In the previous section I tried to drill into your minds the idea that every time you see a fraction, you should do two things: FACTOR and REDUCE! In this section, we'll explore multipying fractions. The same rule applies: FACTOR and REDUCE!

Let's start with a simple example - multiplying two numeric fractions.

Example #1

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Solution #1

Hopefully you remember that multiplying fractions involves multiplying straight across the numerators, and straight across the denominators. Doing this right away, however, results in:

 · = , which is definitely not in simplest form. So let's do our rule - every time you have a fraction, factor and reduce. You're going to do that to each fraction, before you multiply them.

= =

= =

 · =  · = =

Example #2

Multiply  ·

Solution #2

First we factor each fraction, which gives us:

 · =  ·

Interestingly, there's nothing we can cancel out of either fraction. So let's combine those two:

 · = 

Ah! Now there's something that'll cancel; we have a factor of 3 and a factor of 5 in the numerator and denominator. So before we multiply that all out, we'll cancel those:

=

If we remember that at every step we should reduce, we keep things from getting too ugly. Later you'll see some examples where forgetting to FACTOR and REDUCE makes the problems HORRIFYINGLY UGLY!

I should point out now that you may have been taught that when multiplying, you can cancel from the numerator of one fraction and the denominator of another. And that is true. So why didn't I do it that way? Why did I wait until I'd combined them into a single fraction before canceling? There are two reasons. First, I wanted you to see why you can cancel across fractions. Second, I have a lot of students who forget that you can only do that when you're multiplying; they want to do it when dividing, or adding, or subtracting. THAT DOESN'T WORK! If you're concerned that you'll forget when you can cancel across fractions, then avoid doing it altogether. Combine things into a single fraction, and THEN cancel. If you're confident that you won't forget the circumstances when you can cancel across fractions, then by all means do it.

Example #3

Multiply:  ·

Solution #3

As always, we factor and reduce: 

 · =  ·

In the first fraction, we can't cancel anything, but in the second fraction there is a factor of 2 in both the numerator and denominator:

= 

Now we multiply across:

  · =   

And now we find there is a 3, an 11, an x and a y that will cancel:

 = =

Example #4

Multiply:  ·

Solution #4

Let's not forget to stick parentheses around our binomials and trinomials to remind ourselves that the fraction bar is grouping them, and we can't cancel pieces out of them:

 · =  ·

By now you are hopefully getting sick of hearing "FACTOR and REDUCE." But hopefully you're getting used to it. We can factor both the numerator and denominator of the first fraction:

=

Do we have a common factor in the numerator and denominator? YES! (x + 2)

=

Now we multiply, since we can't factor and reduce the second fraction:

  · =

Now we can cancel an x:

=

The Rules So Far

So far, we really only have three rules for fraction manipulation

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.

Of course, you need to remember that once you've done step #3, you now have a new fraction, and you need to apply steps 1 and 2 all over again, to make sure it's in lowest terms!

In the next section, we'll add one simple rule, and we'll be able to divide algebraic fractions as well.

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