# Where’s the other dollar?

Three guys in a hotel call room service, place an order for 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 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 wil keep two dollars for a tip, and just give them back three dollars. So far so good! He knocks on the door and one fellow answers. He explains there was a mix up in the bill, and hands the guy the three dollars, then departs with his two-dollar tip in his pocket. Now the fun begins! Remember! \$30-\$25=\$5. Right? \$5-\$3=\$2. Right? 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 multiplied by three is \$27.00. The delivery boy kept \$2.00 for a tip. \$27.00 plus \$2.00 equals \$29.00.

Where is the other dollar?

There is no missing dollar:

The error is in the last statement. It says to add the \$2 that the delivery boy kept to get 29\$. But the delivery boy is keeping the \$2. Therefore there was \$27 spent by the men in the problem and so we subtract the \$2 kept by the delivery boy for a total of \$25 for the pizzas (the actual cost).

It is a common mistake in mathematics courses to start applying operations out of the blue, such as addition or multiplication, without any real reason to do so. In this case, the statement to add \$27 and \$2 is unmotivated. The real problem is trying to figure out what total are we trying to determine.

• Is it the total that was spent? Well, the three guys spent a total of 27\$ after the return of the 3 bucks. Of that \$25 is for the pizza, \$2 is for the delivery boy. No reason to add the 2 bucks.
• Is it the total that the will be given to the pizza place? \$30 was originally, but the delivery boy returns \$3 dollars and keeps \$2 so we have the problem of \$30 – \$5 – \$2. Again, no reason to add the 2 bucks.
• Is the total that the delivery boy goes home with? Clearly that is \$2, from \$30 – \$3 – \$25, where the last \$25 is what he must pass on to the pizza place. Again, no reason to add 2 bucks.

If that doesnâ€™t make sense, let me word it another way: The 30\$ was not paid all to the hotel keeper. The \$30 breaks down into \$25 to the Pizza Place, \$3 to the men and \$2 to the delivery boy for a total of \$30. (not 29)