Answer to old chestnut: horseman and 40 mile army:

The answer can be found from a simple bit of algebra.

Let x be the distance in miles that the army marches during the horseman's run forward. Then the horseman traveled 40+x miles while the army marched x miles. On the return trip, the horseman marched x miles (because the back of the army is 40 miles closer than when the trip started) and the army marched 40-x miles, for total distances of 40+2x for the horseman and 40 for the army.

Since both the horseman and the army travel at constant rates, the ratio of their rates is also constant, so we have:

(40+x)/x = (40+2x)/40
40(40+x) = (40+2x)x
1600 + 40x = 40x + 2x2
x2 = 800.

So x = sqrt(800) = 20sqrt(2).

Thus the horseman has traveled 40 + 40sqrt(2) or about 96.6 miles.

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