Eureka!

So I'm a little embarrassed about this, but here goes. I was speaking with a friend about a junior high math problem I was having trouble remembering. You've heard it, the one that asks if it takes Billy a certain amount of time to paint a house and Sally a certain amount of time, how long would it take them together? When trying to recall the answer to the question, I thought I remembered that there was some sort of catch. I was using the times of one hour apiece and telling myself that it could not be 1/2 an hour total, that I was missing a step. It was driving me crazy.

And then, as Einstein said, a storm broke in my mind. Now when this happened to him, he had just discovered the theory of relativity; my accomplishment was slightly less than that. I realized, as you probably have already figured out, that I was completely wrong--there is no catch when they are working at the same rate. If Billy and Sally are working at the same rate of 1 house per hour, when they paint together their time will indeed be reduced by half, resulting in 1/2 an hour to paint one house.

Now then, the "catch" if you will, comes when they would happen to work at different rates. If Billy works at a rate of 1 house/3 hours and Sally paints at a rate of 1 house/5 hours, how long would it take them together? I remembered that some people would just say 3 + 5 = 8 and 8/2=4, so 4 hours is the answer right? Wrong, you need to remember the rate. 1/3 + 1/5 = 8 houses in 15 hours, reduced down by cross multiplication to 1 house in 1 hr 52 min and 30 sec. That is how long it would take them to paint the house together.

Einstein may rest easy. I don't think I'll be challenging his genius quite yet.