Multiplying and Dividing Fractions
Examples with Solutions
Example 1:
Multiply and simplify the result.
solution: Although the expressions are rather
more complicated here than in any previous examples, the same old
strategy still applies. We need to simplify the product of the
two fractions:
Even though the numerator and denominator here contain
products of binomials, there is a temptation to multiply both of
the products out separately. Here this would give us something
like
which can be simplified, but surely has to look like
we’re going in the wrong direction. Since simplifying
fractions occurs via cancelling common factors between the
numerator and the denominator, it is always far better to attempt
more factorization than to consider multiplying to\par remove
whatever factors are already present. So, we will abandon (at
least for the present) continuing with this last monstrosity just
above, and attempt to factor the original product. Considering
existing factors one at a time, this turns out to be quite easy.
Thus
as our final answer. It is tempting to cancel the (a + 2b) in
the numerator with the (a – 2b) in the denominator, but
these two expressions are not identical, so it would be an error
to cancel them. (If you enjoy a challenge, or if you are a
glutton for punishment, you could attempt to simplify the gross
“multiplied out” expression above. If you follow the
strategies for factoring, it’s not impossible to do, and you
will end up with the same final answer. Along the way, you may
realize that some of the work you’re having to do is simply
undoing the initial expansion operation – hence that initial
step really was no progress at all!)
Example 2:
Simplify the result of dividing .
solution: The overall work to complete this
problem is very similar to what appeared in the previous example,
so you might try to do this one by yourself first, before looking
at the solution to follow.
To avoid trying to do so many things at once that making
errors becomes almost inevitable, we’ll do the factoring of
each bracketed expression first. It’s probably prudent to do
these factorizations in the stepwise fashion described earlier
in these notes, since there is a fair amount of detail to keep
track of here.
So
To save space, we’ll just list the final results of the
factoring of the remaining three expressions:
So,
as the final result.
This problem, like the one in the previous example, looked
horrible at a first glance. But you should spend enough time
studying it stepbystep to see that as bad as it may look as a
whole, the final result shows up almost automatically if you just
carefully employ the basic strategy for simplifying fractions.
Anyone can do the individual steps oneatatime required for
solving this problem.
Example 3:
Simplify
solution: This looks too simple. We get
Even though there appears to be a lot of similar things in the
numerator and denominator, no further cancellations are possible,
and so this is the final answer. Probably most people would write
it as
How do you know that, for example,
is wrong? Because in the first step, the a’s that are
cancelled are not factors in either the numerator or the
denominator. If what you’re cancelling are not common
factors, then you’re making a mistake. The same
applies here to suggestions that we cancel the b’s or the
5’s. In both cases, although one is a factor, the other is
not, and so cancellation produces an invalid result. This is an
example where, once we’ve done the initial division of one
fraction by another, no further simplification of the result is
possible.
Example 4:
Simplify:
solution: Although we’ve only looked at
products of two factors so far, it is obvious that products of
three or more fractions work the same way – just multiply
the numerators together and multiply the denominators together,
and then simplify the results. So here, we get
There was a lot of bookkeeping of factors here, so we took a
few extra steps to ensure no errors were made. Recall that any
number to the power zero just gives 1, and so x^{ 0} = 1
in the last line above.
In a similar fashion, the rule for division can be extended to
situations involving more than two fractions. However, in such
cases, we would be dealing with some variation of a mixed
multiplication and division problem. You should be able to sort
such problems out if you pay careful attention to brackets or
other symbols of grouping.
