# Charlie's money

How much money did Charlie have to begin with?

Charlie spent $\frac14$ of his money on a book, gave his brother $\frac23$ of what was left, and still had £9 remaining.

How much money did he start off with?

*This problem is taken from the World Mathematics Championships*

**Answer**: £36

**Splitting the money into parts**

Charlie spent $\frac14$ of his money on a book:

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He then gave $\frac23$ of his remaining money to his brother:

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Then he had £9 left. So each of the equal parts is worth £9.

So he must have started off with 4$\times$ £9 = £36

**Working backwards**

After giving $\frac23$ of his money to his brother, Charlie has £9. So £9 is $\frac13$ of the amount of money Charlie had before he gave money to his brother.

So before he gave money to his brother, Charlie had £9$\times$3 = £27.

That means that after spending $\frac14$ of his money on a book, Charlie had £27. So £27 is $\frac34$ of the amount of money Charlie had before he bought the book.

So $\frac14$ of the amount of money Charlie had before he bought the book is £27$\div$3= £9.

So before he bought the book, Charlie had £9$\times$4 = £36.

**Finding what fraction of his money Charlie has left**

After he spent $\frac14$ of his money on a book, Charlie had $\frac34$ of his original money left.

Then he gave $\frac23$ of this $\frac34$ to his brother, leaving him with $\frac13$ of $\frac34$ of his original money.

$\frac13$ of $\frac34$ is equal to $\frac14$. So Charlie has $\frac14$ of his original money left.

So $\frac14$ of Charlie's original money is £9, so Charlie must have started off with 4$\times$ £9 = £36.

**Using algebra**

Let the amount of money Charlie had at the beginning be $c$.

So he spent $\frac14c$ on a book, which left him with $\frac34c$.

Then he have $\frac23$ of $\frac34c$ to his brother, leaving him with $\frac13$ of $\frac34c$, which was $ £9.$

So $\frac13\times\frac34c= £9\Rightarrow\frac14c= £9\Rightarrow c= £9\times4= £36.$