PRACTICE WITH THE FORM   [beautiful math coming... please be patient] $\displaystyle a\cdot\frac{b}{c}$

One pattern that arises frequently in working with fractions is [beautiful math coming... please be patient] $\displaystyle \,a\cdot \frac{b}{c}\,$.

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

[beautiful math coming... please be patient] $$ a\cdot\frac{b}{c} \ =\ \frac{ab}{c} \ =\ \frac{ba}c \ =\ b\cdot\frac{a}{c} \ =\ ab\cdot\frac{1}{c} \ =\ ba\cdot\frac{1}{c} \ =\ a\cdot\frac{1}{c}\cdot b \ =\ \frac{1}{c}\cdot ba \ =\ b\cdot\frac{1}{c}\cdot a \ =\ \frac{1}{c}\cdot ab \ =\ \cdots $$

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 [beautiful math coming... please be patient] $\displaystyle\quad a\cdot\frac{b}{c}\quad$ and [beautiful math coming... please be patient] $\displaystyle\quad\frac{ba}{c}\quad$ are ALWAYS EQUAL.

That is, no matter what numbers are chosen for [beautiful math coming... please be patient] $\,a\,$, $\,b\,$, and $\,c\,$,
substitution into these two expressions yields the same number.
(Note, of course, that $\,c\,$ is not allowed to equal zero.)
The expressions [beautiful math coming... please be patient] $\displaystyle\quad ab\cdot\frac{1}{c}\quad$ and [beautiful math coming... please be patient] $\displaystyle\quad a\cdot\frac{1}{bc}\quad$ are NOT ALWAYS EQUAL.

Note that there do exist choices for which these two expressions give the same value:
when [beautiful math coming... please be patient] $\,b = 1\,$ 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.

Compare these two expressions:
  and  
ALWAYS EQUAL
NOT ALWAYS EQUAL
    
(an even number, please)