At the end of the last section I said that one simple addition to our rules allows us to do algebraic fraction division problems. Since that one rule turns division problems to multiplication problems, and you already know how to do those kinds of problems, this will be a short section. Here's the rule:

Example #1

Divide ÷

Solution #1

We rewrite this as a multiplication problem:   · 

And you know where to go from here. We factor all numerators and denominators, giving:

 ·

In the second fraction we cancel the common factor:

=

Now we multiply across:

 ·  =

And finally, we cancel common factors (2² and 7):

= =

As I mentioned in the previous section, when multiplying fractions, you can cancel across fractions (for example, you can cancel the 7 from the numerator of the first fraction with the 7 in the denominator of the second fraction) before combining them into a single fraction. However, if you're concerned that you'll forget the circumstances under which it's allowed to cancel that way, I encourage you to combine them into a single fraction before canceling.

Example #2

Divide  ÷

Solution #2

Rewrite as multiplication:  ÷ =  · 

Factor:  ·  =  ·

Cancel (x + 3) out of the first fraction's numerator and denominator, and then multiply across:

 · =  · =

Finally, cancel factors from the resulting fraction. Interestingly, both factors occur in both the numerator and denominator, which means we are left with = 1.

The Rules So Far

Now we have four rules

1. Any time you see a fraction, FACTOR the numerator and denominator.
2. Any time you have a fraction with factored numerator and denominator, REDUCE, by canceling common factors.
3. Multiplying fractions involves multiplying across the numerators, and multiplying across the denominators.
4. Dividing fraction is the same as multiplying the first fraction by the reciprocal of the second fraction.

 

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