The Missing Dollar

Three guys walk into a hotel and they are tight for cash so they decide to split one room. The concierge at the counter checks them in and when it comes time to pay he asks them for $30. Since there are three of them they decide to split the bill evenly and they each pay$10. Later that night the concierge realizes he has made a terrible mistake. He charged them for a luxury suite when they only got a regular suite. The price of the regular suite was only $25. He realizes that if he gives them back a$5 bill they will not be able to split it evenly. His solution to this problem is to give them each back $1 in the morning (giving back a total of 3$) and then he pockets the other $2 as a tip. Here is where the problem arises. If they each originally paid 10$ but they each got 1$back, then it is as if they have each now paid 9$. If they each paid 9$and there are three of them, and$9 times 3 is equal to $27. Then it would seem they paid$27 for the room. Of course the concierge also took a $2 tip. So$27 plus $2 gives a grand total of$29. But originally they had paid \$30. Where is the missing dollar.

Can you discover where the error occurs?

Note by Brody Acquilano
1 year, 11 months ago

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The 2 dollar tip is on top of the 25 dollar actual price, so the concierge ends up with 27 dollars, and each guy ends up with 1 dollar, for a total of 30 dollars. The steps are shown below.

- 1 year, 11 months ago

As Steven chase pointed out the proper way to approach this problem is:

$10+10+10= 30$ $10+10+10-3= 30- 3$ $(10-1)+(10-1)+(10-1)= 30-3$ $9+9+9=27$

The error arises as follows. Notice in step three we forget to subtract the three dollars from the right hand side of the equation In step five we add two dollars back to the left hand side, but again we don't do anything to the right hand side. If we subtracted three dollars from the left hand side and then add two dollars to the left hand side then the left hand side is off by one.

$10+10+10= 25 +5$ $10+10+10= 25 + 3+ 2$ $10+10+10-3=25+2+3$ $(10-1) + (10-1)+ (10-1) = 25+2+3$ $9 + 9+ 9 +2 = 25+2+3$ $29 = 30$

- 1 year, 11 months ago