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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.

Gaurang Tandon
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salman
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  • Are you looking for something more like an intuitive explanation or a rigorous definition? – platty Aug 14 '17 at 15:31
  • @platty intuitive more than anything, I want to understand what I'm doing instead of just doing the calculation. – salman Aug 14 '17 at 15:40

2 Answers2

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Intuitively, taking $\frac{3}{4}$ of $20$ means that you're dividing $20$ objects into $4$ equal-sized groups, and taking three of them (you can think of it as multiplication $3 \times \left(\frac{1}{4} \times 20\right)$). So here, you would take the $20$ objects, divide them into $4$ groups of $5$, and choose $3$ of these groups to get $15$ objects total.

In general, once you start working with division or rational numbers, these quantities won't necessarily be integers. For example, if we were to take $\frac{3}{4}$ of $15$, we'd split up the $15$ objects into 4 groups of $\frac{15}{4} = 3 \frac{3}{4}$ objects each. Then taking three of these gives us $\frac{45}{4} = 11\frac{1}{4}$ objects.

platty
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  • Small addition:where $11\frac{1}{4}$ means $11$ whole (full, undivided) parts and the $\frac{1}{4}$ means 1/4th of another whole part. – Gaurang Tandon Aug 14 '17 at 15:52
  • @platty ok that helps, so if I was to take a quarter of 20, I would be breaking 20 into 4 equal parts and just taking 1 part.

    Likewise if we consider, 20% of 10. Here, am I saying that: Break 10 into 100 equal parts and take 20 of those equal parts? Or is there another way to look at it?

     – salman Aug 14 '17 at 16:10
  • Yep, that's exactly it! – platty Aug 14 '17 at 16:10
  • @platty sorry I edited it. – salman Aug 14 '17 at 16:12
  • Yes, that works. Another way to think about it is to see that $20% = \frac{1}{5}$ and proceed accordingly. You'll get the same answer both ways. – platty Aug 14 '17 at 16:13
  • @platty ok thank you very much, so all taking fractions OF something is, breaking the target into x parts and taking y parts of the x parts? Like that's all there is to the topic? – salman Aug 14 '17 at 16:15
  • That's the practical way to think about it. The theory goes a lot deeper than that (in that, how do we define what it means to "break into parts"), but you can think of fractions as essentially corresponding to a division and a multiplication. – platty Aug 14 '17 at 16:17
  • @platty I've currently revisited my interetest in mathematics after a few years. I'm stuck in this cycle, where I HAVE to learn the theory behind everything or else I don't feel satisfied inside. I find myself having to define every single term. How do I remedy this and not become obsessed with it? – salman Aug 14 '17 at 16:24
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$\frac34$ 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.

Χpẘ
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  • Can you suggest where I can learn these topics completely from the ground level? I want to know exactly what I'm doing. What actually piqued my interest in learning topics in depth was my inability to explain fractions to my younger sibling. – salman Aug 14 '17 at 16:53
  • Sounds like you may have two goals: 1. Understand math topics in depth, 2) Be able to teach them to your sibling. For #1 when I google "fraction multiplication" the first hit is Khan Academy. I've never used their resources before, but have heard good things about them. I suggest that as a starting point. Beyond that, there are numerous websites (which you can find by using Google) that teach elementary to intermediate math topics. For #2, there is a mathematics educators stack exchange website. You might try going there and then searching for a specific topic, like "fraction multiplication". – Χpẘ Aug 14 '17 at 17:01
  • In regards to teaching, I'm able to teach my brother through example and repetition (he usually picks it up eventually, although I would love for him to understand it intuitively I think that is reserved for those who are naturally better at understanding).

    I just want to be able to understand these topics for my own satisfaction.

     – salman Aug 14 '17 at 17:06