X, Y and Z start at some point and same time in the same direction to run around a circular stadium. X completes a round in 252 seconds, Y in 308 seconds and Z in 198 seconds. After what time will they meet again at the starting point?
46 minutes 12 seconds
The correct answer is 46 minutes 12 seconds (Option 4).
Concept used:
When some people start running on a circular path from the same point, they will meet again at the initial point at the same time which will be the Least Common Multiple (LCM) of the time taken by them to complete one round.
Calculation:
Time taken by X to complete one round = \(252\) seconds
Time taken by Y to complete one round = \(308\) seconds
Time taken by Z to complete one round = \(198\) seconds
Time to meet again at the initial point = \(\text{LCM}(252, 308, 198)\)
On performing prime factorization:
\(252 = 2 \times 2 \times 3 \times 3 \times 7 = 2^2 \times 3^2 \times 7\)
\(308 = 2 \times 2 \times 7 \times 11 = 2^2 \times 7 \times 11\)
\(198 = 2 \times 3 \times 3 \times 11 = 2 \times 3^2 \times 11\)
\(\text{LCM} = 2^2 \times 3^2 \times 7 \times 11\)
\(\text{LCM} = 4 \times 9 \times 7 \times 11 = 2772 \text{ seconds}\)
Converting seconds to minutes:
We know that \(1 \text{ minute} = 60 \text{ seconds}\)
\(\frac{2772}{60} = 46 \text{ minutes and } 12 \text{ seconds remaining}\)
Thus, all three will meet again at the initial point after 46 minutes 12 seconds.
Therefore, option 4 is the correct answer.
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