A, B and C start at the same time in the same direction to run around a…
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A, B and C start at the same time in the same direction to run around a circular stadium. A completes a round in 252 seconds, B in 308 seconds and C in 198 seconds, all starting at the same point. After what time will they again be at the starting point?
Answer: D. 46 minutes and 12 seconds — Concept: When two or more bodies start together and repeat their motion in fixed cycle times, they return together to the starting point after a time equal to…
- A.
26 minutes and 18 seconds
- B.
42 minutes and 36 seconds
- C.
45 minutes
- D.
46 minutes and 12 seconds
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Show answer & explanation
Correct answer: D
Concept: When two or more bodies start together and repeat their motion in fixed cycle times, they return together to the starting point after a time equal to the Least Common Multiple (LCM) of their individual cycle times — the smallest time that is simultaneously divisible by every period.
Application:
Write each lap time as a product of prime powers: 252 = 22 × 32 × 7 seconds; 308 = 22 × 7 × 11 seconds; 198 = 2 × 32 × 11 seconds.
For each prime that appears, take the highest power seen across the three factorizations: 22, 32, 7, and 11.
Multiply these highest powers to get the LCM: 22 × 32 × 7 × 11 = 4 × 9 × 7 × 11 = 2772 seconds.
Convert 2772 seconds into minutes and seconds: 2772 ÷ 60 = 46 remainder 12, so 2772 seconds = 46 minutes 12 seconds.
Cross-check: 2772 ÷ 252 = 11, 2772 ÷ 308 = 9, and 2772 ÷ 198 = 14 — each division gives a whole number, confirming A, B and C each complete an exact whole number of laps at this instant, so 46 minutes 12 seconds is indeed the first time all three are together again at the starting point.