##### Jul 06, 2019

**RAY:** Recently, the Car Talk staff decided to go out for a few drinks and some buffalo wings. The gang goes to a local eatery. We have drinks and some snacks. And just before the bill comes... my brother and I each go to the bathroom and climb out the window.

The bill comes, and it's $63. Bugsy says, "Those guys stiffed us -- again! But look. If everyone throws in an extra two bucks, we'll cover the bill."

The question is, how many people were in the original group? And, prove that there's only one answer.

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

So, we're going to write two equations. We'll say x equals the number of people and y equals the amount paid. So, x times y equals 63.

But you also know that if you add two to the number of people it also adds up to $63, so x plus two times, in parentheses, y minus two, which is the number of people that we're short has got to equal 63 also.

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?

And all you have 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!

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

And when you factor that, it comes out to (x + 7)(x - 9) = 0. Right?And so, one answer is x = 9. And the other one is x = -7. There were nine people in the original group. And we left and we stiffed them, so they had to come up with two more bucks apiece.

So, each of the seven people pays nine bucks, whereas each of the nine people would have paid seven bucks.