Applesellers' Problem/Variant

Problem

 * Three women, $A$, $B$ and $C$, carried apples to a market to sell.
 * $A$ had sold $20$,
 * $B$ had sold $30$,
 * and $C$ had sold $40$.
 * They sold at the same price, the one as the other,
 * and, each having sold all their apples,
 * brought home as much money as each other.


 * How could this be?

Solution
During the first part of the day, they sold their apples at $1$ penny each.

During the second part of the day, they sold them at $3$ pence each.

$A$ sold $2$ at $1$ penny and $18$ at $3$ pence each.

$B$ sold $17$ at $1$ penny and $13$ at $3$ pence each.

$C$ sold $32$ at $1$ penny and $8$ at $3$ pence each.

Hence each one made $56$ pence.

Proof
In order for this to make sense, the two separate prices for the different times of day needs to be assumed.

Hence we need to determine:


 * the two price tiers
 * the total sum made
 * the number each sold at each tier.

This is a Diophantine equation which has a number of solutions.