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!
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