Henry Ernest Dudeney/Modern Puzzles/91 - Counting the Wounded/Solution

by : $91$

 * Counting the Wounded
 * When visiting with a friend one of our hospitals for wounded soldiers, I was informed that
 * exactly two-thirds of the men had lost an eye,
 * three-fourths had lost an arm,
 * and four-fifths had lost a leg.


 * "Then," I remarked to my friend, "it follows that at least twenty-six of the men must have lost all three -- an eye, an arm, and a leg."


 * That being so, can you say exactly how many men were in the hospital?


 * It is a very simple calculation, but I have no doubt it will perplex a good many readers.

Proof
Let $n$ be the total number of soldiers.

Let $e$, $a$ and $l$ be the number of soldiers that have lost an eye, an arm and a leg respectively.

Let $f$ be the number of soldiers that have lost both an eye and an arm.

Let $g$ be the number of soldiers that have lost both an eye and a leg.

Let $f$ be the number of soldiers that have lost both an arm and a leg.

We are given that at least $26$ have lost all three.

We have: