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.
How much did each start with?
Let Amy start with £A and let Beth start with £B. The sequence of transactions is summarised in the table.

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


  (1)
Obviously    (2)
(1)+ 3(2) gives    then from (2)  .