Associated Topics || Dr. Math Home || Search Dr. Math

### Tricky Word Problem - The Missing Dollar

```
Date: 2/26/96 at 1:7:14
From: Robert S Moran
Subject: Impossible math problem?

A friend of my Mom's gave us this problem.  It has had engineers
stumped at a nearby Martin-Marietta plant.  I would appreciate your
helping us solve this problem.

Here it is: Three men go to stay at a motel and the clerk
charges them \$30.00 for the room. They split the cost ten
dollars each. Later the manager tells the clerk that he over-
charged the men and that the actual cost should have been
\$25.00. He gives the clerk \$5.00 and tells him to give it to the
men. But he decides to cheat them and pockets \$2.00. He then
gives each man a dollar. Now each man has paid \$9.00 to stay in
the room and 3 X \$9.00 = \$27.00.  The clerk pocketed \$2.00.
\$27.00 + \$2.00 = \$29.00.  So where is the other \$1.00?

Thanks,
Mark Moran
```

```
Date: 2/26/96 at 10:43:55
From: Doctor Ethan
Subject: Re: Impossible math problem?

Great question.

When I was in the sixth grade I was asked this one, and I think
that I was stumped for the next 3 years.  (Although I didn't
work on it that whole time.)

The problem is that the question is always cleverly phrased to
conceal what is really going on.  Since I don't want to just
give you the answer, I will tell you how I think about it and
see if you understand it.

First let's locate all that money.

There are two ways to think about how much money is out there to
be found.The way that this question is tricky is that it
combines that two ways.

The first way is this.
How much money did the guys pay originally?

The second way is this.
How much money did they end up paying?

So if it is the first way, then clearly the total we need to
acccount for is 30.  So let's see what happens to that 30
dollars.  The owner keeps 25, the clerk gets 2 and the guys get
3 back.

Now let's look at the second way.

How much money did they end up paying?  Well 27 dollars, and 25
of it went to the owner and 2 to the clerk.

The problem with the question is that the the two dollars that
the clerk gets are contained in the 27 dollars that they end up
paying so we shouldn't expect them to add to anything
meaningful.

Hope this helps some.

-Doctor Ethan,  The Math Forum
```

```
Date: 07/10/97 at 09:19:00
From: Conor Faughnan
Subject: Puzzle

I have a conundrum which I cannot solve but you probably can:

Three guys share a meal in a restaurant and split the bill.
The waiter charges them \$30, so they pay \$10 each. However,
the waiter comes back afterwards and says he has overcharged
them and the bill should have been \$25 dollars. Of the \$5 they
are owed, they agree to take \$1 dollar each and tip the waiter
the remaining \$2 to thank him for his honesty. They have now
each paid \$9 for the meal.

But: three nines are \$27, plus the \$2 which the waiter got
is \$29. Where did the extra dollar go?
```

```
Date: 07/11/97 at 11:38:07
From: Dr. Math
Subject: Re: Puzzle

Be careful about accepting what you are told! The flaw is in
the phrase "plus the \$2 which the waiter got." This should not
be added, it should be subtracted, since the \$2 the waiter got
is part of the \$27 dollars the three guys spent altogether.
If you subtract the \$2 from the \$27 you get the \$25 that goes
into the restaurant till.

These guys are real cheapskates. Originally I suppose they weren't
going to tip the waiter at all, and then they tip him a paltry \$2
on a \$25 bill!

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

```
Associated Topics:
Middle School Puzzles

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search