Practice with the Form $\,a\cdot\frac{b}{c}$

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 $\displaystyle\quad a\cdot\frac{b}{c}\quad$ and $\displaystyle\quad\frac{ba}{c}\quad$ 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 $\displaystyle\quad ab\cdot\frac{1}{c}\quad$ and $\displaystyle\quad a\cdot\frac{1}{bc}\quad$ 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.