Getting Lapped

Number Theory Level 4

Alice and Carla are swimming in a pool when they decide to have a race. Carla is pretty fast and swims at a rate of 81m/min, while Alice only swims at a rate of 64m/min. Both girls swim down and back multiple times, and because Carla is faster than Alice, Carla laps Alice a few times during the race. If $t$ is the time it takes for Carla to lap Alice (in minutes) for the first time, what is the minimum length of the pool $l$ such that $l$ and $t$ are positive integers?

The answer is 17.

1 solution

Garrett Clarke
Aug 8, 2015

Where $n$ is the number of times Alice has swam the length of the pool, and $x$ is the distance that Alice is away from the pool when Carla laps her:

$64t=nl+x,\,81t=(n+2)l+x$

$81t-64t=(n+2)l+x-nl-x$

$17t=2l$

$t=\frac{2l}{17}$

If $17|2l$ , then $17|l$ . The smallest positive integer that can be divided by $17$ is $17$ itself, therefore our answer must be $\boxed{17}$ .

We could also model the race as swimming around a circular pool of length $2l$ , thus the first lapping occurs when $81t-64t=2l$

Xuming Liang - 5 years, 10 months ago

That would also work!

Garrett Clarke - 5 years, 10 months ago

Getting Lapped

The answer is 17.

1 solution

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