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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?

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.

Hey, that adds up fine.

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.

Hey, that adds up too.

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 

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
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