Henry Ernest Dudeney/Modern Puzzles/2 - Pocket-money

by : $2$

 * Concerning a Cheque


 * I went down the street with a certain amount of money in my pocket,
 * and when I returned home I discovered that I had spent just half of it,
 * and that I now had just as many shillings as I previously had pounds,
 * and half as many pounds as I then had shillings.
 * How much money had I spent?

Solution
The amount spent was $\pounds 9 \ 19 \, \mathrm s.$

Proof
Let $S$ be the amount I started with: $S_L$ pounds and $S_s$ shillings.

Let $F$ be the amount I finished with: $F_L$ pounds and $F_s$ shillings.

We have that:
 * $2 F = S$

and so the amount spent is equal to $2 F - F = F$, that is, the amount I finished with.

We recall the conversion factors:


 * $20$ shillings make one pound.

Hence any shilling quantities in either $S$ or $F$ cannot be greater than $19$.

That is:
 * $S_s < 20$
 * $F_s < 20$

It is assumed that $S_s$ and $F_s$ are both integers, that is: both $S$ and $F$ are a whole number of shillings.

We are given that:

We have that:

The smallest values of $S_L$ and $S_s$ that satisfy the above equation are:

As $S_s \le 19$ it follows that there can be no other solution.

Hence:

which is equal to the amount spent.