# The missing penny

Here is an ancient puzzle that has always perplexed some people. Two market women were selling their apples, one at three for a penny and the other at two for a penny. One day they were both called away when each had thirty apples unsold: these they handed to a friend to sell at five for 2¢. It will be seen that if they had sold their apples separately they would have fetched 25¢, but when they were sold together they fetched only 24¢.

“Now,” people ask, “what in the world has become of that missing penny?” because, it is said, three for l¢ and two for l¢ is surely exactly the same as five for 2¢.

Can you explain the little mystery?

The explanation is simply this. The two ways of selling are only identical when the number of apples sold at three for a penny and two for a penny is

in the proportion of three to two. Thus, if the first woman had handed over 36 apples, and the second woman 24, they would have fetched 24¢, whether sold separately or at five for 2¢. But if they each held the same number of apples there would be a loss when sold together of l¢ in every 60 apples. So if they had 60 each there would be a loss of2¢. If there were 180 apples (90 each) they would lose 3¢, and so on. The missing penny in the case of 60 arises from the fact that the three a penny woman gains 2¢, and the two a penny woman loses 3¢. Perhaps the fairest practical division of the 24¢ would be that the first woman receives 9.5¢ and the second woman 14.5 , so that each loses 0.5¢ on the transaction.

### Your Answer

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