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

Modern Puzzles by Henry Ernest Dudeney: $91$

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?

$120$ men are in the hospital.

We know that:

exactly two-thirds of the men had lost an eye.

We also know that:

three-fourths had lost an arm.

$\dfrac 2 3 + \dfrac 3 4 - 1 = \dfrac 5 {12}$

of the men had lost both an eye and an arm.

Similarly, since we know that:

four-fifths had lost a leg

$\dfrac 5 {12} + \dfrac 4 5 - 1 = \dfrac {13} {60}$

of the men had lost all three.

Dudeney equates this to $26$.

Hence there are exactly:

$26 \div \dfrac {13} {60} = 120$

men in this hospital.

$\blacksquare$