Teaser 2502

[Cactus]

Sunday Times Teaser 2502 by H Bradley and C Higgins
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John and Pat were swimming at a pool, starting at the same time from
opposite ends and each swimming lengths at their own constant speed.
They passed each other on their first lengths and again on their
second lengths, the length of the pool being an integer multiple of
the distance between these two passing points. Pat completed a whole
number of length at the same time that John completed 48 lengths. How
many lengths had Pat completed?

Cactus

[Garry]

Hello all

Via a distance-time diagram and some algebra, the only valid solution
I found is that Paul swims at 5/3 of John's speed, the 1st crossing is
3/8 of the length from John's initial end, the 2nd one is 7/8 of the
length from the same end, so the distance between is half the length.
So the length is twice this distance and Pat does 80 lengths in the
time John does 48.

Garry

[Peter]

Hello All,

from a graph and a little calculation I found that the speed ratio is
r = (2m + 1)/(2m - 1), where L/m is the distance between the two
passing points. It turns out that Pat is faster than John, while the
other way around would not give a valid integer solution for m.

Peter