## Simple puzzle will confuse all but the accountants

This is a famous brain teaser which supposedly first appeared in R. M. Abraham’s *Diversions and Pastimes* in 1933. The version below was taken from Answers.com

Three ladies go to a restaurant for a meal. They receive a bill for $30. They each put $10 on the table, which the waiter collects and takes to the till. The cashier informs the waiter that the bill should only have been for $25 and returns $5 to the waiter in $1 coins. On the way back to the table the waiter realizes that he cannot divide the coins equally between the ladies. As they didn’t know the total of the revised bill, he decides to put $2 in his own pocket and give each of the ladies $1.

Now that each lady has been given a dollar back, each of the ladies has paid $9. Three times 9 is 27. The waiter has $2 in his pocket. Two plus 27 is $29. The ladies originally handed over $30. Where is the missing dollar?

I’ve seen lots of longwinded explanations for this puzzle; can anyone post concisely what really happened to the missing dollar?

Tags: missing-dollar-problem

Vishvas Vasuki| Guest January 30th, 2007 - 12:01 amTotal bill + The money the waiter pocketed = the money the ladies spent.

Indeed, we find:

25 + 2 = 3 * 9

-Vishvas Vasuki

david| Guest January 30th, 2007 - 12:58 am5$ change..

waiter takes two dollars = 3$ change …

each lady recieves 1$ back…each lady paid 9$…

9×3=27…+2=29…

……..

25$ bill + 5$ change…(equals 30$ total).. waiter takes two dollars… (equals 28$ total)each lady is given back 1 dollar …(equals 25$,the bill)

well it gets screwed up when you multiply, but not when you subtract.

Keni Neal| Guest January 30th, 2007 - 1:03 pmThe puzzle is given in a somewaht misleading manner. To think about it as each lady having given 9 dollars is a little off the mark. In reality each lady gave 8.33 towards the actual bill of 25 (8.33×3=25) and when the waiter gives each a dollar back we have accounted for 9.33 each which equals 28 dollars plus the waiters two and you have 30.

Matthew| Guest January 30th, 2007 - 1:20 pmThe statement “Two plus 27 is $29.” is misleading. As claimed, the ladies actually paid 3 x $9 = $27 total: $25 for the bill plus $2 that the waiter pocketed, which = $27. They also got $3 in change, which adds up to $30. To further explain, the “27″ I quoted above includes only the bill + the waiter’s take. It does not include the change, which was $3. To be accurate, that statement should have read “Three plus 27 is $30.” The missing dollar was the extra $1 in change the ladies received that was not represented in the final paragraph of the riddle.

Ram N| Guest February 2nd, 2007 - 6:07 pmThere is a fallacy in the question posed:

>Now that each lady has been given a dollar back, each of the ladies has >paid $9. Three times 9 is 27. The waiter has $2 in his pocket. Two plus 27 >is $29.

My question is : why are we adding this $2 to the $27. The $27 already includes the $2 that the waiter pocketed.

To put it another way, say that the ladies paid $90 for the bill. The cashier gives back $55 (with the actual cost being $35). Say, the waiter pockets $10 (because $55 is not a nice multiple of 3) and returns back $45. Then according the logic in the puzzle, the ladies spent $30 – $15 = $15 each. And according to the puzzle, the total amount spent would be $15 * 3 + $10 = $55. The we have to ask this absurd question : what happened to the missing $35?

The ladies paid $90 and got back $45. So they paid a total of $45 including the $10 that the waiter pocketed!

Scott| Guest February 10th, 2007 - 5:43 pmThe manager took it.

Scott| Guest February 10th, 2007 - 5:44 pmAs Ram N said, the 27 includes the 2$ the waiter pocketed, so we dont need to add that to the 27, instead we would add the 3 dollars they recieved back, which WOULD make it 30$

didier| Guest February 12th, 2007 - 8:18 pmehehhe…the bill was 25!!!

30 was the wrong price

then the waiter takes 2 $ more

so the final bill is 27

hehehe 9*3

Jean Paul| Guest February 15th, 2007 - 2:29 amIt simply is the wrong focus. 2$ have already been added to 25$ (the bill). You add 3$ (the change) to 27$ (the bill + the waiter’s “pocket money” and you get back to the original 30$.

Geoff| Guest February 15th, 2007 - 8:46 pmalright im not sure if it has been stated already (which im assuming it is), but heres how i saw it. The actual bill is 25 dollars yes? well you then have to divide that but 3 (because of the number of people) rather than dividing the 30 dollars by 3. So what do you get when you divide 25 by 3? 8.3333333 and so on, which is what each person paid. Then add 1 dollar to each amount which then gives us 9.33333333 and multiply that by 3 which gives us a 27.9999999, plus the 2 bucks the waiter kept for himself and you get 29.999999999, not quite the 30 bucks back but its close enough and makes sense doesnt it?

Kyle| Guest February 16th, 2007 - 9:45 pmI found an explanation a lot like these at another website. Click here to see it. I’td be great if the doc could tell me if it’s a good explanation or not. Thanks much.

Koolaid| Guest April 4th, 2007 - 2:04 pmThe Real Bill was $24. The Cashier Kept a Dollar and Gave the waiter $5 so the Waiter kept 2 dollars and gave each lady $3.

The 3 ladies each got $3 dollars back meaning they all paid 8$, soo with there bill being $24 and they got back $3 that equals $27. The Cashier kept $1 when she gave the waiter change and the waiter kept $2. soo

24+3(each lady)=27+2(waiter)=29+1(Cashier)=30

the Cashier screwed it all up by giving him $5 instead of $6

tina| Guest April 10th, 2007 - 10:44 pmwell the extra $5 was already included in the tip so you dont include it when u multiply so its actually $30

David| Guest April 19th, 2007 - 5:06 pmThe real deal is the transaction.

Bill: $-25.00 ($25.00/3= $8.333)(Here is the missing dollar!)

Tendered: $ 30.00

Change: $ 5.00

There is no $9*3 = $27. The charge would have really been $25.00/3= $8.33, not $9.00. The change, to be split equally would have been roughly $1.67 a person. But the change is not at issue. Changing $8.33 to $9.00 produces the $.666 per person deficit, this shows up when the deficit is accounted. The $5.00 change has already been accounted for appropriately as the women and waiter have taken their share, fair or not! What you are witnessing is “misdirection”. They had you looking at the “change” but the real solution is in the “bill”.

Now do the math:

$8.333 x 3 = $25.00 (Bill)

$1 x 3 = $3.00 (Change received)

$ .666 x 3 = $2.00 (Waiter)

————————-

$ 30.00

Twilight_Magester| Guest April 27th, 2007 - 2:52 amnothing. there is no missing dollar. puzzle solved

Adam| Guest May 10th, 2007 - 11:31 amThere is no missing dollar. The ladies paid a total of $27 dollars, $25 that went to the cashier and $2 that went to the waiter.

So, the $2 paid to the waiter is already included in the $27 paid, and the remaining $3 is what went back to the ladies.

Yehuda Groden| Guest May 10th, 2007 - 7:18 pmSice they paid $5 extra, they really only paid $25. the waiter returns $3, which accounts for $28. He stole 2 which accounts for a total of $30.

when the 3 dolllars are returned it accounts for $28 (they spent $25 and are given 3 backe = $25)

Edwin_Bruin| Guest May 12th, 2007 - 5:15 amThe error is not “$9*3 = $27″

That’s fine…

Here lies the error, “Two plus 27 is $25.”

It should’ve been, “Two MINUS $27 is $25″ since they also “paid” the waiter $2; the waiter didn’t contribute to the sum. Wrong concept.

They paid $25 ok.

Each lady gives $1 back ($3 bucks) on the table, and the waiter gives his back ($2), then it’s $5. Then if the restaurant gives its $25 that it kept, then voila= $30 on the table.

Now if we were to say each lady gives $1 back to the waiter, (9+1)+(9+1)+(9+1), each of the ladies will pay $10 to the restaurant. Three times 10 is 30.

Yet, the waiter still has $2 bucks in his pocket?

Loz| Guest May 14th, 2007 - 8:41 amThere is no missing dollar, the equation is just written the wrong way.

It is not how much the women paid + what the waiter kept. How much the waiter kept is irrelevant.

It is how much the women paid – what the waiter kept.

Just as when you give a tip ontop of a bill, the total is bill + tip

THEMAN| Guest May 15th, 2007 - 6:09 pmThis problem is simple. At the beginning all three women paid 10 dollars each which equals to 30, after taking 5 dollars that leaves 25 dollars for the three of them. The waiter got the 5 and took 2 dollars for himself and gave each women a dollar. 3+25=28. When you add 28 plus the 2 dollars the waiter took that equals 30. People assume that each women is going to get 9 dollars each but thats wrong.

wizzard| Guest May 24th, 2007 - 4:13 pmA missing dollar there is not. 25 dollar bill, 2 dollars for the waiter, 1 dollar per 3 ladies. 30 bucks

Talia| Guest May 28th, 2007 - 3:54 amthis should be what happened to the missing cent more then dollar

There is a misconception that each woman has paid 9 dollars, when infact each has paid 9.3333333 etc ( as 25 divide by 3 = 8.333333 etc) so 9.33 x 3 = 27.99 + 2 = 29.99.

I assume that as my calculator and myself can not be bother seeing whats at the end of the 3′s, it in fact would make up the extra cent between the 3 women.

kits| Guest June 9th, 2007 - 2:36 pmit’s not missing…

MJ| Guest June 24th, 2007 - 6:10 amThis question is poorly worded, that’s all. There’s no missing dollar.

$30.00 (what the ladies paid) – $25.00 (what the actual cost was) = $5.00. $5.00 – $2.00 (which I hope the waiter doesn’t spend all in one place… = $3.00. $3.00 divided by 3 ladies = $1.00 each.

Brian M| Guest June 27th, 2007 - 8:21 amThe ladies started out with $10 each ($30)- the boss took $25 the waiter swiped $2 (total $27) and the ladies got $1 each ($3) $27 plus $3 is $30. No missing dollar

Abhijit| Guest June 28th, 2007 - 8:14 pmThere is no missing dollar . If we approach the problem from the angle of the ladies , we have to stick to it . Since they have been returned $3 by the waiter the total is Paid amount + returned amount

= 27 +3

=30

So there is no missing dollar.

Corey911| Guest July 5th, 2007 - 2:01 pmThe women only paid the restaurant $25. Each of the women paid in $9, $9*3=$27. The waiter took two of the dollars so that didn’t go towards the bill so $27-$2=$25. The $5 dollars they got off is made of each of the women having $1 back and the waiter having $2 dollars so, $1+$1+$1=$2=$5. and $5+$25=$30.

Daniel| Guest July 13th, 2007 - 10:05 pmDidn’t read the other explanations, so don’t know if this is more concise:

Perspective of first paragraph:

Ladies gave $30 to cashier (quantity A), who gives $5 to the waiter (quantity B), leaving $25 in the register (quantity C). Waiter gives $3 back to ladies (quantity D) and keeps $2 to himself (quantity E).

So the following is known:

A = B + C

B = D + E

therefore A = C + D + E

Perspective of second paragraph:

Ladies ‘effectively’ give $27 to cashier (C + E = A – D); the waiter has $2 (quantity E). The money in the waiter’s pocket was PART of the $27 originally given to the cashier (C + E), so it’s counted twice; the two quantities never existed at the same time. Once the waiter deducted his $2 from (C + E), it became C = $25–so this is adding quantities that existed at different times.

When the waiter had $2, the ladies had $3, not $27,and theregisterhad $25–and $2 + $3 + $25 still equals $30.—

The problem we have in processing it is that we add quantities without considering that they don’t exist simultaneously. Consider the following story with the same distortion, but magnified:

Three ladies sit down in a restaurant and pay $10 each for a $30 meal. The cashier decides it should only cost $5, so the waiter decides to give each woman back $1 and to keep $22 for himself.

Ladies: $3

Waiter: $22

Cashier: $5

Now that each lady has been given a dollar back, each lady has paid $9. $9 x $3 = $27 paid. The cashier has $22 in his pocket. $27 + $22 = $49. Where are the missing $19?

This should make the distortion clear; the waiter’s $22 have been counted twice, while the ladies’ $3 (which is what they had when the waiter had $22) wasn’t counted at all.

Robert L| Guest July 14th, 2007 - 7:16 amyou have to understand that the cashier is one where the ladies are 3. SO when the money went to the cashier he/she divided what was not possible by the 3 ladies therefor even though it looks like a paradox, what you do not see is what the cashier did.

Ian| Guest August 13th, 2007 - 2:32 pm25 in cashier

3 with ladies

2 with waiter

KC| Guest August 21st, 2007 - 2:23 pmThis puzzle is cleverly phrased.

But basically, you can’t add what was paid out (by the ladies) to what was gained (by the waiter) because they are not on the same side of the accounting column.

dave| Guest August 26th, 2007 - 11:03 amHere is another way of looking at it:

Before: 3 ladies each have a $10 bill. There are 5 $1 coins in the register.

Total amt: 10 + 10 + 10 + 5 = 35$

After: There are 3 $10 bills in the cash register, the waiter a 2 $1 coins in his pocket. 3 ladies each have a 1$ coin.

Karthik Ramachandran| Guest November 19th, 2007 - 5:38 amThe question is misleading us in this part

“Now that each lady has been given a dollar back, each of the ladies has paid $9. Three times 9 is 27. The waiter has $2 in his pocket. (Two plus 27 is $29). The ladies originally handed over $30. Where is the missing dollar?

”

they already gave 27 and this includes the what waiter took i.e. 25+2=27.

it should be 27 + 3 what the waiter handed over to the ladies and not

27 + 2 what the waiter took.

Hey got u……………ha ha ha

The One Eyed Man| Profile March 25th, 2008 - 3:51 pmBefore paying:

$10 per lady ($30 total)

After paying:

Cashier: $25

Waiter: $2

Ladies: $1 each ($3)

25 + 2 + 3 = 30.

The misdirection in the puzzle is that the amount kept by the waiter, $2, was being ADDED TO instead of subtracted from the total amount paid. $27 (total paid by all 3 ladies) – $2 (kept by waiter) = $25 (amount paid to cashier). The puzzle makes it sound like $29 was paid, which is not the case.

marjer427| Profile February 10th, 2009 - 8:06 pmThe waiter’s 2 dollars was already included in the 27 dollar amount, thus it’s being counted TWICE. By saying 9 x 3 = 27. the 2 dollars is already included in that $27 total. The only money left out of the equation is the $3 from the ladies. DUH!

ksharp| Profile February 10th, 2009 - 9:47 pmActually what you end up with is this:

(30 – 2)/3 = 9.33.. — What the ladies each paid

or this

(25/3) + (3/3) = 9.33..

narbat1| Profile March 25th, 2010 - 6:11 pmI dont see the problem…. 25 with the cashier, 2 in the waiter’s pocket and 3 with the ladies…. the question setting up a wrong equation. If the ladies paid 9 each, that means that they spend a total of 27, but you don’t add 2 (what the waiter takes) to that amount, because the waiter takes the money from THAT amount, in other words you would subtract, but that’s not what we want, it’s only to show that that the equation is incoherent, the bill was for 25, plus 2 that the waiter takes makes 27. If each lady spends 9 dollars, for an amount of 27 (including what the waiter takes) you will only be missing 3 dollars, which are in each lady’s pocket! The question is only trying to trick us by setting a wrong equation and making it look really hard when is only simple math.

Eddy

joyo036| Profile December 23rd, 2010 - 1:09 pmIt is not missing. Basically, there was 30 in the till -5 taken back by waiter so 25 in till 5 not in till. that five is split into a set of 3 for the women, and 2 for the waiter so

25=till

2=waiter

3=women

equals $30

fanfan| Profile February 23rd, 2011 - 5:49 pmWell. Ladies paid 27, of which 25 went to the restaurant, and 2 to the waiter. There is no missing dollar. Put another way: 30 paid= 25 to restaurant, 2 to waiter, and 3 back to ladies. the misdirection is in the statement “Three times 9 is 27. The waiter has $2 in his pocket. Two plus 27 is $29″; the 27 includes the $2! Adding it to the 27 is double counting.

Nadim.bakry| Profile November 12th, 2011 - 3:52 amIt’s too simple guys, the new real price is 25 $, plus the 2$ tips for the waiter, equal 27 $ soo, 30-27= 3 $, this 3$ are giving to the ladies, so, we attend this result: It’s a fake, no missing coins

koolviet| Profile November 16th, 2011 - 12:59 am3 times 8.33 is 25. 3 times the missing 33 cent is 1$ rounded off. that is where the missing dollar is at. the missing dollar got spited 3 ways to the customers.

CJ_S| Profile February 8th, 2012 - 4:44 pmNow, I have no verification as to the right answer. Never looked into replies about the brainteaser question. Though, I believe I have an explanation.

what matters little in the question is the the social situation provided and what matters most is the math, specifically in the division of the bill. If it was 30/3 than ten is correct, if a revised bill is $25.00 than that should be divided also by three to obtain a split or average. 25/3 = 8.3333 repeating. So, the missing dollar is the difference between what they were said to have been charged ($9.00) by the waiter tipping himself and the actual amount they we’re sharing, 3 ladies, .33 1/3 cents in each of their pockets. Voila, the missing dollar.

salih509| Profile February 17th, 2012 - 3:42 pm30/10 = (10+10+10)/3 = (25+5)/3 = 25/3 + 5/3 ? 8.33 + 1.67

waiter :3 times 0.67 ? 2 (each lady has been paid 0.67 for the waiter)

then, 8.33 + 1.67 – 0.67 ? 8.33 + 1 ? 9.33

9.33 – 8.33 = 1 dollar per 3 ladies .No missing dollar.

Roza| Profile June 1st, 2012 - 1:45 pmthere is no missing dollar,

they paid $25 the waiter took $2 and they got back $3

25+2+3=30

Scholar gypsy| Profile January 23rd, 2013 - 8:20 amHv no idea..what would be the answer is?

ryhiah| Profile February 17th, 2013 - 1:48 pmIf each lady were to split $25 and there were three ladies then you would say $25/3 = $8.33333333 so then if you add $8.33333333 three times you get exactly $25 dollars with the waiter returning each lady $1x 3 ladies= $3; $25 + $3= $28 and then the waiters $2 + $28= $30

What I’m trying to say is they didn’t split the bill evenly math is wrong cause its rounded to $9 each

So, there you go

Dannyboy| Profile February 28th, 2013 - 6:20 pmThere is a missing dollar. It being a puzzle, you would assume all the facts given are correct, but it appears as though the aim of this puzzle is to find the problem within the puzzle itself. ”each of the ladies has paid 9$” is an incorrect statement. Each of the ladies has actually paid 9.33$. For the revised bill, each lady has paid 8.33$. You therefore add the dollar received to the 8.33$ paid. You do NOT subtract the dollar from the original 10$ Paid, which is what the puzzle has done. The aim of the puzzle was to discover that the puzzle was lying to you. There is a missing dollar with the way the puzzle is phrased, or ”incorrectly phrased” if you’d prefer.

JayofMaya| Profile April 4th, 2013 - 6:32 pmOnce the waiter took the 2 coins, he was officially involved in the transaction and for the sake of ease will become a payee (Who for some reason gets his whole input back). So he must be involved now into the equasion too.

The bill is 25 coins. The ladies PLUS the waiter paid a total of 10 coins each, even if the waiter only put a small amount in (2 coins) it must be divided in that way. So the three ladies paid 8.333* in the bill,

and the waiter kept an equal fraction of the 3 way split change .666*.

(It’s basicaly about reversing what was done in the question and this is a totally irrelevant and long answer)

The ladies split the 28 coin bill, (Due to the waiter stealing 2 coins) so that now must be divided by three, which equals 9.333*. The waiter won’t divide his 2 dollars because it’s unnecessary.

Well, of course there is an infinitely small fraction lost in the time and space continuum, but we shall forget about that. The basic problem is the way in which the question was asked and the fact that you can not divide a round number by three.

Lurch666| Profile April 10th, 2013 - 4:35 pmHeard this puzzle years ago and it took me a while to figure it out but alot of you have got it the wrong way round.

the answer is simple:

The meal DIDN’T cost $30 it cost $25 so the correct way of saying the answer is three ladys paid $9 which means they paid $27 in total LESS the $2 the waiter kept equals $25 which was the price of the meal.

You subtract the amount the waiter kept from the total not add it because it’s money that was taken

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