# 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.

## Sources

- c. 800: Alcuin of York:
*Propositiones ad Acuendos Juvenes* - 1992: John Hadley/2 and David Singmaster:
*Problems to Sharpen the Young*(*Math. Gazette***Vol. 76**,*no. 475*: pp. 102 – 126) www.jstor.org/stable/3620384 - 1992: David Wells:
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