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 | \[\frac{2A}{3} + \frac{1}{3} (B+ \frac{A}{3})\] | \[\frac{2A}{3}(B+ \frac{A}{3})\] |
\[\frac{2A}{3} + \frac{1}{3} (B+ \frac{A}{3}) = \frac{2}{3}(B+ \frac{A}{3})\]
\[6A+3B+A=6B+2A\]
\[5A=3B \rightarrow 5A-3B=0\]
(1)Obviously
\[A+B=100\]
(2)(1)+ 3(2) gives
\[8A=300 \rightarrow A=37.5\]
then from (2) \[B=100-A=100-37.5=62.5\]
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