Why if we want to know what percentage of 16 is 4 we do
4/16
and not
16/4
?
4/16 gives you the answer, because it's equal to 0.25, that is equal to 25%, while 16/4 doesn't (I guess); but I don't understand the logic. It seems more natural to me to do 16/4, because if I want to know the percentage I would split the 16 by 4 to know how many parts does the 4 create (in my logic the number of parts created define the percentage). So if I try to do 16/4 I get 4, but then I don't know how to go on to get 25%. Maybe there's a way to obtain the result in the way I'm proposing but I don't know it.
I hope my question doesn't sound stupid, but this is one of the many basic things that I'm lacking in math to fully understand its logic.
Percentages are always expressed as a part of 1.
As you say $r = {p\over q}$ gives you the number of parts of $q$ in $p$, but then you have to compute the number of parts of $r$ in $1$ to get a percentage. So your percentage is indeed ${1\over r}$.
And
$${1\over r} = {1\over {p\over q}} = {q\over p}$$