Slow down

A friend recently e-mailed me this question, with the subject line, ‘Splain this please.  It’s messin’ me up!

I recognized it from my younger days.  It messed me up quite a bit at the time as well…

Three guys walk into a hotel. The room is $30, so they each pay $10. After they get to their room, the receptionist realizes that the room was only supposed to be $25, so he gives $5 to the bellboy and tells him to give it back to the 3 guys. On the way to the room, the bellboy takes $2 and puts it in his pocket, and tells the guys that they were overcharged by $3. So each guy gets $1 back, which means they paid $9 each. 9 \times 3 = 27, plus the $2 the bellboy took makes 29. What happened to the other dollar?

The problem here is that the explanation at the end sounds right, so if someone says it to you fast, you get tricked and think to yourself, “What the…?”  But if you slow down and follow the money, you realize that there is no contradiction:

1) Of the 30 dollars the men originally paid, 25 went to the hotel, 3 came back to them, and 2 went to the bellboy.

Equivalently:

2) Of the 27 dollars the men eventually paid, 25 went to the hotel and 2 went to the bellboy.

In other words, the reason we are tricked is because the puzzle inappropriately adds that 2 dollars to $27 to make $29 instead of subtracting the 2 dollars to make $25.

The trick works so well on us because the amounts we are dealing with are so small.    We probably would be much more suspicious if each person had paid $50 ($150 total), the hotel returned $40, and the bellboy took $10, returning $10 to each person.  $40 \times 3 = $120 and $120 + $10 = $130.  What happened to the other 20 dollars???

In fact, a false argument like the one above can allow us to lose an arbitrary amount of money!  (A new theory on the origins of the current economic crisis?)

The moral: don’t believe statements just because they sound right.  Check them out for yourself!

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