# Dudeney's Modern Puzzles/Arithmetical and Algebraical Problems/Money Puzzles/2 - Pocket-money

## Modern Puzzles by Henry Ernest Dudeney: $2$

Pocket-money
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:

 $\ds S$ $=$ $\ds 2 F$ ... and when I returned home I discovered that I had spent just half of it, $\ds F_s$ $=$ $\ds S_L$ and that I now had just as many shillings as I previously had pounds, $\ds 2 F_L$ $=$ $\ds S_s$ and half as many pounds as I then had shillings.

We have that:

 $\ds S$ $=$ $\ds 20 S_L + S_s$ where $S$ shillings is the money I started out with $\ds F$ $=$ $\ds 20 F_L + F_s$ where $F$ shillings is the money I came home with $\ds$ $=$ $\ds 20 \dfrac {S_s} 2 + S_L$ as $2 F_L = S_s$ $\ds$ $=$ $\ds 10 S_s + S_L$ $\ds \leadsto \ \$ $\ds 20 S_L + S_s$ $=$ $\ds 2 \paren {10 S_s + S_L}$ as $2 F = S$ $\ds \leadsto \ \$ $\ds 18 S_L$ $=$ $\ds 19 S_s$ simplifying

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

 $\ds S_L$ $=$ $\ds 19$ $\ds S_s$ $=$ $\ds 18$

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

Hence:

 $\ds F$ $=$ $\ds \dfrac {\pounds 19 \ 18 \, \mathrm s.} 2$ $\ds$ $=$ $\ds \pounds 9 \ 19 \, \mathrm s.$

which is equal to the amount spent.

$\blacksquare$