Three men go to stay at a hotel and they are charged \$30 for the room.

They split the cost with ten dollars each.

Later the manager finds out that the rate is \$25 and gives the bellboy \$5 to return to the guests. On the way to the room the bellboy reasons that \$5 would be difficult to split 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 another \$2, adding up to \$29. Where is the missing dollar?

The problem is that the question is cleverly phrased to conceal what is really going on.

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 account for is the \$30.

So let’s see what happens to that \$30. The manager keeps \$25, the bellboy gets \$2 and the guys get \$3 back. 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 bellboy. Now, that adds up too.

The problem with the question is that the the \$2 that the bellboy gets are contained in the \$27 that they end up paying so we shouldn’t expect them to add to anything meaningful.

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