Propositiones ad Acuendos Juvenes/Problems/46 - De Sacculo ab Homine Invento

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Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $46$

De Sacculo ab Homine Invento
A Man Finding a Purse
A man walking along a road, found a purse containing $2$ talents.
Others saw this and said: "Friend, give us a portion of your find."
He refused to give any to them.
So they set upon him and took the bag from him,
and each one took $50$ shillings.
Seeing that he could not stop them, he reached out and snatched $50$ shillings for himself.
How many men were there?


Solution

$216$


Proof

There are $75$ (troy) pounds in a talent.

Apparently, in this context, there are also $72$ gold shillings in a troy pound.

Hence there are $75 \times 72 = 5400$ shillings in a talent.

Let $x$ be the number of men that were there.

Then we have:

\(\ds 50 x\) \(=\) \(\ds 2 \times 5400\) as each man obtains $50$ shillings
\(\ds x\) \(=\) \(\ds \dfrac {10 \, 800} {50}\) simplifying
\(\ds \) \(=\) \(\ds 216\)

$\blacksquare$


Historical Note

David Singmaster refers to similar problems by Mahaviracharya and al-Karkhi.

He also references The Bloom of Thymarides, given by both Diophantus and Iamblichus, still to be investigated by $\mathsf{Pr} \infty \mathsf{fWiki}$.

He also suggests that it is related to $16$: De Duobus Hominibus Boves Ducentibus, classified as a Donkey and Mule problem, but the structure of that and this are considerably different.


Sources

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