Math Puzzle: the Missing Dollar
Here is a puzzle suggested by one of this blog’s readers, Sanket Totewar, which is a variant on the classic “where did that one go?” puzzle. Here goes.
Two women sell apples on a street corner. One sells 3 apples for 1$, the second sells 2 apples for 1$. Putting aside your microeconomics classes for a moment, image the following scenario. Each woman has 30 apples. If they were to sell their merchandise entirely, woman A would make 10$ while woman B would make 15$. Total: 25$. So far so good.
Instead, though, they decide to give their apples to a guy who sells 5 apples for 2$ (which is precisely what they would make if woman A would sell 3 apples and woman B 2 apples, a total of 5 apples). This guy sells the apples and gives the profits to the women. However, note that the 60 apples only made them 60/5*2=24$, one dollar less!
How can this be? Where did the dollar go?
The discrepancy here lies in the ratio of apples the women have. If woman A possessed 36 apples (making $12), and woman B possessed 24 apples (making $12), they would make $24 total. But because they sell at different rates while starting with the same amount of apples, woman A will run out of apples first, leaving woman B with 10 apples to sell. If they sell separately (using only their own apples), they will make more money, but if their goal is to match 3 apples to 2 apples, they must split the remaining 10 apples into groups of 5 sold at $2 ($4 of profit) instead of groups of 2 sold at ($5 of profit), thus resulting in a $1 profit difference.
Of course, in a realistic world, there are probably other advantages to having somebody else sell your apples in different quantities.
Thanks for your insight, MC Safety.
Looks like your puzzle was solved pretty fast, Sanket … I’d better hurry and put a new one up!
X=No of apples sold per dollar = 3A/D
Y=No of apples sold per dollar = 2A/D
Z= No of apples sold per dollar = 5A/2D or 2.5A/D
(1) If they were to sell apples individually:
Total profit = 30/X + 30/Y = 30:3A/D + 30:2A/D =25D
(2) If they were to give all to Z to sell.
Total profit = (30+30)/Z = 60:2.5 = 24D.
Obviously it is more profitable to sell apples by each seller individually because apples were sold at a higher rate.