# Problem of the week #5 - April 30th, 2012

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#### Jameson

Staff member
This problem for some reason has been around for some time but lately I have heard it from many friends and students, so I think it's worth posting for all young students who come by MHB.

Problem:
Three guests check into a hotel room. The clerk says the bill is \$30, so each guest pays \$10. Later the clerk realizes the bill should only be \$25. To rectify this, he gives the bellhop \$5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest \$1 and keep \$2 for himself.

Now that each of the guests has been given \$1 back, each has paid \$9, bringing the total paid to \$27. The bellhop has \$2. If the guests originally handed over \$30, what happened to the remaining \$1?

On top of showing where the accounting error is you should also show the logic error by demonstrated in the problem.

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#### Jameson

Start with writing down the three places where some of the \$30 ended up: the clerk, the bellhop, the 3 guests. The room cost \$25 which goes to the clerk, the guests got \$3 back as a whole and finally the clerk took \$2 so it all adds up correctly: \$25+\$3+\$2=\$30. So why does the problem somehow make it sound like there is a missing dollar? The error in logic and accounting occurs when it's claimed that the \$30 was split into \$27 for the rooms and \$2 that the bellhop stole. The rooms actually cost \$25 so the guests should have payed $$\displaystyle \frac{\30-\5}{3}$$ or about \$8.33 each, but because the bellhop took extra money they overpaid some and each paid \$9 instead. So \$27 for the cost of the room already includes the money to that the bellhop took and the rest of the money, \$3, went back to the guests which totals \\$30.