The Motel Bellboy and the Missing Dollar

Date: 01/28/98 at 23:33:04
From: Cindy Holcomb
Subject: Missing dollar

Do you have a quick answer to this one?

Three men go to a motel and rent a room. The deskman charges them $30 
for the room. The manager of the motel comes in and says that the 
deskman has charged them too much, that it should only be $25. 

The manager then goes to the cash drawer and gets five $1.00 bills, 
and has the bellboy take the money back to the three men. On his way 
up to the room, the bellboy decides to give each of the men only one 
dollar apiece back and keep the other two dollars for himself. 

Now that each one of the men has received one dollar back this means 
that they only paid $9.00 apiece for the room. So three times the 
$9.00 is 27.00 plus the $2.00 the bellboy kept comes to $29. Where is 
the other dollar?

My thinking out loud got me nowhere:

I think it's because the remainder of dividing the difference of 25 
and 30 is 5 divided by 3.  1.66 is what they each should have had 
returned. Now what?

1.66 X 3 is 1.98. They should have had that returned (each), but only 
got 3 dollars among them.

Can you help? I think it stems from the difference between 25 and 30 
and the difference between 27 and 30 but I just can't put it in words.

25 divided by 3 should have been based on the return amount.

They returned three dollars so do you have to take 25 plus 3, not 
9 x 3? Is this just a play on words or is it really math?

I'm a new middle school math teacher - a misplaced English teacher - 
and I hate to get these from students to solve.  I guess it's good for 
me, but I'd appreciate your help very much!

C. Holcomb

Date: 01/29/98 at 08:56:34
From: Doctor Pete
Subject: Re: Missing dollar

Hi,

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 was $5, the 
bellboy taking $22, the men getting $3. Then it becomes clear that the 
$27 that the men wound 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 paid is now in the bellboy's 
pocket, so adding $22 to $27 is in essence counting the bellboy's 
money twice.

I know that was a pretty long answer, but it's a simple argument that 
I wanted to make as clear as possible, with as many different lines of 
reasoning, so you could explain it to your students in whichever way 
they might understand it.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 01/29/98 at 11:42:19
From: Cindy Holcomb
Subject: Re: Missing dollar

Dear Doctor Pete:

Thank you so much for your response. It provides a lot more to the 
answer on the website as far as explaining it to kids, who seem to 
respond well to tables!