Why does flipping the second fraction around and then multiplying work when dividing two fractions?

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Eg. 8 over 3 divided by 1 over 3 = 8 over 3 times 3 over 1

In: 13

Ok, let’s simplify this a bit.

When you take any number and divide it by 1/3, let’s say you’re dividing 5 (instead of 8 over 3, it works the same, 5 is just a simpler example.)

So you have 5 divided by 1/3.

This is like taking 5 apples, and dividing them into groups where each group only has 1/3rd of an apple.

Which is like cutting an apple into thirds.

Which is like taking 1 Apple and making it 3 pieces. One piece for each group.

So for each apple, you get 3 pieces, so if you divide all 5 apples, you get 15 pieces. 5 divided by 1/3 is 15.

Just like multiplying 5 by 3.

Multiplication and division are just opposite functions.

For example, you could do this in reverse. Say you want to take 10, and divide it by 2. That gives you 5. Or you could take 10, and multiply it by 1/2 (or 0.5). Which also gives you 5.

Division is the opposite of multiplication.

So dividing by a dividing thing is a double-division, so the opposite of the opposite of multiplication; which is just multiplication.

Because division and multiplication are the same kind of thing(ish), we can always do them in any order. So:

> 8/3 divided by 1/3

can be split up into:

> 8 divide by 3, divide by 1, divide by “divide by 3”

but that “divide by divide by 3” is a double division, so a multiplication:

> 8 divide by 3, divide by 1, multiply by 3

We can now re-order that:

> 8 divide by 3, multiply by 3, divide by 1

and then join these bits up together again to get:

> 8/3 multiply by 3/1

It is a deceptively complicated process but it only relies on division being the opposite of multiplication, and that you can re-order (and regroup) division and multiplication.

And understanding that fractions are the same as division.

[This also means that phrasing and brackets are important. 8/1/3 could be “8 divide by 1, divide by 3”, or it could be “8 divide by (1 divided by 3)” (so 8 multiplied by 3). ]

Say that you have

> (a/b) / (c/d) = e

You can multiply both sides by (c/d):
> ((a/b) * (c/d)) / (c/d) = e * (c/d)

You can then simplify the left side since (c/d) / (c/d) = 1
> (a/b) = e * (c/d)

You can now multiply both sides by d, and then divide by c. c cannot be 0, or you would have a division by 0 to begin with.
> (a/b) * d = e * (c/d) * d = e * c

> (a/b) * (d/c) = e * (c/c) = e

And both (a/b) / (c/d) and (a/b) * (d/c) equal e, which means they are equal.

Division does not exist. There is only Multiplication and the Multiplication of Reciprocals. This is why you cannot divide by Zero, Zero has no Reciprocal

khalamar’s answer is correct and probably the simplest one so far.
But we can do even simpler, in the same spirit.

Start with a fraction:

> (a / b) / (c / d)

Multiply both sides of the fraction by (d), you get:

> (d * a / b) / ( c )

Multiply both sides again by (b) , you get:

> (d * a) / (c * b)

Notice that this is equivalent to the flip you were mentioning.

Division fundamentally means splitting things up into groups. 6 divided by 2 is like saying “if I have 6 bananas and I split them up into groups of 2, then how many groups will there be?” How many 2s go into 6? 3. So you get 3 groups.

Now apply this to a fraction, say 6 divided by 1/2. Well now I’m splitting them up into groups of a half each. Well, how many 1/2s go into 6? 12. It’s the same principle.

However, as this gets more complicated and the fractions get harder, this gets harder to reason through intuitively. So maths is all about creating shortcuts to reduce mental strain and make life easier. One clever person noticed that 6 divided by 1/2 is the same as 6 multiplied by 2. You can think of 1/2 as 1 divided by 2, so the two divisions sort of cancel each other out to get a multiplication.

Once you have this rule, you can just blindly apply it. It’s therefore much easier to flip a fraction and multiply than it is to consider how many 2/3rds go into 29/16. Try working that out without flipping the 2/3 fraction and it’ll take forever.

Basically, someone noticed that there was a trick, and ever since we’ve just used that trick. This isn’t necessarily a why it works, it’s more of a why we do it – sometimes, I’d argue that the latter is just as important.