Propositiones ad Acuendos Juvenes/Problems/40 - De Homine et Ovibus in Monte Pascentibus

by : Problem $40$

 * De Homine et Ovibus in Monte Pascentibus: A Man and some Sheep Grazing on the Hillside
 * A man looked at some sheep grazing on the hillside, and said,
 * "I wish I had these,
 * and as many again,
 * plus half of half that total,
 * plus half of that last amount;
 * then, counting myself, I would take $100$ to my home.


 * How many sheep did he see?

Solution

 * $36$.

Proof
Let $x$ be the number of sheep seen.

Then we have:
 * $x + x + \dfrac {x + x} 4 + \dfrac 1 2 \paren {\dfrac {x + x} 4} + 1 = 100$

That is, after algebra:
 * $\dfrac {11 x} 4 = 99$

Hence the result.