Henry Ernest Dudeney/Modern Puzzles/5 - Doubling the Value/Solution

by : $5$

 * Doubling the Value

Solution

 * $\pounds 2 \ 17 \shillings$

which, when multiplied by $6$, becomes:


 * $\pounds 17 \ 2 \shillings$

Proof
It is assumed that when says number he means natural number

Let $k$ denote the shilling value of the original sum of money.

Recall there are $20$ shillings to the pound.

Let $s$ and pounds $p$ denote the number of shillings and pounds that $k$ consists of such that $s < 20$.

That is:
 * $k = \pounds p \ s \shillings$

where:
 * $0 \le s < 20$

and of course trivially $s \ne 0$.

Let $n$ be a number which switches the shillings and pounds when you multiply $k$ by it.

Hence we want to find values for $n$ that will result in:
 * $k = 20 p + s$

and:
 * $n k = 20 s + p$

where both $p < 20$ and $s < 20$.

We have:

It remains to substitute values for $n$ and see where that gets us.

Clearly $n < 20$ in order for $\paren {20 - n} s > 0$.

$n = 2$:

which is the answer we have been provided.

$n = 3$:

$n = 4$:

$n = 5$:

$n = 6$:

Hence we have that:
 * $k = \pounds 2 \ 17 \shillings$

and we check that:

It remains to be shown that this is the only solution.

$n = 7$:

$n = 8$:

$n = 9$:

$n = 10$:

$n = 10$:

and at this stage $\dfrac p s > 20$ and so there can be no solutions such that $p < 20$.

That exhausts all possibilities for $n$.

It is interesting to see what happens when $n = 1$ is entered:

which is of course obvious.