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The Extra Dollar in the Hotel

Logic puzzles require you to think. You will have to be logical in your reasoning.


Puzzle ID:#110
Fun:*** (2.74)
Difficulty:** (2.05)
Submitted By:giraffe**
Corrected By:Winner4600




Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room, the bellboy reasons that $5 would be difficult to share among three people, so he pockets $2 and gives $1 to each person. Now, each person paid $10 and got back $1. So they paid $9 each, totalling $27. The bellboy has $2, totalling $29. Where is the remaining dollar?

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Mar 27, 2002

An old riddle, but a good one.
Jul 11, 2002

good one
Sep 02, 2002

Dec 29, 2002

g8t teaser old but i cudnt reember the answer until now thanxs lol
Feb 19, 2003

Could someone explain WHY you can`t do the math 27 + 2, other than because it doesn`t work... WHY doesn`t it work???
Apr 05, 2003

It's a cash flow thingie. You look at who paid out what and where it went. The men paid $30 out but got $3 back so they really only paid out $27. Now where did the $27 go? $25 went to the manager and $2 to the bell-hop.
Apr 06, 2003

I understand the "cash flow" analysis, but it doesn't answer why, mathematically, it's wrong to calculate the problem as presented. I seem to recall something about the problem, as stated, mixes the associative and distributive properties, which is improper, but I don't really understand what that means and why. Does anyone else know?
May 18, 2003

I can field this one - the $27 + $2 math doesn't work because the $2 is PART OF the $27 they paid ($25 to the hotel, and $2 to the bellhop). "They each paid $27, plus the $2 for the bellhop..." - well, that counts the $2 twice now, and doesn't count the money in the guests' pockets. The riddle leads you to try to add part of the payment to the total payment, instead of adding what was paid to what was not paid. That's why it doesn't work; it has nothing whatever to do with associative or distributive anything!
Aug 08, 2003

Jan 22, 2004

This is a classic case of misdirection. You are given a scenario with a bunch of numbers. You are then given a series of statements that use those numbers, and are intentionally misdirected to add the bellboy's $2 to $27, thus creating the missing dollar. Not realizing that you have been tricked, you spend the rest of your life wondering where the dollar went.
Jan 23, 2004

old but nicely put nice teaser
Jan 28, 2004

The answer it gives seems a little confusing to me. I look at it like this: In the end they paid $25 for the room, the Bellboy has $2 and each person has $1 dollaer, 1+1+1=3. Thus the $30 is accounted for!! Just my way of looking at it anyway.
Mar 29, 2004

ok you understand the cash flow thingie, but you want to know why you cant just add 27 and 2 as stated in the problem. Most simply put your adding when you should be subtracting. the two dollars should be subtracted from the 27 to give you 25, the actual rate for the room.
Jun 22, 2004

Classic 1 but I still don't get it.
Jun 23, 2004

It confused me but I got it. Pretty funny!
Dec 31, 2004

I think I have a different answer. When the guys pay the 30 dollars, the lady realizes they only should have paid 27 dollars, so she take out an extra 5 dollars, which makes the total 35 dollars, gives it to the bell hop, he pays the guys each a dollar... and ends up with 32 since he's giving money back... and keeps the last 2 dollars... which would be taking money away as a tip, so theres the last 2 dollars subtracted to get to 30 dollars. Am I a genuis or what? Just kidding... but it makes sense if you understood that. It kinda makes sense.
Jan 31, 2005

Cool teaser
I guess you are deliberately misdirected from figuring the money that the manager has into the equasion... either way it's mathematics like that that have lead to me losing SO MUCH MONEY over the last couple of years. I need to keep a better eye on my budget
Mar 24, 2005

Wow,that was good!!!I think ive haerd it before.But not on this website.
Apr 20, 2005

That was a nice one.

You really had me thinking there.

Jun 07, 2005

why does the bellhop get $2? good teaser tho
Sep 14, 2005

i am a 6 grader and they gave me this exact one. what is the longest time it took to find this out
Oct 31, 2005

Actually, by definition, the answer to this question is wrong.

The answer to this question deals with the mediumof money. Money only uses two decimal places. In order to account for the full extra dollar, one must calculate the division of the dollars, observe the decimals, and NOT round. This way, the full value of the missing dollar is represented. It has to do with theoretical math and imaginary numbers.

But technically, this is impossible... because you need all... eh... twelve decimal places to make up for the missing dollar.

PS- My college Calc class worked this problem out today using the imaginary numbers theorum.
Nov 06, 2005

yes, the question was where was the extra dollar, not how to figure out-well, i don't exactly get what answer is being answered
Dec 21, 2005

Jan 27, 2006

i'm not good with numbers...
Jan 27, 2006

My dad has been telling this one for years....and I still don't get it!
(user deleted)
Feb 02, 2006

This was a great teaser. Thumbs up. I had to really think about it. I emailed it to my mother, and she drove me crazy trying to figure it out.
Feb 09, 2006

WOW . I'm confused. Too hard for me. Great one though!
Feb 11, 2006

It is an oldie but still clever and very fun. Thanks for the teaser!
Mar 12, 2006

The problem may be hard and confusing but it has a simple answer. No theoretical math is required, certainly not imaginary numbers. I have no idea how this problem could have anything to do with imaginary numbers or calculus.
Mar 18, 2006

Mar 18, 2006

Great teaser, keep these coming.
Mar 25, 2006

Awesome teaser.
It never struck my mind that the bellboy's 2 dollars counted toward the cost of the room & must be added to the amount that the manager was holding.
Very nice.
Apr 12, 2006

intresting this guy ended up telling this at a camp i went to but got the answer all messed up and said it was a mathmatical error so i asked my dad and he explained it to me like this. awesome
May 13, 2006

wow that made my brain hurt!
great teaser!
May 19, 2006

good one, i worked it out
Jun 01, 2006

my grandpa tought me this when i was really little. i got it right!
(user deleted)
Jun 05, 2006

so where is the extra dollar actually then?

i understand how to figure out the details, but wheres the extra dollar?

how would u answer that?
Jun 07, 2006

hotrodmeg - there was no extra dollar. The men paid 30 dollars and got 3 back. The manager recieved 30 dollars, but paid 5 back. The bellboy got 2 dollars.

The men are at -27 and the manager is at +25 and the bellboy is at +2. The 27$ that was paid is still present.

Make sense?
Jun 29, 2006


I'm usually good at math, but I got stumped
Jul 11, 2006

Aug 08, 2006

$30 a hotel room! Where's that at? It's cheaper then the rent at my house!
Aug 22, 2006

to answer this, you would divide the total of the room, $25, by three. therefore each man originally paid $8.33 for the room. add the extra dollar they got back and the real total per man is $9.33, not $9 each as the question states. times that by three is $27.99, plus $2 is $29.99. so where did the extra penny go?
Aug 30, 2006

the following day two men go and stay at the same hotel and pay $30 and go off to their room. the manager forgot again that it was only $25 and gave the bellboy $5 to give to them. the bellboy this time keeps $3 for himself and gives $2 back to the men. take tha away from the $30 they originally paind makes $28 plus the bellboy's $3 equalling $31!
There's your missing dollar!
Sep 03, 2006

lol@Battery I think this was a good teaser. It is in the trick category so it is meant to attempt to get you thinking in the wrong direction. No calculus or imaginary numbers, maybe heard before, but still what a trick teaser should be.
Oct 13, 2006

I like the way you explained it. I never actually understood that riddle before but now I do. Thanks!
Nov 01, 2006

A classic example of a "nonsense" sum. Just because one can make a statement that such and such equals something, does not make it true. It has to make sense at each step along the way and in the entirety.

The hotel has $25. The bellboy has $2 and each of three guests have$1. 25+2+1+1+1=30. Nothing is missing.

In fact, they "paid" $27 for the room -- $2 to the dishonest bellboy and $25 to the manager. Adding the $2 to this $27 AGAIN is just nonsense -- so of course it produces a meaningless total of $29.

To the folks who still had trouble understanding... the trick is really about the language of the problem and NOT the arithmetic.
(user deleted)
Dec 18, 2006

I heard this riddle when I was living in Germany but the man didn't know the answer. That was 6 years ago and I continued to ask people this riddle to try to get the answer but to no avail...I'm glad I finally have the answer I can now let my husband know too because we were both stumped!
Feb 08, 2007

same here, my daddy has told this story, but he doesn't say anything about the bellhop having 2 dollars
Aug 07, 2007

Very tricky teaser, but still VERY FUN!!!
Sep 28, 2007

i honestly DONT GET IT.
Jan 09, 2008

i'm 32 years old and i first heard this when i was in 5th grade (does that tell you something?)... simply put, there is no missing dollar... it's called ORDER OF OPERATIONS... you cannot take an addition/substraction equation and turn it into a multiplication equation...

$30(original payment) - $5(refund)=$25(actual payment)

they did not pay $27 for the room, they paid $25... it's not their fault the bellhop ripped them off $2 ($25 + $2=$27) and only gave them each $1 back ($27+$1+$1+$1=$30, the original payment)

anymore questions???
Jan 16, 2008

here' something i thought of last nite...

if the bellhop is as much a thief so as to take $2, then he should just stick all $5 in his pocket and not say anything to anyone...

problem solved
Apr 15, 2008

Yeah, it's just the way it's worded, but it's worded so well that you just don't notice at all...

They get back the dollar each, and then they would say "We each now have paid 9$ making a total of $27." Right, $27 is how much they believe they paid for the room now, because they got a dollar back each from the bellboy. The manager has $25, and the bellboy still has $2, which makes the $27 they think they paid. They first each gave ten making it a total of thirty. Then they each got back a dollar, making the total they paid 27. The bellboy has two, so the total money is now not 30, but 27. the manager has 25 and the bellboy has 2. The trick is that the thirty, changes when they each get back the dollar! So the total is no longer 30. It was only 30 until they each got the dollar back.

I'm not gonna think about this anymore, that's it! Wow, that is one tricky teaser!

Apr 20, 2008

The title is a mistake. It should say "The Missing Dollar in the Hotel."
(user deleted)
Aug 07, 2008

The $1 was lost in the maths. Like people said the bellboys $2 should have been subtracted from the $27 not added, as that is double counting it as the bellboys $2 is already included in the $27.


To answer the actual question of 'Where is the "missing" dollar?' The difference is between the $3 returned and the $2 the bellboy has. $3 - $2 = $1.

e.g. Change the question to the room rate was $26, the manager returns $4 but the bellboy keeps $1. Then you have 3 x $9 = $27. The bellboy has $1, so $28. Where are the missing $2 dollars?

Answer: the difference between the $3 returned and the $1 the bellboy has. $3 - $1 = $2.

e.g.2. The room rate was $24 dollars. Manager returns $6. Bellboy keeps $3 (because he's a thief!) and returns $3. 3 x $9 = $27. Add the $3 with the bellboy = $30.

Here there is no missing $ because there is no difference between the amount the bellboy kept and the amount he returned.
Jul 02, 2011

So, the three men payed $30, $10 each. The room only costed $30. The manager gave the bellhop $5 to return to the men. The bellhop thought it was difficult to share $5 between 3 people and gave them $3, keeping $2. Now, the three men payed $10 each, getting $1 each back. So they payed 3x$9, or $27. Now the bellhop has $2, making the total $29. Adding the $3 they got back, it equals $32. Where did the extra $2 come from?

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