## Teaser 2749: Round the river

**From The Sunday Times, 31st May 2015** [link] [link]

My school holds “Round the river” runs — a whole number of metres to a bridge on the river and then the same number of metres back. Some years ago I took part with my friends Roy, Al, David and Cy. We each did the outward half at our own steady speeds (each being a whole number of centimetres per minute). For the return half I continued at my steady speed, Roy increased his speed by 10%, Al increased his speed by 20%, David increased his by 30%, and Cy increased his by 40%. We all finished together in a whole number of minutes, a little less than half an hour.

What (in metres) is the total length of the race?

[teaser2749]

## Jim Randell 8:46 am

on27 September 2022 Permalink |The speeds are whole numbers of centimetres per minute, so we will work in units of centimetres and minutes.

If the distance to bridge is

xcentimetres, thenxmust be divisible by 100.And the time is

tminutes (less than 30).If the 5 speeds are:

a, b, c, d, e, then we have:From which we see that

xmust also be divisible by 11, 6, 13, 7.Placing some sensible limits on distance and speeds, we can suppose

xis less than 10km (= 10,000m = 1,000,000cm), and that speeds are less than 15mph (≈ 40,000cm/min).This Python program runs in 61ms. (Internal runtime is 129µs).

Run:[ @replit ]Solution:The total length of the race is 6006m.And the race took exactly 25 minutes.

The outward speeds are:

Note that the speeds on the return leg are not all whole numbers of cm/min.

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