# Restaurant Bill

**RAY:** Well, it doesn't take too much. The only two numbers I know that make 63 when multiplied together are nine and seven.

**TOM:** So, that must be the answer, so you can work backwards and write the equation.

**RAY:** So, we're going to write two equations. We're going to write --

**TOM:** You have to write two.

**RAY:** If x equals the number of people --

**TOM:** Yeah. Let me write these down. Go ahead. x is the number of people.

**RAY:** And y equals the amount paid. So, x times y equals 63, the original number of people.

**TOM:** Ah! So, if we hadn't sneaked out the window, then --

**RAY:** x times y --

**TOM:** x times y would have been 63.

**RAY:** -- would have been enough and you wouldn't have been able to figure it out.

**TOM:** x times y is 63.

**RAY:** Right.

**TOM:** We can solve that.

**RAY:** So, then, but you also know that if you add two to the number of people, so x plus two --

**TOM:** Yeah.

**RAY:** -- times, in parentheses, y minus two, which is the number of people that we're short --

**TOM:** Yeah.

**RAY:** -- has got to equal 63 also.

**TOM:** Oh, how sweet it is. So, you just go multiply stuff together.

**RAY:** So, you multiply it all out, and what you get is - you get a long expression, and what you end up doing is you substitute. You get rid of the y, and you substitute the 63 over x for y. And when you do that, you wind up with something called a quadratic equation. Is that right?

**TOM:** Of course. Everyone knows. And all you got to do is give them those two equations, and any kid will figure this out in five minutes. Just give them those two equations. (x + 2)(y - 2) = 63, and xy = 63. Solve that. You can't do that; you're out! That's it!

**RAY:** Well, it finally ends up with x^2 - 2x - 63 = 0.

**TOM:** Oh, yeah.

**RAY:** And when you factor that, it comes out to (x + 7)(x - 9) = 0. Right?

**TOM:** Yeah. I see it.

**RAY:** Right.

**TOM:** Yes, that's correct. I see it.

**RAY:** And so, one answer is x = 9.

**TOM:** I like it.

**RAY:** And the other one is x = -7.

**TOM:** That's how many people there were originally, minus seven. No one showed up.

**RAY:** Well, in the anti-matter universe, the minus seven answer is valid.

**TOM:** Would be significant, yeah.

**RAY:** But, otherwise, there were nine people in the original group.

**TOM:** Right. And we left.

**RAY:** And we left and we stiffed them, so they had to come up with two more bucks apiece. If you think you have any questions about that, write to my brother. Who won?

**TOM:** So, each of the seven people pays nine bucks.

**RAY:** Right. Whereas each of the nine people would have paid --

**TOM:** Seven bucks.

**RAY:** Seven bucks.

**TOM:** Oh, man!

**RAY:** Right, and if you hadn't climbed out the window with me, they never would have figured it out.

**TOM:**The winner is Tucker Matsik from Lombard, Illinois.

[ Car Talk Puzzler ]