## Sharing Money Equally

Suppose two people Amy and Beth start with an a total of £100. Amy gives one third of her money to Beth and Beth gives one third of the money she now has to Amy. As a result, Amy and Beth now have the same amount of money each.
 Transaction Amy Beth $A$ $B$ Any gives one third of her money to Beth $\frac{2A}{3}$ $B+ \frac{A}{3}$ Beth gives one third of her money to Amy $\frac{2A}{3} + \frac{1}{3} (B+ \frac{A}{3})$ $\frac{2A}{3}(B+ \frac{A}{3})$
The amounts are now equal so
$\frac{2A}{3} + \frac{1}{3} (B+ \frac{A}{3}) = \frac{2}{3}(B+ \frac{A}{3})$

$6A+3B+A=6B+2A$

$5A=3B \rightarrow 5A-3B=0$
(1)
Obviously
$A+B=100$
(2)
(1)+ 3(2) gives
$8A=300 \rightarrow A=37.5$
then from (2)
$B=100-A=100-37.5=62.5$
.