[zephen]

> Sorry to say, the first half of this comment is almost gibberish.

In what way?

> It became algebra when I introduced an unknown variable and wrote down an equation.

But a lot of people, including me, can solve it without writing down any equation.

> I can't see a way to solve the problem without doing that.

Ah. So it's gibberish because of your limitations? That's... not how this usually works. (Although, to be fair, it's often the case that people of limited intellectual means lash out with unkind comments such as "gibberish" so maybe this is how it works.)

> How did you get this equation from the problem statement?

"Mary is 24 years old. She is twice as old as Ann was when Mary was as old as Ann is now. How old is Ann?"

We know that there is a delta between Mary's current age (24) and what I called X in my previous comment (Ann's current age AKA Mary's prior age). That would be 24 - X.

We also know that at some previous time, Ann was 12 (half of 24) when Mary was (Ann's current age AKA Mary's prior age).

Some of us just take the mental shortcut that the in between age has to be the average of 24 and 12 (because the delta doesn't change[1], so the delta must have been the same before as it is now), but if forced to write it down into algebra, we can say that this second statement is X - 12.

And then of course, again, because the delta doesn't change, and the delta is equal to both x-12 and 24-x, those expressions must be equal to each other.

[1] Except of course, the delta could wobble a bit for birthdays being on different dates within the year. In simple puzzles like this, of course, that +/- 1 year possibility is usually ignored.

[tempfile]

I'll answer in reverse order.

I think what you suggest works! Putting it in the wording of my original comment: Mary IS 24, Ann WAS 12, and the same amount of time gets you from 12 to Ann's current age, and from Ann's current age to 24. I think it is obvious from there that you're half-way between, so Ann is 18. Thank you!

The part of your earlier comment I consider gibberish has nothing to do with the problem.

> It's slightly different, because you've supercharged your hammer.

This does not mean anything.

> Everything you ever understood about how to solve word problems has been subsumed into what your brain labels as "algebra."

This is bizarre speculation. Why are you analyzing my brain? I just asked for an explanation!

> Did you use algebra to convert the problem to algebraic form?

As you say, this is impossible to do, because it doesn't make sense. So why bring it up? It sounds like you are trying to convince me the problem is not algebra - or that I am in some way only seeing it as algebra because I am not thinking about the problem clearly. But I never said it was! I just said I can't see how to do it without algebra. Since you mixed those things up, I clarified by saying exactly what sense I was talking about algebra in the first place. Clearly that didn't work!

Again, thank you for explaining your reasoning, I appreciate it. Just leave out the analysis of my mind!

[zephen]

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.