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

Question

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

In: 13

Answer 1

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.