[Math Lair] The Missing Dollar

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Three men decide to spend a night in a hotel room. They pay $30 for the room (assume that this took place a long time ago, when hotel rooms might have actually cost $30). They split the cost evenly amongst themselves, with each man paying $10. As they are about to leave, the manager of the hotel, realizing that the three men are frequent patrons of the hotel, decides to give the men $5 back. He gives $5 to the bellhop and asks him to give the money to the three men. The bellhop, not wanting to split $5 among the three men, decides to pocket $2 of the money and gives each of the three guests $1. Each of the guests has now paid $9 for the room, for a total of $27, and the bellhop has $2, for a total of $29. Where is the missing dollar?

Many people find this problem rather frustrating. The key to it is to look at it in a different way. The three men each paid $30. Out of that, $25 is in the hotel's cash register, $2 is in the bellhop's pocket, and each of the three men have $1, for a total of $30. Looking at it this way, there is no missing dollar. So why is there a dollar missing in the analysis above?

The problem contains a trick. Each of the guests paid $9, for a total of $27. That $27 includes the $2 in the bellhop's pocket. You can't add that $2 to the $27 and expect to get a meaningful result, but that's what the problem does. The sum in the problem is a red herring that doesn't correspond to any real-life amount.