PRACTICE WITH THE FORM   $\displaystyle a\cdot\frac{b}{c}$
by Dr. Carol JVF Burns (website creator)
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One pattern that arises frequently in working with fractions is $\displaystyle \,a\cdot \frac{b}{c}\,$.

It's important to realize that this expression can be written in many different ways:

$$ \cssId{s6}{a\cdot\frac{b}{c}} \cssId{s7}{\ =\ \frac{ab}{c}} \cssId{s8}{\ =\ \frac{ba}c} \cssId{s9}{\ =\ b\cdot\frac{a}{c}} \cssId{s10}{\ =\ ab\cdot\frac{1}{c}} \cssId{s11}{\ =\ ba\cdot\frac{1}{c}} \cssId{s12}{\ =\ a\cdot\frac{1}{c}\cdot b} \cssId{s13}{\ =\ \frac{1}{c}\cdot ba} \cssId{s14}{\ =\ b\cdot\frac{1}{c}\cdot a} \cssId{s15}{\ =\ \frac{1}{c}\cdot ab} \cssId{s16}{\ =\ \cdots} $$

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.

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.
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
More Practice with the form $\,a\cdot\frac{b}{c}$


Assume that all variables are nonzero, so there's no concern about division by zero.

Compare these two expressions:

(an even number, please)