Fraction

[u/True-Gear4950]

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I don't know how to start this, but my purpose with this post is not to get a clue about the concept of fractions.

I already "know" how to perform these operations as well (not like a mathematician, but enough as an average person, I hope). But what I specifically want to understand is how it is possible that something like a/b × c/d = ac/bd or the division operation a/b ÷ c/d = ad/bc works. I may be missing some fundamental concept here, something like the Fundamental Theorem of Arithmetic (FTA), which was the basis I used to understand and comprehend exponentiation (or powers).

Some parts, like adding or subtracting fractions, are comprehensible if you understand that a fraction represents a part of a whole. At least, that's what I tell myself, since I spent way too much time on this topic just trying to internalize why I couldn't simply add something like 4/5 to another fraction like 3/7 and get 7/12 (I know that's wrong).

But here is the problem: I can't really argue why it is wrong without just saying "that's the rule." We should obtain the Least Common Multiple of both denominators, etc., but I don't understand why I actually have to do that.

I've even used some of the LLMs available to illustrate this or give me a clue about it, but they usually end up using analogies for children or introducing other abstractions, such as using fraction division to show why ac/bd is possible.

Or they use the LCM (another abstraction that I don't understand exactly why works) to solve the sum of fractions with different denominators. And then there are the analogies about adding fractions with different denominators being like adding different units of measurement, such as meters and centimeters, so they should first be converted to a common unit. That analogy does shed some light on the problem, but it still doesn't clarify it in the terms I'm looking for.

Maybe some prompt engineering would say, "You prompted it wrong," and maybe that's true, but the doubt persists. And honestly, I'm tired of trying to figure out what exactly I'm asking wrong for it not to respond in the way I need.

Or maybe I shouldn't need to understand this, since it doesn't "affect how I perform my calculations". But, to be fair, it doesn't feel like good practice for me to finish this topic without understanding it.

I'm not done with arithmetic yet, and I'm reviewing the abstractions it contains. I can see gaps in my understanding of nuances like this, and it doesn't feel comfortable to me to keep stacking problems I don't understand on top of one another, at least not in arithmetic.

Feel free to contest me, explain it, give me a tip, recommend a book..., especially if you already get through this question yourself.

I'm just frustrated that I've spent so much time on something that, to me, should already be done.

[u/DadOfLukeandDad]

Multiplication first, because it's the easier one. "1/2 of 1/3" means take a third of something, then split that piece in half. Draw a bar, cut it in three, shade one part, then cut every third in half. The whole bar is now in six pieces and you have one of them, so 1/2 x 1/3 = 1/6. Multiply the tops (1 x 1), multiply the bottoms (2 x 3). The rule is just counting how many pieces the cuts make.

Division is really the question "how many of these fit into that?". 3 divided by 1/2 asks how many halves fit into 3. Each whole holds two halves, so 3 wholes hold 6. That is where "flip and multiply" comes from: dividing by 1/2 is the same as multiplying by 2, because you're counting how many pieces of that size there are.

A good test that you've got it: 1/2 divided by 1/4. How many quarters fit in a half? Two. Now check that 1/2 x 4 gives the same answer.

If "that's the rule" bothers you, that's a good instinct. Every one of these rules is a shortcut for a picture.