Propositiones ad Acuendos Juvenes/Problems/39 - De Quodam Emptore in Oriente

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Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $39$

De Quodam Emptore in Oriente
An Oriental Merchant
A man in the east wanted to buy $100$ assorted animals for $100$ shillings.
He ordered his servant to pay $5$ shillings for a camel,
$1$ shilling for a donkey,
and $1$ shilling for $20$ sheep.
How many camels, donkeys and sheep did he buy?


Solution

$19$ camels, $1$ donkey and $80$ sheep.


Proof

Let $c$, $d$ and $s$ denote the number of camels, donkeys and sheep respectively.

We have:

\(\ds 5 c + d + \dfrac s {20}\) \(=\) \(\ds 100\) the amount spent
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 100 c + 20 d + s\) \(=\) \(\ds 2000\)
\(\text {(2)}: \quad\) \(\ds c + d + s\) \(=\) \(\ds 100\) the total number of animals
\(\ds \leadsto \ \ \) \(\ds 99 c + 19 d\) \(=\) \(\ds 1900\) $(1) - (2)$
\(\ds \leadsto \ \ \) \(\ds 1900 - 99 c\) \(=\) \(\ds 19 d\)

Note that both $c$ and $d$ need to be (strictly) positive.

We need to find possible values of $c$ such that $2300 - 71 c$ is divisible by $19$.

This can happen only when $c$ itself is divisible by $19$.

\(\, \ds c = 0: \, \) \(\ds 1900 - 99 \times 0\) \(=\) \(\ds 1900\) \(\ds = 19 \times 100\)
\(\, \ds c = 19: \, \) \(\ds 1900 - 19 \times 99\) \(=\) \(\ds 19\) \(\ds = 1 \times 19\)


It is implicit that there are at least some camels are bought, so the solution:

$c = 0, d = 100, s = 0$

is usually ruled out.

Hence we have:

$c = 19, d = 1, s = 80$

$\blacksquare$


Historical Note

This is numerically the same as a problem given by Abu Kamil $\text {c. 900}$, except that the latter uses ducks, hens and sparrows.

Abu Kamil discusses the One Hundred Fowls problems at some length, even investigating $4$ and $5$ different types of bird.

Thus it may be that this problem was circulating in both Europe and Arabia for some time before Alcuin of York presents it here.


David Singmaster points out that Alcuin may very well never have seen a camel, unless when he visited Rome.

However, he would of course have read about them in the Bible.


Sources

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