Index

The Missing Dollar Problem

Three students checked into a hotel and paid the clerk \$30 for a room (\$10 each). When the hotel manager returned, he noticed that the clerk had incorrectly charged \$30 instead of \$25 for the room. The manager told the clerk to return \$5 to the students. The clerk, knowing that the students would not be able to divide \$5 evenly, decided to keep \$2 and to give them only \$3.

The students were very happy because they paid only \$27 for the room (\$9 each). However, if they paid \$27 and the clerk kept \$2, that adds up to \$29. What happened to the other Dollar?

There is a saying that you cannot add apples and oranges. If you have 3 apples and 2 oranges do you have 5 apples? No. Do you have 5 oranges? No. You have five fruits, but the number of apples and oranges has not changed. Similarly, you cannot add real money and "what they think they paid".

When we count only real money, the students have \$3, the clerk has \$2, and the manager has \$25. That is \$30 total.

Accounting of Real Money
Before   After
Student #1    \$10.00  \$1.00
Student #2 \$10.00  \$1.00
Student #3 \$10.00  \$1.00
Clerk   \$0.00  \$2.00
Manager   \$0.00 \$25.00
Total \$30.00  \$30.00

One reader sent the following comment:

There is no missing dollar. The students have paid \$27 and have kept \$3 for themselves and that adds up to \$30. Not a real puzzle... just a mistake in math! You cannot tally the clerks total twice, it's already included in the \$27 the students have paid. The equation is incorrect...if they paid \$27 and the clerk kept \$2...then that does not add up to anything.

Another said:

Err... what? That is worded so incorrectly, it should be pretty obvious... but, alas...
"...they paid \$27 and the clerk kept \$2..."
WRONG... The \$27 includes the \$2 that the clerk "kept." They simply paid \$27 and were each given \$1 back, thus making \$30. That shouldn't be classified as a "brain teaser," but "totally misleading." So, the answer as to why most people make this error is because they were incorrectly informed.

These readers are right that, as far as the students are concerned, they paid \$27 and received \$3 change which adds up to \$30, but the \$27 Dollars obtained from subtracting \$3 from \$30 is just the result of a calculation and not an accounting of the real money. The confusion about the \$2 Dollars kept by the clerk can be avoided by tracking down the real money.

Swiss friends of mine told me that all the missing dollars go to Swiss bank accounts :-)

But seriously: From my point of view, it is misleading to talk about "calculated" money being different from "real" money. If you do the calculation right, the two are the same.

In the case of the riddle, a sleight of hand is used to make the calculation wrong: The riddle states "The students were very happy because they paid only \$27 for the room (\$9 each). However, if they paid \$27 and the clerk kept \$2, that adds up to \$29. What happened to the other Dollar?"

The trick is that you are not supposed to ADD the \$2 to the \$27 -- you have to SUBTRACT them! If you do this, you arrive at \$25, which is what the hotel received as payment. The way the riddle is formulated is purposefully misleading -- and very effective in baffling people.

The students may TRULY think that they collectively paid \$27 Dollars by looking at their wallets. The manager is SURE that he only charged \$25 Dollars after issuing a \$5 Dollar credit. As external observers, we know that the students got cheated out of \$2 Dollars. Reconciling these perspectives is the job of the accountants. In real life, we seldom know how much we are overcharged when we pay for products like gasoline. It is only when the companies report record profits that we realize that we probably paid too much.