Where on Earth did the \$1 bill go?

by Paul Michael on 1 December 2007 11 comments

I'd like to start by saying I'm awful at math. I mean, really bad. So it will come as no surprise to learn that I just scratched my bald head in wonder when I was told the following story. It may be one you've heard before, or a variation of something that's really old, but I'll tell it anyway. And then maybe one smart WB reader can show me how this puzzle actually is solved, because I'm clueless.

The story starts with three guys (although it could have been three girls, hey, I'm not being sexist here). They're all out on the town and get quite merry in a local bar. Instead of attempting to drive (bad idea guys) or paying for a pricey cab, they decide to share a cheap hotel room and split the bill.

The price for the room, they are told by the receptionist, is \$30 for one night. No problem for our guys, they each cough up \$10 and go sleep off the booze. In the morning, the receptionist gets the bill and realizes the price for the room is actually only \$25. As this guy is as bad at math as I am, he decided to give each guy \$1 back and pocket the remaining \$2 cash.

The guys originally paid \$10 each for the room.

They get \$1 back each, meaning they paid \$9 each for the room.

And the receptionist has \$2.

\$9 x 3 = \$27

Add the \$2 from the receptionist.

\$27 + \$2 = \$29

But they originally paid \$30. So, where on Earth did the \$1 bill go? Help!

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It makes sense if you think about it a different way.

They each payed \$9 for a total of \$27.
\$27 - \$2 (the guy kept) = \$25 paid for the room. You have to subtract that \$2 from what they paid, not add it.

That simple huh? And it made so much sense (without making any sense) before.

Yeah, I have heard it before and I still had to think about it for a minute. The \$2 is taken out of the \$27 that they paid, so it would make no sense to add it.

\$30 = \$25 (room) + \$2 (taken) + \$3 (given back)

Yeah, I came up with the same (correct) result in a different sort of way.

Think of it as if all three guests (nice sexism dodge) paid with ten one dollar bills.

Desk clerk now has 30 \$1 bills. Returns \$1 each to guests, now has \$27 in \$1 bills. Clerk puts \$25 in the till and pockets the remaining \$2.

\$9 x 3 = \$27 is telling you how much the guys paid the receptionist for the room.

\$27 - 2 = \$25 is how much the hotel is getting after the receptionist pocketed 2 dollars.

The confusion arises because the first \$27 ignores the \$3 refund and instead tries to add the receptionists \$2 which is already included in that figure.

[)amien

I Love this one !!! It's all in the presentation. 9 x 3 Is indeed 27 +2 = 29. If you take it apart another way the buck is still there you are just being told otherwise.

25 for the room +1 +1 +1 FOR EACH GUY + 2 IN HOMEY'S POCKET =30

The key to this one is to get the focus on the story math 9x3+2 and not leave room for the real math 25+1+1+1+2

I use this one on the kids at school tell 'em ona friday just to keep 'em going all weekend (It's a wonder they don't hate me LOL)

We considered other systems like "identify the greatest lines from Bob Dylan songs" and "what is the color of Orange"?  But somehow I got out voted.

i'd much rather visualize this story with 3 drunken females sharing the room...

wait...

i guess i am sexist.

I stumped my 11th grade math teacher with this problem. She spent ten minutes reading it over and over before I explained it her her. It was awesome. It was advanced math to boot, so double awesome.

The guy only hands out two dollars, one to each friend. Because he himself is the third guy he is pocketing three dollars!

It's that simple!

My favourite bizarre question is from "Dexter, Boy Genius". In a dream, his sister asks him "What is the purpose of meaning?"

The problem is that while 9x3=27 there is not an extra 2 sitting arround waiting to be added. The \$27 is the amount that is over the counter, \$2 of that \$27 has been pocketd by the clerk. By adding \$2 again to get \$29 you are re-adding the cleark's cut and producing an extra \$2. So really the question isn't where did the \$1 go, it's where did the \$2 come from?