On Aug 6, 2007, at 7:57 AM, gale@Math.Berkeley.EDU wrote:
I'M AFRAID I didn't express myself clearly.I wasn't asking for a solution but rather about the mechanics of finding it. Did either of you use pen or pencil or paper at any point? I had to draw two pictures before I could see what the equations had to be. Maybe you were able to do all these things in your head in which case my speculations are off base.
I think the important part of solving a problem like this comes way before the algebra steps that Dan and Gareth describe. It's the part about going from a disoriented and muddled state to a notion of what to do, or even, what to sketch. This is hard for people like us to dig into, since so much about problems of this sort has become internalized. In this case: it's the kind of thing that throws people because it sounds like there's not enough information. What is it that supports the idea that when they're going opposite directions, the they meet at a frequency determined by the sum of their speeds, and in going in the same direction, the difference of their speeds? There's also the more commonplace issue of getting straight the reciprocal relation between speed and frequency. In my case, I worked it out in my head. I had a mental picture of the two people running in opposite directions around a circle, and then I thought of using moving coordinates as in high school physics class so that one of the runners moved at the sum of speeds, which I said to myself was one lap per minute. The same picture said the difference of speeds was 1 lap per hour. Then I thought of a picture of an interval of length 60, with two points (the two speeds) averaging at the halfway mark but being 1/60th apart: from the picture you see the ratio as 61:59. I have an aversion to writing things like this down in symbolic form, because when I do they become denatured in my head and it's hard for me to keep focused on the whole picture. Actually, I tend to get distracted going back and forth between algebra and the actual situation, and I tend to make algebra errors. But: I've gone through so many math problems that this doesn't reveal much about the important part of solving the problem --- you'd need to go back to algebra students. Bill