I have all my fraction rules written down, but these two confuse me...
$$ \frac{\;\;A\;\;}{\frac{B}{C}} \equiv{\frac{\;AC\;}{B}} $$
and
$$ \frac{\frac{B}{C}}{\;\;A\;\;} \equiv{\frac{\;B\;}{CA}} $$
These do not give the same answers, and I am wondering what the rules are for using them? Are $$A>B>C$$ and $$B>C>A$$ the two conditions?
On
No, there are no such "rules" for using them, $A>B>C$ or $B>A>C$ don't have to be conditions. To make fractions like these seem simpler, think of division as the "opposite" of multiplication. I'll show you what I mean.
$\cfrac{A}{\frac{B}{C}}$
is equal to $\cfrac{A}{1}$*$\cfrac{C}{B}=\cfrac{AC}{B}$
The second fraction you put up is incorrect.
$\cfrac{B}{\frac{C}{A}}$
is equal to $\cfrac{B}{1}*\cfrac{A}{C}=\cfrac{BA}{C}$
Think of division $\cfrac{A}{B}$ as multiplication of $A*\cfrac{1}{B}$
You may have been confused by the implied parentheses in the size of the fraction bars. Your first rule is $$\frac A{\left(\frac BC\right)}=\frac {AC}B$$ which you can establish by multiplying the fraction by $\frac CC$. You have written the second the same way on the left, $$\frac B{\left(\frac CA\right)}=\frac {B}{CA}$$ which is incorrect. I suspect you meant to write $$\frac {\left(\frac BC\right)}A=\frac {B}{CA}$$ which is correct and can be established by multiplying by $\frac CC$. The only clue in what you wrote is the size of the fraction bars and your top one is the largest in both cases. It is hard to see that in my browser, but the strucure of your MathJax makes it clear.