Seema, Meena and Reema begin to jog around a circular stadium and they complete their revolutions in 54 seconds, 42 seconds and 63 seconds respectively. After how much time will they come together at the starting point?

### Concept & Strategy When multiple entities move in a circular path starting at the same time and place, they will all meet again at the starting point after a time equal to the Lowest Common Multiple (L.C.M.) of the times taken by each to complete one full revolution. $$ \text{Meeting Time} = \text{L.C.M.}(T_1, T_2, T_3, \dots) $$ ### Step-by-Step Solution * **Given:** Revolution times are 54 seconds, 42 seconds, and 63 seconds. * Calculate the L.C.M. of 54, 42, and 63. Let's use prime factorization: * 54 = $2 \times 3^3$ * 42 = $2 \times 3 \times 7$ * 63 = $3^2 \times 7$ * Take the highest power of each prime factor present: * For 2: $2^1 = 2$ * For 3: $3^3 = 27$ * For 7: $7^1 = 7$ * Multiply to find the L.C.M.: * L.C.M. = 2 $\times$ 27 $\times$ 7 = 54 $\times$ 7 = 378 seconds. * The three girls will meet again at the starting point exactly 378 seconds later. * Convert 378 seconds into minutes and seconds: * 378 / 60 gives a quotient of 6 (minutes) and a remainder of 18 (seconds). * So, 378 seconds = 6 minutes 18 seconds. ### Exam Strategy & Shortcut Use the options backwards. The correct total seconds must be a multiple of 63 (which means it must be a multiple of 9, so its digit sum is 9). Convert the options to seconds: * Option A: (6 $\times$ 60) + 18 = 378 (Digit sum = 18, div by 9. Works). * Option B: (6 $\times$ 60) + 36 = 396 (Digit sum = 18. Does it divide by 7? 396 / 7 = 56.5. Fails). * Option C: (5 $\times$ 60) + 48 = 348 (Fails digit sum of 9). Option A is confirmed in seconds without calculating the full LCM from scratch. ### Common Pitfall Students sometimes confuse "meeting at the starting point" with "meeting anywhere on the track." If the question asks when they meet *anywhere*, relative speeds are required. Because they specify meeting *at the starting point*, pure L.C.M. of individual lap times is the only correct method. ### Final Answer **Therefore, the correct answer is 6 min 18 sec.**