One pattern that arises frequently in working
with fractions is:
$$a\cdot \frac{b}{c}$$
It’s important to realize that this expression can be
written in many different ways:
$$
\begin{align}
\cssId{s6}{a\cdot\frac{b}{c}}
&\cssId{s7}{\ =\ \frac{ab}{c}}
\cssId{s8}{\ =\ \frac{ba}c}\cr\cr
&\cssId{s9}{\ =\ b\cdot\frac{a}{c}}
\cssId{s10}{\ =\ ab\cdot\frac{1}{c}}\cr\cr
&\cssId{s11}{\ =\ ba\cdot\frac{1}{c}}
\cssId{s12}{\ =\ a\cdot\frac{1}{c}\cdot b}\cr\cr
&\cssId{s13}{\ =\ \frac{1}{c}\cdot ba}
\cssId{s14}{\ =\ b\cdot\frac{1}{c}\cdot a}\cr\cr
&\cssId{s15}{\ =\ \frac{1}{c}\cdot ab}
\cssId{s16}{\ =\ \cdots}
\end{align}
$$
Note that a factor in the numerator
can optionally be centered next to the fraction.
If everything is moved out of the
numerator, then a $\,1\,$ is inserted as a ‘placeholder’.
A factor centered next to the fraction
can be moved into the numerator.
A factor in the denominator must stay in the denominator.
Examples
The expressions
$\displaystyle\, a\cdot\frac{b}{c}\,$
and
$\displaystyle\,\frac{ba}{c}\,$
are ALWAYS EQUAL.
That is, no matter what numbers are chosen for
$\,a\,,$ $\,b\,,$ and
$\,c\,,$
substitution into these two expressions
yields the same number.
(Note, of course, that $\,c\,$ is not allowed
to equal zero.)
The expressions
$\displaystyle\, ab\cdot\frac{1}{c}\,$
and
$\displaystyle\, a\cdot\frac{1}{bc}\,$
are NOT ALWAYS EQUAL.
Note that there do exist choices
for which these two expressions
give the same value:
when $\,a = 0\,,$ or
$\,b = 1\,,$ or $\,b = -1\,.$
However, for all other values of $\,b\,$
(and $\,a\ne 0\,$), they are not equal.
Practice
Assume that all variables are nonzero, so there’s no concern about division by zero.