Mo2men wrote:Three bodies A, B and C start moving around a circular track of length 60m from the same point simultaneously in the same direction at speeds of 3 m/s, 5 m/s and 9 m/s respectively. When will they meet for the first time after they started moving?
A. 30 seconds
B. 60 seconds
C. 15 seconds
D. 10 seconds
E. 25 seconds
OA: A
How to deal with this problem?
Let's analyze the answer choices from the smallest to the largest.
D) 10 seconds
After 10 seconds, A, B, and C have moved 30 m, 50 m, and 90 m, respectively. While A and B haven't completed one lap, C has completed one lap and 30 m more. We see that A and C meet at the same place on the track, but B doesn't meet them at that place .
C) 15 seconds
After 15 seconds, A, B, and C have moved 45 m, 75 m, and 135 m, respectively. While A hasn't completed one lap, B has completed one lap and 15 m more, and C has completed two laps and 15 m more. We see that B and C meet at the same place on the track, but A doesn't meet them at that place.
E) 25 seconds
After 25 seconds, A, B and C have moved 75 m, 125 m, and 225 m, respectively. We see that A has completed one lap and 15 m more, B has completed two laps and 5 m more, and C has completed three laps and 45 m more. We see that each person is at a different place on the track.
A) 30 seconds
After 30 seconds, A, B, and C have moved 90 m, 150 m, and 270 m, respectively. We see that A has completed one lap and 30 m more, B has completed two laps and 30 m more and C has completed four laps and 30 m more. We see that all of them are at the same place on the track.
Alternate Solution:
Suppose that they meet after t seconds for the first time. In t seconds, they have covered a distance of 3t, 5t and 9t meters, respectively. Notice that the distance between A and B is 5t - 3t = 2t, A and C is 9t - 3t = 6t and B and C is 9t - 5t = 4t. If they are at the same spot after t seconds, all of 2t, 6t and 4t must be multiples of 60; say 2t = 60k, 6t = 60s and 4t = 60p. Then, t = 30k = 10s = 15p. Since t is a multiple of 30, 10 and 15; the smallest value of t is given by the LCM of these numbers. Therefore, the smallest value of t is LCM(30, 10, 15) = 30.
Answer: A
