I'm trying to make sense of something in the fairly elementary topic of fractions, I've been out of mathematics study for some period of time.
If we consider $$\frac {3}{4} \text{ of } 20 = 15.$$
Speaking very specifically, what are we saying here or in the case of any given fraction? How do I consider this pictorially? 20 is just a number so what does it mean to take 3 multiples of a quarter of 20?
I hope I'm able to convey my confusion, it's actually causing me a lot of self-doubt.
Many thanks.
Answer
$\frac34$ of $20$ can also be viewed as tripling $20$ to get $60$ and then taking one quarter ($\frac14$) of $60$ to get $15$.
Whether it makes more sense to view it this way or as taking $3$ of $4$ evenly divided parts depends on the context. In cases where the fraction is less than one ($\frac34$ in this case) then thinking of it as taking $3$ of $4$ evenly divided parts probably makes more sense.
On the other hand if you consider $\frac54$ of $20$ then taking $5$ of $4$ evenly split groups is less intuitive (since you don't have $5$ groups). In this case one quarter of $5\cdot 20$ may be more natural.
FWIW, I wouldn't consider thinking these kinds of things through as obsessive. I would think of it as a deeper desire to understand the topic. In school, if you're not so interested in math, you can skip by without understanding the topic to the depth you seem to be interested in, and just apply rules. But I'm guessing that's not your situation, and thinking through these things will serve you well as you get to more advanced topics.
It will enable you to reason, rather than just follow rules.
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