lap times — because that is the smallest time that is a whole-number multiple of every lap time simultaneously.
Write each runner's lap time in seconds: P = 200 s, Q = 300 s, R = 150 s.
Find the prime factorisation of each: 200 = 23 × 52, 300 = 22 × 3 × 52, 150 = 2 × 3 × 52.
The LCM takes the highest power of every prime that appears: 23 × 31 × 52 = 8 × 3 × 25 = 600.
Convert seconds to minutes: 600 ÷ 60 = 10 minutes.
Cross-check: 600 s is exactly divisible by 200 (3 laps), 300 (2 laps) and 150 (4 laps), so all three runners are back at the start together at 600 s — and no smaller common multiple works, since 300 itself is not divisible by 200. This confirms 600 s = 10 minutes is the least time they meet again at the starting point.
So the runners P, Q and R first meet again at the starting point after 10 minutes.