Practice with the form a(b/c)

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:

$$ a\cdot\frac{b}{c} = \frac{ab}{c} = \frac{ba}c = b\cdot\frac{a}{c} = ab\cdot\frac{1}{c} = ba\cdot\frac{1}{c} = a\cdot\frac{1}{c}\cdot b = \frac{1}{c}\cdot ba = b\cdot\frac{1}{c}\cdot a = \frac{1}{c}\cdot ab = \cdots $$

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.

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

$\,b = 1\,$ or $\,b = -1\,$.
However, for all other values of $\,b\,$, 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.

(an even number, please)