Is there any rule in algebra or any related rules for this equations?
if a is not equal to zero
$$
\frac{ab}{c} = \frac{b}{\frac{\displaystyle c}{\displaystyle a}}
$$
if b is not equal to zero
$$
\frac{ab}{c} = \frac{a}{\frac{\displaystyle c}{\displaystyle b}}
$$
On
Note that
$$\frac{ab}{c}=\frac{ab}{c}×\frac {\frac 1a}{\frac 1a}=\frac{ab×\frac 1a}{c×\frac 1a}=\frac{b}{\frac ca}$$
Similarly
$$\frac{ab}{c}=\frac{ab×\frac 1b}{c×\frac 1b}=\frac{a}{\frac cb}$$
Here we used the rule
$$\frac AB×\frac CD=\frac {AC}{BD}.$$
and the following definition
$$\frac AB=A×\frac 1B.$$
I think a common rule encountered at lower levels is $$ \frac{\left(\frac ab\right)}{\left(\frac cd\right)} = \frac ab \cdot \frac dc. $$ (Where I’m from we called it “multiplying by the opposite”). This follows from $$ \frac{1}{\left(\frac cd\right)} = \frac dc, $$ which again follows from $$ \frac cd \cdot \frac dc = \frac{cd}{dc}=1. $$ Does that help at all?
Even more fundamentally, in the end it really boils down to $$ \frac{1}{\left(\frac 1d\right)} =d $$ (together with how fractions and multiplications “play nice” with each other).