Propositiones ad Acuendos Juvenes/Problems/38 - De Quodam Emptore in Animalibus Centum

by : Problem $38$

 * De Quodam Emptore in Animalibus Centum: A Man Buying a Hundred Animals
 * A man wanted to buy $100$ assorted animals for $100$ shillings.
 * He was willing to pay $3$ shillings for a horse,
 * $1$ shillings for an ox,
 * ''and $1$ shillings for $24$ sheep.


 * How many horses, oxen and sheep did he buy?

Solution

 * $23$ horses, $29$ oxen and $48$ sheep.

Proof
Let $h$, $o$ and $s$ denote the number of horses, oxen and sheep respectively.

We have:

Note that both $h$ and $o$ need to be (strictly) positive.

We need to find possible values of $h$ such that $2300 - 71 h$ is divisible by $23$.

This can happen only when $h$ itself is divisible by $23$.

It is implicit that there are at least some horses are bought, so the solution:
 * $h = 0, o = 100, s = o$

is usually ruled out.

Hence we have:
 * $h = 23, o = 29, s = 48$