[Guest post by Dr. Chase]
My wife and I walk on a circular track, starting at the same point. She does m laps in the time that it takes me to do n laps. She walks faster than I do, so m > n. After how many laps will she catch up with me again?
Example: For m = 4, and n = 3, she will catch up when I have finished 3 laps. Reason: When I have finished 1 lap, she finished 1 1/3 laps, so she is 1/3 of the track ahead of me. (But hasn’t passed me yet.) When I have finished 2 laps, she has finished 2 2/3 laps around the track, still ahead of me. When I finish 3 laps, she has finished 3 3/3 laps, which is to say 4 laps. So we are together for the first time since starting. If m = 2, n = 1, she will catch up in just 1 lap. If m = 7, n = 6, she will catch up in 6 laps. Will she always catch up in n laps? In how many laps will she catch up for arbitrary m and n?