The Apple Conundrum

Two women are selling apples. The first sells 30 apples at 2 for $1, earning $15. The second sells 30 apples at 3 for $1, earning $10. So between them they’ve sold 60 apples for $25.

The next day they set the same goal but work together. They sell 60 apples at 5 for $2, but they’re puzzled to find that they’ve made only $24.

What became of the other dollar?

Here, could be that lost dollar! 

Behind The Apple Conundrum

What is the conundrum?

They sell 60 apples at 5 for $2, that means 12 such sets of 5 apples. Suppose, out of each such set, 1 woman takes out $1 for 2 apples and other takes $1 for 3 apples. So, first woman earns $12 by selling 24 apples and second woman sells 36 apples for $12.

In short, first woman gives away 6 apples (from her 30 apples) to second woman increasing her count to 36 reducing her own count to 24. First woman would have made $3 from those but second woman only made $2 from those 6 apples. And there is that lost dollar in earning.

So, 60 apples can't be divided equally to find the earning as they had sold apples at different rates on previous day.

Other way, if they wanted to sell apples together with 30 apples each, then they should have sold apples at average of (1/2 + 1/3)/2  = $5/12 per apple (i.e. 12 apples for $5) instead of $2/5 per apple.

The difference in price per apple (5/12 - 2/5) = (1/60).

So the difference in earning after selling 60 such apples = (1/60) x 60 = 1.

And there is that other dollar! 

What's wrong gone here on next day? Instead of averaging dollars per apple, apples per dollar are added directly which resulted reduced cost of each apple.