Propositiones ad Acuendos Juvenes/Problems/6 - De Duobis Negotiatoribus C Solidos Communis Habentibus

Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $6$

De Duobis Negotiatoribus $\text C$ Solidos Communis Habentibus
$2$ Wholesalers with $100$ Shillings
$2$ wholesalers with $100$ shillings between them bought some pigs with the money.
They bought at the rate of $5$ pigs for $2$ shillings, intending to fatten them up and sell them again, making a profit.
But when they found that it is not the right time of year for fattening pigs,
and they were not able to feed them through the winter,
they tried to sell them again to make a profit.
But they couldn't, because they could only sell them for the price they had paid for them, $2$ shillings for $5$ pigs.
When they saw this, they said to each other, "Let's divide them.
By dividing them, and selling them again at the rate they had bought them for, they made a profit.
How could they be divided to make a profit, which could not be made by selling them all at once?

Solution

First there were $250$ pigs bought with $100$ shillings at the above mentioned rate, for $5$ fifties are $250$.

On division, each merchant had $125$ pigs.

One sold the poorer quality pigs at $3$ for a shilling.

The other sold the better quality pigs at $2$ for a shilling.

The one who sold the poorer quality pigs received $40$ shillings for $120$ pigs.

The one who sold the better quality pigs received $60$ shillings for $120$ pigs.

Then there remained $5$ of each sort of pig.

From these they could make a profit of $4$ shillings and $2$ pence.

Historical Note

According to David Singmaster, this is the first appearance of the type of problem known as the Applesellers' Problem, also known as the missing penny or marketwomen's problem.

It hinges on the fact that a common mean is erroneously being calculated of $2$ for a monetary unit plus $3$ for a monetary unit equals $5$ for $2$ monetary units.

David Singmaster goes on to remark that, while this problem appears everywhere in European puzzle-books since its appearance here, he has never seen a non-European version.

However, Maurice Kraitchik claims in his Mathematical Recreations that it appears in Mahaviracharya's Ganita Sara Samgraha, which dates from c. $850$. However, this has not been corroborated.