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  The Missing Dollar  

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Three men go to stay at a motel, and the man at the desk charges them $30.00 for a room. They split the cost ten dollars each. Later the manager tells the desk man that he overcharged the men, that the actual cost should have been $25.00. The manager gives the bellboy $5.00 and tells him to give it to the men.

The bellboy, however, decides to cheat the men and pockets $2.00, giving each of the men only one dollar.

Now each man has paid $9.00 to stay in the room and 3 x $9.00 = $27.00. The bellboy has pocketed $2.00. $27.00 + $2.00 = $29.00 - so where is the missing $1.00?



This question has been sent to Dr. Math many times. Here's a sampler of answers from a variety of 'math doctors':

  1. From Doctor Ethan:

    The problem is that the question is always cleverly phrased to conceal what is really going on. Since I don't want to just give you the answer, I'll tell you how I think about it and then you can see if you understand it.

    First let's locate all that money. There are two ways to think about how much money is there, and the trick in this question is that it combines the two ways:

    1. How much money did the men originally pay?
    2. How much money did they end up paying?

    For (a), we need to account for $30. The owner keeps $25, the bellboy gets $2, and the men get $3 back. That adds up fine.

    Now let's look at (b). How much money did the men end up paying? $27, of which $25 went to the owner and $2 to the bellboy. That adds up too.

    The problem with the question is that the $2 the bellboy gets is already contained in the $27 that the men end up paying, so we shouldn't expect adding that $2 to anything to be meaningful.



  2. Dr. Rob says:

    Since each man has now paid $9 for the room (3 x 9 = 27), and the bellboy has $2 in his pocket (27 - 2 = 25), the rest of the money is in the hotel till.

    The trick is to realize that the $2 has to be subtracted from the $27, not added to it.



  3. Dr. Wilkinson adds:

    "...three nines are $27, plus the $2 which the bellboy got is $29. Where did the extra dollar go?"

    Be careful about accepting what you are told! The flaw is in the phrase "plus the $2 which the bellboy got." This should not be added; it should be subtracted, since the $2 the bellboy got is part of the $27 dollars the three men spent altogether. If you subtract the $2 from the $27 you get the $25 that goes into the till.



  4. Dr. Pete elaborates:

    Write out a table:

    Deskman     Bellboy     Men
    ----------------------------
       $0         $0        $30   <-- men have not yet paid for room
      $30         $0         $0   <-- men pay deskman
      $25         $5         $0   <-- deskman pays bellboy
      $25         $2         $3   <-- bellboy stiffs men
    ----------------------------
      $25         $2       -$27   <-- what each group of people has 
                                      after all the transactions
                                           
    
    Here, the last row is simply the difference between row 4 and row 1. In all but the last row, the sum of the dollar values along each row is constant and equal to $30. In the last row, the apparent fallacy is that the men and the bellboy should have 30 dollars between them, but this statement is false, as it obviously ignores the question of what the deskman has. In fact, the correct statement about the last row is that the sum of what the deskman and the bellboy have must equal the debt of the three men.

    The men have collectively paid 27 dollars for the room, which is obvious, since the bellboy took $2 and the actual cost was $25. And so we see that there is no missing dollar, because the $27 the men paid is a debt, written as a negative number, and the $2 the bellboy took is a profit, which is a positive number, and the sum is not $29, but a debt of $25, which was paid to the deskman.

    To exaggerate the example, suppose the cost of the room is $5, the bellboy taking $22, the men getting $3. Then it becomes clear that the $27 that the men wind up paying for the room "plus" the $22 the bellboy takes just doesn't equal anything meaningful. What's going on is that $22 of the $27 that the men pay has wound up in the bellboy's pocket, so adding $22 to $27 is in essence counting the bellboy's money twice.



  5. Dr. Rothman numbers the dollars:

    Let's give each of the $30 a number from 1-30, keep track of each individual dollar, and see how the problem works.

    Dollars numbered 1-30 are given to the manager. Then he wants to give $5 back, so he keeps the dollars numbered 1-25, and gives numbers 26-30 to the bellboy in the form of a five dollar bill. The bellboy splits up the five to get 5 one's: numbers 26, 27, 28, 29 and 30. He gives numbers 26, 27 and 28 to the customers and keeps numbers 29 and 30 for himself.

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