Missing Dollar Paradox

Three guests decide to stay the night at a lodge whose rate they are initially told is $30 per night. However, after the guests have each paid $10 and gone to their room, the proprietor discovers that the correct rate should actually be $25. As a result, he gives the bellboy the $5 that was overpaid, together with instructions to return it to the guests. Upon consideration of the fact that $5 will be problematic to split three ways, the bellboy decides to pocket $2 and return $1 each, or a total of $3, to the guests. Upon doing so, the guests have now each paid a total of $9 for the room, for a total of $27, and the bellboy has retained $2. So where has the remaining $1 from the initial $30 paid by the guests gone?! Before perusing further, the reader is encouraged to attempt to work out the resolution to this conundrum.

Of course, no contradiction is actually taking place. The guests have each paid $9 for a total of $27. Of this, $25 has gone to the cost of the room, and $2 has been pocketed by the bellhop. So the relevant numbers to consider are the total amount spent and where it has gone, which are fully accounted for. Similarly, of the original $30 spent, $25 has been retained by the lodge proprietor, $3 has been returned to the guests, and $2 has been pocketed by the bellhop. So again, there is no contradiction here. The fallacy arises in trying to add the $27 paid by the guests to the $2 kept by the bellhop to obtain $30, when in actuality, it is $25 that should be added to the $2 to correctly obtain the total amount spent of $27. It is accountancy, of all things, that supplies a concise answer: "You must not add debits to credits." Money flowing out is a debit, money flowing in is a credit, and they always balance over a transaction.

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Cite this as:

Weisstein, Eric W. "Missing Dollar Paradox." From MathWorld--A Wolfram Web Resource.

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