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+2*x*)/40

or

40(40+*x*) = (40+2*x*)*x*

or

1600 + 40*x* = 40*x* + 2*x*^{2}

or

*x*^{2} = 800.

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

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