Henry Ernest Dudeney/Modern Puzzles/20 - The Seven Applewomen/Solution

Modern Puzzles by Henry Ernest Dudeney: $20$

The Seven Applewomen

Seven applewomen,

possessing respectively $20$, $40$, $60$, $80$, $100$, $120$, and $140$ apples,

went to market and sold all their apples at the same price,

and each received the same sum of money.

What was the price?

Solution

Each woman sold her apples at:

$7$ for $1 \oldpence$

$3 \oldpence$ for the odd ones left over.

Thus each received the same amount::

$1 \shillings 8 \oldpence$

As Henry Ernest Dudeney put it:

Without questioning the ingenuity of the thing, I have always thought the solution unsatisfactory,

because really indeterminate, even if we admit that such an eccentric way of selling may be fairly termed a "price".

It would seem just as fair if they sold them at different rates and afterwards divided the money;

or sold different kinds of apples at different values;

or sold by weight, the apples being of different sizes;

or sold by rates diminishing with the age of the apples;

and so on.

That is why I have never held a high opinion of this old puzzle.

In a general way, we can say that $n$ women, possessing $a n + \paren {n - 1}$, $\paren {a + b} n + \paren {n - 2}$, $\paren {a + 2 b} n + \paren {n - 3}$, $\ldots$, $\paren {a + \paren {n - 1} b} n$ apples respectively,

can sell at $n$ for the penny and $b$ pence for each odd one left over,

and each receive $a + b \paren {n - 1}$ pence.

In the case of our puzzle $a = 2$, $b = 3$, and $n = 7$.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $20$. -- The Seven Applewomen
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $12$. The Seven Applewomen