That bell boy problem

Jason at The Number Warrior recently posted about the nefarious bell boy who causes hotel guests to lose a dollar:

Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room the bellboy reasons that $5 would be difficult to share 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, totaling $27. The bellboy has $2, totaling $29.

Where is the remaining dollar?

Here’s my solution, and thoughts.

The quick answer

The answer, of course, is that there is no missing dollar. There can’t be a missing dollar: all of the dollars are accounted for. Without introducing magical leprechaun money or extra-dimensional denominations, there is always $30. So why does it seem like we lost a dollar somewhere?

A note about basis of comparison

One thing that always trips people up is the basis of comparison. In fact, there is a question related to this one that a first seems entirely unrelated:

A man is at his favorite department store when he sees the following sign: “Take an additional 50% off all clearance items marked 50% off.” The man thinks this is a great deal, for 50% + 50% = 100% off; i.e., everything is free! Explain why the man will be sorely disappointed when he gets to the register.

The trouble is that the basis has changed: the first 50% off uses the original cost as a basis, while the second 50% off uses half of the original cost as a basis. Thus the man will actually pay 50% of (50% of 100%) = 25% of the original price. Still a great deal, but not quite free.

Similarly, the “riddle” of the bell boy problem is that the author changes his basis of comparison, but doesn’t change the other values accordingly.

Let’s do the math

This problem is not tricky if we don’t change the basis—if we keep things in terms of the original $30 it is easy to see where all of the money went: $25 to the hotel, for the room, $3 to the three guests as a partial refund, and $2 to the bell boy for his troubles. $25 + $3 + $2 = $30, and no money is missing.

But that’s not how the solution is given. Instead, the problem’s proposed solution incorporates the guests’ refund by arguing that since each person got $1 back, it’s as though they only paid $9 to begin with. But if each only paid $9 to the hotel, then the basis is only $9•3 = $27, not $30.

In other words, we only need to account for $27 under this formulation. Which is quite easy: $25 to the hotel and $2 to the bell boy!

The takeaway

It’s perfectly reasonable to fiddle around with math problems so that they’re easier to understand and work with. When you do this, however, you have to be sure that you don’t change the meaning of the problem.