Eg. 8 over 3 divided by 1 over 3 = 8 over 3 times 3 over 1
In: 13
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.
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