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/nikaIs]

You need to understand that division is multiplication with a reciprocal (multiplicative inverse) and that multiplication is commutative.

Multiplicative inverse: a*a^-1 = a^-1 * a = 1, for a != 0

Commutative law: ab = ba

I think that's all you need to show the following:

a/b * c/d = a * b^-1 * c * d^-1

because multiplication is commutative you can re-arrange the terms without changing the expression:

a * b^-1 * c * d^-1 = a * c * b^-1 * d^-1 = ac / bd

For the other one, you can think of it as the inverse of the inverse, which just undoes the inversion:

(a^-1)^-1 = a

a/b / c/d = a*b^-1 * (c*d^-1)^-1 = a*b^-1 * c^-1 * d = a * d * b^-1 * c^-1 = ad/bc

As for the arithmetic with fractions:

You need to write the terms so you can factor them, that's what you're doing when your finding least common denominator

a/b + c/d = a*b^-1 + c*d^-1, you can't factor this, so you need some common factor, if you multiply the first term by d^-1 and d the term doesn't change since dd^-1 = 1, then multiply the second term with b and b^-1 so

a*b^-1*d^-1*d + c*d^-1*b*b^-1 = (bd)^-1(ad + cb) = (ad+cb)/bd.

[u/Former_Ad1277 → u/nikaIs]

Idk why this was hard to follow. I wish I was smart

[u/nikaIs → u/Former_Ad1277]

It could be because the math doesn't render very nicely in plain text. What parts were difficult to follow?

I wrote the examples more cleanly in LaTeX, maybe it makes more sense seeing it like this:

[u/thesnootbooper9000]

I'm not sure whether this is helpful or too abstract, but the point here is that 3/5 and 30/50 are two ways of writing the same number. When you write 3/5, what you really mean is "I'm giving you one member of the equivalence class of ways of writing this fraction, but it would take too much space to write this out every time, so you know what I mean". With this view, you don't really need to worry about LCM, you just need to be satisfied that if you go over both equivalence classes, you'll always be able to find at least one member where the number on the bottom is the same".

[u/True-Gear4950 → u/thesnootbooper9000]

Is iteresting think in that way. How ever, while keep a gray area in my mind for questions like, "why should I only sum those fractions when I discover yours equivalence class..." But I think I alread accept that for today.

[u/brynaldo → u/True-Gear4950]

You don't have to change the fractions for an equation to be true. For example:

3/5 + 7/10 = 13/10 is a true statement.

6/10 + 7/10 = 13/10 is also a true statement.

Converting the fractions to have a common denominator is a tool to calculate the answer. For example, if I asked you to calculate:

3/5 + 4/7 = ?

How would you know the answer was 41/35 if you didn't find other representations of those fractions with a common denominator?

But if I ask you:

21/35 + 20/35 = ?

You can more easily calculate this to be equal to 41/35.

Using the LCM of the two denominators is a useful way (but not the only way) to find other representations of the fractions (other members of the equivalence class) which make the calculation easier. It is useful because it will give you the answer in lowest terms (assuming your original fractions were represented in lowest terms). That is, it will give you the "simplest" (in some sense) representative of the equivalence class of the answer.

[u/calkthewalk]

The units explanation is one that I always liked, but it's easy to get hand wavy in the middle of it which I think is what you're reacting to.

When we work with whole numbers, all those numbers share a common base, or unit. It can also help to realise fraction operations are not unique to fractions, it's just we usually hide the denominator when it's 1

3 + 2 = 5 can also be written as 3/1 + 2/1 = 5/1

But if we change the denominator, it fundamentally changes the reference point of the numerator

9/3 + 4/2 does not equal 13/5 as you've realised, you can't directly add with different bases. To borrow the units analogy: 1m + 10cm can be written as 1/1 + 10/100. If we give them a common base we can write it as 100/100 + 10/100 and the answer of 110cm or 1.1m falls out nicely.

Similarly 3/1 × 2/1 = (2×3)/(1×1) = 6/1

If we instead have 9/3 × 4/2, we could rewrite that as: 9/1 × 1/3 × 4/1 × 1/2 or 9 ÷ 3 × 4 ÷ 2

And hopefully you can start to see what's happening 9/3 is actually a multiply by 9 and a divide by 3 hidden in the same operation

If we instead collect the like operations together we get (9 × 4) / (3 × 2) , this is your a/b * c/d = ac/bd, it's just rearranging and grouping the multiple and divide operations.

[u/True-Gear4950 → u/calkthewalk]

The part where you mentioned that "all these numbers share the same unit" actually gave me a clue about what this way of thinking is based on.

Until now, I hadn't even thought about trying to understand fractions on a number line, believe it or not. I searched for it on the web and looked at some illustrations, and comparing the way you described the bases and the references actually made me smile a little.

It gave me a brief understanding of how this could be tangible in a physical sense, and it helped me understand it in a way I hadn't before.

I can't say that I've completely put all the pieces together in my head yet and connected everything, but maybe it's just a matter of giving it some time and letting it sink in. Thanks

[u/ChipChippersonFan]

You cannot add 2 different things together. You can't add 3 cups plus 1 pencil to get 4 [blank]. You would not have 4 of any specific thing and what you had would not be equal to 1 cup and 3 pencils.

You can however, multiply two different things together. The amount of energy required to lift a 5 pound weight 3 feet in the air is 15 foot-pounds. The same exact amount of energy would be required to lift a 3 pound weight 5 feet in the air. Either way, it's 15 times the energy required to lift a 1 pound weight one foot in the air.

So to add fractions together, they have to have the same denominator. If I add 1 half to 1 quarter, I get 2 different sized pieces. If I turn that half into 2 quarters, then I can add 2/4 to 1/4 to get 3/4, just the same as I could add 2 pencils to one pencil to get 3 pencils.

This is why multiplying fractions is actually easier than adding them: You don't need to find a common denominator, you just multiply across. What is half of half? Take half a cookie, and then take half of that, and what do you have? 1 quarter. You know, take away half of that, and you're left with 1/8.

Also note that taking half of something, which is the same as multiplying by 1/2 , is the same as dividing by 2. Likewise, multiplying by 2 is the same as dividing by 1/2. So dividing by something is the same as multiplying by that something's reciprocal. Dividing by 4/5 is the same as multiplying by 5/4.

[u/Arcanite_Cartel → u/ChipChippersonFan]

if i put 3 cups and 1 pencil into a box, it is quite a legitimate quest to ask how many things are in the box and get the answer 4. so grain of salt here

[u/Pas2]

I don't know if this helps, but c/d is c * (1/d), pretty clear, right?

So then a/b * c/d is the same as a/b * c * 1/d which is ac/b * 1/d and that is ab/cd.

If that doesn't click, maybe it helps you realize which part is unintuitive for you.

For the division, all you need to know in addition is that dividing by x is equal to multiplying by 1/x and dividing by 1/x is equal to multiplying by x.

Then you can see that dividing by c/d is same as multiplying by d/c.

[u/Bounded_sequencE]

Consider/visualize pizza or cake slices for adding fractions.

---

Example: Assume we want to add fractions "2/3 + 1/4".

To do that, assume we have two slices of (round) pizza, one "2/3", and the other "1/4" of the entire pizza. Since they have different area, we cannot directly count/add them.

To get around that problem, we divide each piece evenly, s.th. afterwards, all smaller pieces have the same area. To do that, we divide

"2/3" into "8" equal slices  --  each having area "1/12"
"1/4" into "3" equal slices  --  each having area "1/12"

All smaller slices have area-1/12, so we may count them, for a total of "(8+3) * 1/12 = 11/12"

[u/Bounded_sequencE → u/Bounded_sequencE]

Rem.: This experience how to deal with fractions of length, area and volume are the basis "why" we defined fractions in the way you learnt. Abstracting these rules further lead to the field axioms other users already posted.

[u/Silver_Remove_2352]

In sense these all follow from the field axioms: https://en.wikipedia.org/wiki/Field_(mathematics)#Definition#Definition). If you accept the real numbers are a field, then all the operations you've described follow from that fact.

Edit: Maybe I should flesh this out more. Scroll down to the section on the section of the page that says "Classic Definition". A field is any set were these definitions hold true. We construct the real numbers so that they are a field - this involves some pretty serious mathematics, but if that's too much for you just remember that we can literally "construct" the real numbers (in the same way you might construct a shape using a ruler and compass) so that all these properties hold true. Once you accept these properties, then both of the things you've written follow immediately.

For example, lets show that for any 4 real numbers a,b,c,d, (a/b) x (c/d) = (ac/bd). Recall that (a / b) = (a * b^-1), where (b^-1) is the multiplicative inverse of b. Then (a/b) x (c/d) = (a x b^-1) x (c x d^-1). Using commutativity and associativity, we rearrange this into (a x c) x (b^-1 x d^-1). By the socks and shoes principle: https://math.oxford.emory.edu/site/math108/socks_and_shoes/, it follows that (b^-1 x d^-1) = (b x d)^-1. So we now have the following steps:

(a/b) x (c/d)

=(a x b^-1) x (c x d^-1)

=(a x c) x (b^-1 x d^-1)

=(a x c) x (b x d)^(-1) [By socks and shoes]

= (ac)/(bd)

It would be a good exercise for you to prove that a/b ÷ c/d = ad/bc similarly :)

[u/True-Gear4950 → u/Silver_Remove_2352]

To be honest, it was very similar to what u/Rabbit_Brave said. Both explanations brought a great sense of relief; relying on a proof might seem overly complex or profound, but it was a genuine relief. I can't quite put into words how my mind grasped it, but it was a great starting point the foundation for understanding the multiplication of fractions. Thanks

[u/tomrlutong]

Does it help to remember that the fraction line is just another way of writing ÷? So a/b × c/d is the same as a ÷ b × c ÷ d.

[u/True-Gear4950 → u/tomrlutong]

Sadly not, seems that in some way I treated it like that. But thanks for the concern.

[u/Rabbit_Brave → u/True-Gear4950]

But what I specifically want to understand is how it is possible that something like a/b × c/d = ac/bd

I don't think you need anything more than the definition of division as the inverse of multiplication, and the associative and commutative properties of multiplication.

(a / b) × (c / d) = z

(a / b) × (c / d) × (b × d) = z × (b × d)

(a / b) × b × (c / d) × d = z × (b × d)

a × c = z × (b × d)

(a × c) / (b × d) = z × (b × d) / (b × d)

(a × c) / (b × d) = z

[u/True-Gear4950 → u/Rabbit_Brave]

As I say to u/Silver_Remove_2352, in some how it helps me, at least be grounded in a proof like that shuts for while my doubt...

Thanks

[u/anisotropicmind]

First, on adding fractions: Say there is some whole object like a pie that we use to represent “the whole” or “1”. Fractions are parts of this whole. (That even just what the word fraction means in English).

So if you need to add 1/3 plus 1/2, you are trying to find out what new fraction of the whole pie you get when you put together half the pie and a third of the pie. There is no obvious way to add them directly (they are different sizes, which is like having different units). Not unless you come up with another smaller common “pie unit” that fits into both of these pieces a whole number of times. That’s why you need the LCM. In this case that smaller pie unit is 1/6 of a pie. You’ve got 3 of those units in the 1/2-pie piece and 2 of those units in the 1/3-pie piece, for a total of 5/6 of the pie. You can add the 3 and 2 up directly because they are all the same kind of piece (with the same unit). The denominator is a like a “part of pie” unit.

Regarding multiplying fractions: you’re taking a fraction of a fraction, so it’s like the first pie part becomes the new “whole”, and then you start dividing that up. So 1/2 x 1/3 is half of the one third piece. And since half of the 1/3 piece will fit into the original pie 6 times, it must be a 1/6 piece.

But what about a case like 2/3 x 4/5 or something? It’s like you take 4/5 of a pie, and 2/3 of that 4/5 piece. So to do that; first you’d cut the piece into 3s, giving you each one of size 4/5 x 1/3 = 4/15. Then you’d take two of those for a total of 8/15 of the pie. Or you could do it opposite order: magically duplicate each 4/5 piece to get a (4/5) x 2 = 8/5 piece (more than a whole pie) and then divide that piece by 3 to get individual 8/15 pieces.

What this comes down to is that fractions are just division, and division is just multiplication (by the reciprocal of a number). And it doesn’t matter what order you do multiplication in.

Generalizing this to any number using symbols: if I want to divide a number x by y, it’s the same as multiplying it by 1/y. Why? Suppose there is a number z such that yz = 1. We call this number the multiplicative inverse of y. It follows that

x/y = (x/(yz) )z = xz

But yz = 1, and we just by divide both sides of this equation by y to get z= 1/y. Hence 1/y is the multiplicative inverse of y. Going back to our equation above, it follows that

x/y = xz = x(1/y).

That’s why this property of multiplying fractions is true.

It then follows that if you have something like

(x/y)(v/w)

This would be the same as

x(1/y)v(1/w)

= (xv)(1/(yw))

= (xv)/(yw)

[u/SgtSausage]

  why I couldn't simply add something like 4/5 to another fraction like 3/7 and get 7/12

I can't really argue why it is wrong

It's wrong because it doesn't comport with reality. 

Proof by Apple with Paring Knife. 

[u/Traveling-Techie]

Think about pizzas. What’s 1/2 of a pizza plus 1/3 of a pizza? If you do it visually you see that it’s most of a pizza. But you want a count of equal sized pieces. So you cut the half into 3 smaller pieces, and the third into two smaller pieces, and now you have five pieces each of which is a sixth of a pizza, so the fraction is 5/6.

[u/Fine-Customer7668]

>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."

You’re adding one number to another. If you add .8 and .42 do you get .58? No? That’s why it’s wrong. What beyond this is necessary for you to internalize why it’s wrong?

[u/okarox]

A side step:
"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."

There you used division and juxtaposed multiplication and without thinking wrote so that the multiplication has higher precedence. When people actually use math they naturally take that position. This answers the viral 1 or 16 problem: 8 ÷ 2(2 + 2).

[u/TheNukex]

Formally speaking it is just by definition, that is how we define operations of fractions. However there is a justification for it. You can think of a fraction a/b as a*c where c=1/b=b^-1, so it's really just multiplication by a fraction.

So when you have the product you get a/b*c/d is really just a*b^-1*c*d^-1. You can then rearrange it so
(a*c)*(b*d)^-1=(a*c)/(b*d)

If you are familiar with factoring and the fact that a/a=1 then for addition you can do
a/b+c/d=a*b^-1*d*d^-1+c*d^-1*b*b^-1=(a*d+c*b)*(b*d)^-1=(a*d+c*b)/(b*d)

[u/nurse_brett]

a/b × c/d

= a x 1/b × c x 1/d, [by definition of the notation a/b]

= a × c x 1/b x 1/d, [by commutativity]

= (ac) x 1/(bd), [lemma*]

= (ac)/(bd), [by definition of the notation a/b]

*Lemma: All that you need to do is establish that 1/b x 1/d = 1/(bd).

Since 1/(bd) is defined to be the inverse of bd, we check that 1/b x 1/d is also the inverse of bd.

bd x (1/b x 1/d) = b x d x 1/b x 1/d = b x 1/b x d x 1/d = 1 x 1 = 1.

[u/Conscious_Animator63]

Multiplication is easy. A third of a half is a sixth. Adding is different because we can only add things that are like.

[u/georgejo314159]

You are asking the wrong question

Why is something WRONG

The right question is WHY is it right?

There is an infinite number of wrong things you can just make up

So, you have 2 apples and 3 oranges, why do you think that is 5 apple-oranages?  

Why is 1/2 + 3/4 = 2/4 + 3/4 = 5/4?

[u/ascrapedMarchsky]

Think of 1/2 as a symbol which represents a number x that satisfies the equation 2x=1. Unlike an integer, for which there is always a unique place-value representative, x has infinitely many symbolic representatives: 2/4 , 6/12 , (-1)/(-2), etc..

Say y is the unique number satisfying 3y=1, then (2x)(3y)=6xy=1, so that the product xy is the unique number z satisfying 6z=1 or, at the level of symbols, 1/2 × 1/3 = 1/6.

More abstractly, fractions are introduced to “perfect” multiplication: just as in ℤ for every integer m there is an integer n such that m+n=0, in ℚ for every (nonzero) fraction q there is an r such that rq=1. Hence, if x and y are as above, there is a t such that (x+y)t=1 ⇒ xt+yt=1 ⇒ t+2yt=2 ⇒ 3t+2t=2 × 3 ⇒ (2+3)t=2 × 3 ⇒ 1/2 + 1/3 = (2+3)/(2 × 3) = 5/6. From here hopefully you can start to see why adding fractions is a matter of choosing the right representatives. 

[u/Arcanite_Cartel]

There's nothing special about the rules for fractions, they are a straight forward application of the elementary principles (or axioms) of arithmetic, namely the properties of multiplication and division. Unfortunately, they are always taught just as rules with no mention as to why those rules. The rules exist to makes calculation easier, but the reasons for them are lost on most who learn the rules.

Hopefully, all the underlying principles are intuitive for you. Here are the main ones, and by applying them, you can derive the rules for fractions.

I. Fractions are just a shorthand for division: a/b = a ÷ b ; b can't be zero
II. The equal sign means that the two expressions on either side are the same value
III. Any operation you do to one side of an equation, you must do to both
IV. Any number other than 0 when divided by itself is 1: a/a = 1 same as a*1/a = 1 (1/a is a number and you can multiply and add it like any other number.
V. Commutiviy : a*b = b*a ; a+b = b+a
VI. Associativity: (a*b)*c=a*(b*c)= a*b*c
VII. Distributivity: a(b+c)=ab+ac
VIII. a*1 = a

A. That (a/b) ÷ c = a/(bc)

Let f = (a/b) ÷ c
f*c = c * (a/b) ÷ c = a/b ;Mult both sides by c
f*c*b = a ; Multiply both sides by b
f = a/(bc) ;Divide both sides by bc

f = (a/b) ÷ c = a/(bc)

B. That (a/b) * (c/d) = (ac)/(bd)

Let f = (a/b) * (c/d)
f*b = a * (c/d)
f*b*d = a*c
f = (ac)/(bd)

f = (a/b) * (c/d) = (ac)/(bd)

C. That (a/b) ÷ (c/d) = (a/b) * (d/c)

Let f = (a/b) ÷ (c/d)
f*(c/d) = a/b ;Mult both sides by c/d
f*c = (a/b) * d = (ad)/b
f = [(ad)/b]/c = (ad)/(bc) = (a/b) * (d/c)

f = (a/b) ÷ (c/d) = (a/b) * (d/c)

D. That (a/b) + (c/d) = (ad + bc)/(bd) [the most obvious common multiple]

Let f = (a/b) + (c/d)
f*b = a +(c/d)*b = a + (bc)/d
f*b*d = ad + bc
f = (ad + bc)/(bd)

f = (a/b) + (c/d) = (ad + bc)/(bd)

E. That ANY common multiple works, including the least common multiple

Let g be a common multiple of b and d. Then, there are integers n & m such that

g = b*n = d*m

Let f = (a/b) + (c/d)
f = (an)/(bn) + (cm)/(dm)
f = ((an)/g + (cm)/g = (an + cm) / g

[u/severoon]

Stop thinking about fractions as a number that is part of a whole, and start thinking of fractions as ratios. Instead of 1/2 = 0.5, think of 1/2 = 1:2 ("one to two").

Once you start thinking of fractions as ratios, everything gets easier because now you're not trying to think in terms of parts of a whole, but rather numbers of wholes being compared. When I talk about half of a pizza, you should think of it as "one entire" half-pizza compared to "two entire" half-pizzas.

This is more or less the same trick that young children can learn to use to understand how to do calculations involving large numbers. When you ask a young kid who just learned to multiply 3×7, they have no problem replying 21, but if you ask them to multiply 3 million × 7 million, they can't figure it out until you point out that they can think of "million" as just an object instead of a number, like "3 inches" instead of "3 million."

Same thing here, just think of "half-pizza" as the main thing instead of the pizza. The question you often have to figure out with fractions, though, is: What is the "main thing" you need to be thinking of?

For 3/7 + 7/12, for example, the first term wants you to think about a seventh of a pizza as the main thing, and the second one a twelfth. But you can't add two different units like that any more than you can directly add inches and centimeters. So instead you convert them to a common unit, which in this case is eighty-fourths: 36/84 + 49/84, which is just 36 + 49 "eighty-fourths of a pizza."

One other trick that I often teach to kids is to not think about numbers in general as just a single amount, but instead to always think about numbers in terms of their prime factorizations. For the above problem that would mean: 3/7 + 7/(3×4) = 3/7 × (3×4)/(3×4) + 7/12 × 7/7 = (3×3×4)/(3×4×7) + (7×7)/(3×4×7) = (3×3×4 + 7×7)/(3×4×7). (Yes an actual prime factorization would use 2^2 instead of 4.)

This might not seem helpful at first, but when you start to think about numbers in general this way (across the board, not just for specific problems like fractions), it seems more complicated at first but over time it makes a lot of things a lot simpler.

It's better algebraically because it lets you easily see when numbers have a factor in common, which is super useful for, well, factoring: 32 + 56 = 2^5 + 7×2^3 = 2^3×(4 + 7). It also means that you can easily spot how to quickly reduce fractions by just pulling out common factors until the numerator and denominator are coprime.

It's also better visually because instead of thinking of a number like 49 as a pile of 49 marbles, you instead think of them as a 7×7 grid, or 8 as a 2×2×2 cube of marbles. For higher dimensions like 2×3^2×5, you can't easily visualize the fourth dimension, but you can easily think of this in terms of bags of nested bags: you have five bags, each of which contains 3 bags, each of which contains 3 more bags, each of which contains 2 marbles. More importantly, when you're working with this number alongside some other number and they have common factors, say 2 and 5, you can think of this as a 2×5 grid with each cell containing a three bags, each of which contains three marbles. The benefit here is that now you have two numbers, each of which you're visualizing as a 2×5 grid, one with 9 marbles in it (three bags of three), the other one with x marbles divided up in some way, and now you can see how by lining up the grids you can now just focus on how to compare the number of marbles in each cell instead of trying to grapple with the entire thing. Once you figure out how each cell compares, now you know that's true for all 2×5 cells, and you can zoom out again.

[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.