Sunday, April 12, 2009

4-12-09: Product Over Sum

There's a formula I learned in my undergraduate days in my electrical engineering studies known as the "Product Over Sum" method. It's used to calculate the equivalent resistance of two resistors in parallel.

Last night I was somehow reminded of this formula. I don't know why, but it came up in a random thought. I first remember learning an example of a parallel system back in, I think, middle school with the "two painters" example:

Two painters are hired to paint a house. One can paint the house on his own in 12 hours, whereas the other is twice as fast and can paint the same house on his own in 6 hours. How long would it take them to paint the house if they worked together?

The answer isn't the average, or 9 hours. It's actually 4 hours, which is the product (72) over the sum (18). The two men are painting the house in parallel. I didn't realize it was a parallel system at first; I learned this years later with parallel resistors.

But last night I thought of a different way to look at it, using logic gates. Does this make sense?

If you think about it as a ratio of the rates at which the two painters do their work, you could think of it as: (Worker 1 AND Worker 2) / (Worker 1 OR Worker 2). When you "AND", you are multiplying the two; "OR" means you add them. I never really thought of this way, but does it make sense? You're comparing how fast both of them work together (Worker 1 AND Worker 2) to how fast either of them works (Worker 1 OR Worker 2). It's simply a ratio.

Would a kid understand this better than saying "it's a parallel system"? Maybe. I'm just thinking out loud.

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