## 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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