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Saturday, July 26, 2008

All Apples

On the market, mrs. Jones and mrs. Smith sell apples. Mrs. Jones sells her apples per two for 0.50 euro. The apples of Mrs. Smith are a bit smaller; she sells hers per three for 0.50 euro. At a certain moment, when both ladies have the same amount of apples left, Mrs. Smith is being called away. She asks her neighbour to take care of her goods. To make everything not too complicated, Mrs. Jones puts all apples to one big pile, and starts selling them for one euro per five apples. When Mrs. Smith returns at the end of the day, all apples have been sold. But when they start dividing the money, there appears to be a shortage of 3.50 euro.

The Question: Supposing they divide the amount of money equally, how much does mrs. Jones lose with this deal?

Answer: The big pile of apples contains the same amount of large apples of a quarter of a euro each (from mrs. Jones), as smaller apples of one sixth euro each (from mrs. Smith). The average price is therefore (1/4 + 1/6) / 2 = 5/24 euro. But the apples are sold for 1/5 euro each (5 apples for 1 euro). This means that per sold apple there is a shortage of 5/24 - 1/5 = 1/120 euro. The total shortage is 3.50 euro, so the ladies together started out with 420 apples. These are worth 1/5 × 420 = 84 euro, or with equal division 42 euro for each. If Mrs. Jones would have sold her 210 apples herself, she would have received 52.50 euro.

Conclusion: Mrs. Jones loses 10.50 euro in this deal.

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