RAY: Hi. We're back. You're listening to Car Talk with us, Click and Clack, the Tappett Brothers, and we're here to talk about cars, car repair, and, duh, the answer to last week's Puzzler. This is an algebraic Puzzler for the vacationing school kids with atrophying brains, didn't you think?
TOM: Mmmm. Mmm-nnn-uhg-hm.
RAY: Berman spent two weeks on it.
TOM: I don't really care. I'm watching TV. What do I care about atrophy, huh?
RAY: Recently, after we finished one of our Car Talk shows, we and our staff decided to go out for a little, you know, snack. So, the whole gang goes to the Ptomaine Deli down the street, and we order a bunch of food and whatever. And as luck would have it, just before the bill comes, my brother and I excuse ourselves and go to the bathroom and climb out the window.
TOM: I mean, what else would you do when the bill was going to come?
RAY: It's interesting that I met you there. We hadn't planned it in advance. Anyway, the bill comes, and it's 63 bucks. Everyone figures after a while that we ain't coming back, so Bugsy says, "Oh, those guys, they stiffed us. But, look, if everyone throws in an extra two bucks, we'll cover the bill." So, the question, very simply, was how many people were in the original group? And show your work. Right?
TOM: Yeah.
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 isÉ 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. And, Tucker, for having your answer selected at random from among the thousands of correct answers that we got, you will get a 25 --
[ Car Talk Puzzler ]
