math problem

This might have been posted before, and I usually don’t ask questions like this, but I got it by email and it’s really bothering me. So here goes:

Three guys in a hotel call room service and order two large pizzas. The delivery boy brings them up with a bill
for exactly \$30.00. Each guy gives him a \$10.00 bill, and he leaves. That’s fact! When he hands the \$30.00
to the cashier, he is told a mistake was made. The bill was only \$25.00, not \$30.00. The cashier gives the
delivery boy five \$1.00 bills and tells him to take it back to the 3 guys who ordered the pizza. That’s fact! On
the way back to their room, the delivery boy has a thought. These guys did not give him a tip. He figures that
since there is no way to split \$5.00 evenly three ways anyhow, he will keep two dollars for himself and give
them back three dollars. OK! So far so good! He knocks on the door and one fellow answers. He explains
about a mix up in the bill, and hands the guy the three dollars, and then departs with his two-dollar tip in his
pocket. Now the fun begins! Remember \$30-\$25=\$5, right? \$5-\$3=\$2, right? So what’s the problem? All
is well, right? Not quite. Answer this: Each of the three guys originally gave \$10.00 each. They each got back
\$1.00 in change. That means they paid \$9.00 each, which times three is \$27.00. The delivery boy kept \$2.00
for a tip. \$27.00 plus \$2.00 equals \$29.00. Where the heck is the other dollar?

This is one of those classic misdirection ploys.

It’s better to look at it as follows.

The original 30 dollars comprises of Pizza=25 and tip = 2
refund = 3.

No missing three dollars. If they each gave 9 dollars for a 25 dollar pizza, he would have a 2 dollar tip, or have to give 2 dollars in change.

This is because you’re adding together the wrong dollars.

The \$27 that the guests paid for the pizza consists of the \$25 that room service has plus the \$2 that the delivery boy kept. To that, your re-adding the \$2 that the delivery boy kept. This doesn’t represent the total.

Or algebraically: Divide the \$30 into three parts:
Let
A = Final charge for pizza (\$25)
B = Amount that delivery boy kept (\$2)
C = Amount refunded to men (\$3)

so A + B + C = 30

(A + B) + B <> 30

Of course it doesn’t, because it doesn’t include C and it double-includes B.

That seems to make sense, but I can’t see exactly where the disinformation occurs in the original message. Can you show me where it happens?

It occurs in the “They paid \$9 each, and the delivery boy kept \$2, that’s \$29”

The problem is, the \$2 of the delivery boy is part of the \$9 that each paid, see? The ‘counting’ that occurs above counts that two dollars twice, and fails to count the three dollars given back to the men.

Search on “bellboy” (hey! search is working pretty good…)

It’s like the question: “If you have five apples and I take two, how many do I have?” which kinda implies that you have to do arithmetic in order to get the answer, when you don’t.

OK, taking from the OP:

1. Each of the three guys originally gave \$10.00 each.

OK, given

1. They each got back \$1.00 in change. That means they paid \$9.00 each, which times three is \$27.00.

Given. they’re out \$27 for services rendered.

1. The delivery boy kept \$2.00 for a tip.

Right…

1. \$27.00 plus \$2.00 equals \$29.00.

BUZZ!!! Here’s the misdirection. The \$2 of the delivery boy’s “tip” is part of the \$27 in sentence 2. The real money is: \$25 that the hotel charged + \$2 “tip” + \$3 that the boy did return = \$30.

If you took the hotel’s \$25 and broke it down: \$12 to the food supplier, \$3 tax, \$10 stockholders. Then if you said: the men paid \$27, the boy’s tip was \$2, the food supplier got \$12, the gov’t got \$3, and the stockholders got \$10. 27 + 2 + 12 + 3 + 10 = \$54! Wow, they overpaid! :D:D

Here’s what Cecil had to say on the “puzzle”: The bellhop, the three guys, and the missing buck

And here’s where this was discussed ad nauseam in this very forum: Is the missing dollar unanswerable?