Numerator and Denominator
The number or algebraic expression that appears on the
top line of a fraction is called the numerator
of the fraction.
The number of algebraic expression that appears on the bottom
line of a fraction is called the denominator of
Expressed in symbols, the rule for adding fraction is as
Lets break this down to see everything that is expressed
in this rule. The numerator of the sum is a · d + b · c. You
can remember the numerator without having to memorize this
particular formula by remembering the pattern of cross-multiplying.
To create the numerator, you multiply each numerator by the
opposing denominator, forming a cross pattern.
To get the denominator of the sum, you just multiply the two
denominators ( b and d ) together.
Work out each of the following sums of fractions.
Often it will be possible for you to simplify your fractional
expressions by combining like terms just as you do
when FOILing a polynomial. Although this kind of simplification
is not always needed just to get the right answer, if can make
your fractional expressions much easier to deal with. Remember to
keep the numerator and denominator separate when combining like
In Example (b), note how when the cross-multiplication is
done, the 7 from the numerator of the first fraction
multiplies the entire quantity ( x + 1) that is in the
denominator of the second fraction, not just the x . Also notice
that when the two denominators are multiplied to create the
denominator of the sum, the 10 from the denominator
of the first fraction multiplies everything (i.e. the entire
quantity ( x + 1)) that appears in the denominator of the second
When simplifying fractions, simplify the numerator and
denominator separately. You cannot combine like terms from the
numerator with like terms from the denominator (or vice versa).
Often you will need to FOIL when simplifying the numerator and
denominator of fractions that involve algebraic expressions such
as x .
This answer is not the simplest one that is possible. If you
look closely at the middle fraction above, you can see that every
single term in the numerator has at least one factor of ( x + 1).
The denominator also has a factor of ( x + 1). These
common factors can be factored out of the numerator
and the denominator as shown below.
When you have a common factor that you have pulled out of
every term in the numerator, and it matches a factor that shows
up in the denominator, you can almost always cancel this factor
from both the numerator and the denominator.
, provided x -1.
The only situation when it is not okay to cancel the factor of
( x + 1) from the top and bottom is when you have the x -value of
x = - 1 (i.e. the particular x -value that makes the factor of (
x + 1) equal to zero).