Three friends A, B and C regularly jog in a circular jogging track every morning. The track is 1000 m in circumference.

A takes 8 mins to complete one lap, B takes 10 mins and 40 secs and C takes 12 mins. All of them setout from a same point but C runs in the opposite direction from starting point.

Another friend D joins this run. D normally takes 13 mins and 20 secs to complete one lap. He decides to run in the same direction as C but from a point 600 m ahead of C in his jogging direction. When will they all meet each other on the circular track & where?

The angular velocities of A, B, C and D are 1/480, 1/640, -1/720 and -1/800 rev/sec respectively, with the negative signs representing the fact C and D go the opposite direction from A and B.

D starts out .6 rev ahead of C. C is overtaking D at a relative rate of 1/720 - 1/800 = 1/7200 rev/sec, so it takes C 4320 seconds to catch up to D. A, C and D will all be at the starting point at that time, but B will be 1/4 revolution away, having gone 6 3/4 revolutions by that time.

The LCM of 480, 720 and 800 is 7200, so once A, C and D meet (that's also the LCM of 288 and 300--the relative period of A vs C and of A vs D, as the three always meet at the same spot), they meet every 7200 seconds thereafter. During 7200 seconds, B advances 11 1/4 revolutions, so one period of 7200 seconds will be enough to bring B into line with A, C and D.

That then is a total of 4320+7200=11,520 seconds or 3 hours, 12 minutes.

Posted by Charlie on 2006-04-13 14:59:38