Three friends A, B and C regularly jog in 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. One day they decided to find out if they set out together in the same direction from a point what would be the fastest time for all of them to meet at any point on the track. What did they conclude?

The second day C decides to run in the opposite direction from the starting point. When will they all meet? Will this be sooner than the time taken on day one? Where do they meet in both cases?

Well, the three of them are obviously going to meet at the starting point at the lowest common multiple of their three times.  It takes A 480 secs to make one lap, B 640 secs and C 720 secs.  So after 5760 seconds they will all meet at the starting point (A would have made 12 laps, B 9 laps and C 8 laps).  I don't know if they would all meet at any point before this happens, but my guess is that they don't.

For the second part, I'm not sure that it changes anything, I'm guessing that at 5760 seconds they will meet for the first time, once again at the starting point.  But that I really have no idea about.

Edited to add:

This is, of course, assuming that they all follow the same track, i.e., they are all running exactly 1000m.  It would be possible, for example, to widen the track enough so that even at their different speeds they all stayed together around the track.  For example, widen the track so that the outside lane of the track actually has circumference of 1500m and put person A in that lane.  Make the second lane 1125m circumference and put person B there, and leave person C on the inside track with circumference of 1000m.  This way their varying speeds are offset by the distances they are running.  (I rushed the math on that, so there's probably a good chance I messed it up, but you get the idea...)

Edited on April 12, 2006, 10:03 am

Posted by tomarken on 2006-04-12 09:56:18