Three runners A, B, and C start together from the same point around a circular track. Their lap times are A: 252 s, B: 308 s, and C: 198 s. After how much time will they next be together at the starting point?
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A26 min and 18 s
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B42 min and 36 s
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C45 min
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D46 min and 12 s
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E44 min
Answer
Correct Answer: 46 min and 12 s
Explanation
Introduction: When several runners start together and repeat laps with different lap times, their next common meeting at the start occurs after a time equal to the least common multiple of their lap times.
Given Data / Assumptions:
- A lap: 252 s
- B lap: 308 s
- C lap: 198 s
- All start together at the same point
Concept / Approach: Compute LCM(252, 308, 198). Convert the resulting seconds to minutes and seconds for a readable time format.
Step-by-Step Solution:
LCM(252, 308, 198) = 2772 s 2772 s = 46 min and 12 sVerification / Alternative check: Prime factorizations confirm that 2772 is divisible by 252, 308, and 198, and there is no smaller positive time that satisfies all three simultaneously.
Why Other Options Are Wrong: 26 min 18 s, 42 min 36 s, 45 min, and 44 min are not common multiples of all three lap times.
Common Pitfalls: Taking a simple sum or pairwise LCM instead of the LCM of all three values. Always combine all periods.
Final Answer: 46 min and 12 s