# Riddle (simple arithmetic problem/illusion)

I'm not sure how well known this "riddle" is but here it goes.

3 people go to a restaurant, each buy food worth 10.00. When they're done, they give 30.00 to the waitress. She gives the money to the manager, manager says they paid too much, gives 5.00 back to the waitress. She comes back to the table, gives a dollar back to each person and then puts the remaining 2.00 in her pocket for tip.

So now, each person has paid 9.00. 9.00 x 3 = 27.00. Plus the 2.00 in the waitresses pocket is 29.00. What happened to the 30th dollar?

So what's the issue with this way of calculating (since if you backward it works fine), that doesn't give us the correct result?

• This is a very old and well-known puzzle indeed... – Hans Lundmark Dec 25 '10 at 22:50
• A humorous but disparaging view of this problem is given in a two-part comic on Spiked Math, here and here. – Mitch Schwartz Dec 26 '10 at 0:52
• lol, thanks for those mitch! – Sev Dec 26 '10 at 11:44
• – Bill Dubuque Aug 1 '11 at 2:10

I am not sure how 'mathematical' the riddle part of this story is. :) Each guest paid \$9 because together they paid \$30, and got back \$27. Of those \$27, the manager got \$25, and the waitress got the other \$2 -- there is no sense/reason in adding \$27 to \$2 -- though you might have a gut feeling that you are headed in the 'right direction' of getting the initial \$30 that way. But take a look -- there really is no extra dollar left, that's all it should be -- \$25+\$2. In other words, each guest paid 9 dollars with the tip included into that. Another, possibly more insightful way to look at things: The guests initially paid 30 dollars. The waitress returned them 3 dollars. So the guests ended up paying \$27 (not thirty!). Of those 27 dollars, 2 dollars were pocketed as a tip by the waitress, so you could also say that they paid \$8.333... each and then added a two dollar tip. I think it's a question of order of operations (and using it with respect to appropriate quantities) that could be confusing here. Punchline: the waitress' 2 dollars were part of 27 dollars paid by the customers. They saved the other 3 dollars of the initial \$30 because of what the manager said, so if you are wondering 'what happened to \$30?' it's more of 27+3, or 25+2+3, where \$25 is what they paid without a tip, \$2 is the tip, and \$3 is the amount they saved.
• Had to remove $signs as that triggered the LaTeX code. – InterestedGuest Dec 25 '10 at 22:51 • If you type a dollar sign, a backslash, and then two more dollar signs, you'll get the desired symbol within LaTeX,$\. – anon Aug 1 '11 at 1:01