The missing penny

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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?

John123 Expert Asked on 30th July 2015 in

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.

ravi Expert Answered on 30th July 2015.

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