| You're welcome. > Just leave out the analysis of my mind! I should have added weasel words like "maybe" and "probably" because I don't know you. Yet, everything I wrote, I have observed numerous times in other people. And (and of course, maybe I'm wrong here) if you hadn't learned algebra, this solution would have been more obvious to you. Obviously we can't run that experiment directly. But think back to your childhood. Did you ever intuitively know the answer to a problem that others struggled with? Does that happen as often lately? Formal methods like algebra are powerful and allow us to document, step by step, transformations that would be impossible to hold in our heads informally. We can solve problems that were unapproachable before. But we can get so used to using them that we forget intuitive tricks that we used to use. Maybe this didn't happen to you. You're right. I don't know you. I'm only extrapolating. It really is the sort of simple problem that many people (possibly most of them younger) can solve immediately without assigning variable names or even writing anything down. If this describes your capabilities when you were younger, then something changed. What is it? Let me give you an example of my own. When I was a child I would play around with electronics. I intuitively knew that if I put two resistors in parallel, the amount of resistance would go down, and by how much, and could easily extend that to 3 or 4 resistors, rummaging through my collection to find resistors that would parallel to give me the value I wanted. Once I learned the parallel resistor formula, I got slower at everything above two resistors. Which brings us to: > As you say, [using algebra to convert the problem to algebraic form] is impossible to do, because it doesn't make sense. So why bring it up? I brought it up because, although we agree that it is not strictly part of algebra, it is something that you had to learn in order to use algebra effectively. To me (and of course, again, I'm still speculating here) the fact that you're smart enough to use intuitive methods to set the problem up in algebraic form means that you were probably smart enough to use intuitive methods to simply solve the problem directly, if you weren't so used to directing your brain activity towards doing things using algebra. |