# Here's an ancient puzzle [duplicate]

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 1¢ is surely exactly the same as five for 2¢.

Can you explain the little mystery?

The stock

Woman A: 30 apples (originally sold at two for a penny)
Woman B: 30 apples (originally sold at three for a penny)

Indeed, if each sold their apples individually, there be a total income of 25 cents (Woman A = 30/2 = 15 cents; Woman B = 30/3 = 10 cents)

It would seem that this is the same result you get by combining the stock and selling five for two pennies. However:

Consider that each woman contributes apples in sets according to their original price (Woman A contributes sets of 2 apples, and Woman B contributed sets of 3 apples). Woman B is actually contributing one extra apple to each sale. Thus, once 10 sets of five apples are sold (yielding 20 cents), Woman B is out of apples. At this point, Woman B has sold 30 apples and has earned her expected 10 cents (a penny per three apples) and Woman A has sold 20 apples and has earned 10 cents (a penny per two apples).

There remain 10 apples from Woman A. At two cents per five apples, this brings in five additional pennies, thus yielding a total of 25 cents.

• Not the way I solved the problem but you're still right.....I think. Commented Aug 1, 2020 at 23:17