Section 6.6 Complex Fractions
TWO WAYS TO WRITE DIVISION
8
You know that 8 ÷ 2 = 4. You also know that 2 can reduce to 4. Both of these forms are used to express
“8 divided by 4.” In other words, we can use the division symbol ÷ or the fraction bar to represent division.
2
5
When the division is a little more complex, like one fraction divided by another, as in 3 divided by 4 , it’s
2
5
easier to write it as 3 ÷ 4 . In algebra, however, we could see that division written with the division bar, as in
2
3
5
4
2
5
. This looks, of course, more complex than just 3 ÷ 4 , and for this reason, we call such a fraction as
2
3
5
4
a complex fraction.
A complex fraction is any fraction in which the numerator or the denominator, or both, are themselves
fractions. Other examples of complex fractions are
5
2
3
1
7
8
6
5
2
x
2y
3
y2
2 5
3+6
5 1
4–2
1
x
3
– 2x
5
1 + x2
Notice that each has more than one division bar, but the largest division bar is the main operation, the one that
is applied last. It is also the main division bar in each expression.
The first four examples of complex fractions might be called “simpler” complex fractions because the only
operation is division.
The last two are not as simple; they might be called “higher level complex fractions” because they contain
other operations, such as addition and subtraction. In any case, the large division bar is always the main
operation. (You may want to refer to Section 1.6 to remind yourself about the main operation.)
SIMPLIFYING SIMPLER COMPLEX FRACTIONS, METHOD 1
To evaluate—or simplify—a complex fraction, we need to recognize it for what it is, a division. Therefore,
we can rewrite the simpler complex fractions using the division symbol, ÷.
For example,
multiplying:
2
3
5
4
2
4
3 · 5
Complex Fractions
2
5
can be rewritten as 3 ÷ 4 . This can be evaluated by inverting the second fraction and
8
8
= 15 . Of course, 15 cannot be simplified any further.
page 6.6 - 1
Example 1:
Simplify each simpler complex fraction by first rewriting it using the division symbol.
5
a)
Procedure:
b)
7
8
2
3
c)
6
1
x
2y
3
y2
d)
5
2
First rewrite each using the division symbol and then “invert and multiply.”
5
a)
b)
7
8
7
= 5 ÷ 8
2
3
2
6
= 3 ÷ 1
 Rewrite as division 
5
6
6
( Think of 5 as 1) (Think of 6 as 1 )
5
8
= 1 · 7
 Invert and multiply 
2
1
= 3 · 6
40
7
 Simplify if possible 
2
= 18
=
1
c)
Exercise 1
a)
2
3
7
5
Complex Fractions
x
2y
3
y2
d)
5
2
1
= 9
5
= 1 ÷ 2
 Rewrite as division 
x
3
= 2y ÷ y2
1
2
= 1 · 5
 Invert and multiply 
x
y2
= 2y · 3
2
= 5
 Simplify if possible 
=
xy2
6y
xy
= 6
Simplify each simpler complex fraction by first rewriting it using the division symbol.
b)
5
8
3
2
page 6.6 - 2
Exercise 2
Simplify each simpler complex fraction by first rewriting it using the division symbol.
Simplify completely.
a)
4
9
8
3
9
4
b)
c)
6
d)
e)
x
y
3x
y2
f)
g)
4a2
b
2a3
Complex Fractions
h)
10
5
4
6
1
5
4c
8c2
3b
5xy2
4w
10x2y
3w2
page 6.6 - 3
SIMPLIFYING SIMPLER COMPLEX FRACTIONS, METHOD 2
The second method of simplifying simpler complex fractions requires that we remember that we can multiply
7
any fraction (any value) by 1, like 7 , and the value of the fraction will not change. The benefit of multiplying a
fraction by 1 is that we can change the way it looks without changing its value.
For example,
5
we can multiply 7
2
by 2
10
and get a new looking fraction, 14 , but the value of the
5
fraction hasn’t changed; it’s value is still 7 .
3x
2c
6cx
We can multiply 4y by 2c and get a new looking fraction, 8cy , but the value of the
6cx
3x
fraction hasn’t changed; after we reduce 8cy , it’s value is still 4y
We can also be very creative and multiply
written as
10
1
10
1
Multiplying by
10
10
by 10 . However, 10 might be better
so that we can more easily multiply:
4
5
3
2
10
1
10
1
4
5
3
2
·
10
1
10
1
=
4 10
5· 1
3 10
2· 1
doesn’t change the value of the complex fraction, but it does make it look different.
Let’s finish this multiplication:
4
5·
3
2·
10
1
10
1
=
40
5
30
2
which simplifies to
(
10
Notice that, in that last example, multiplying by 10 as 1
8
1
15
1
8
= 15 .
) allowed us to reduce the numerator and
4
3
denominator fractions completely. This is because 10 is the common denominator of 5 and 2 .
common denominator
The idea of multiplying by 1 as common denominator is the backbone of Method 2.
Complex Fractions
page 6.6 - 4
What makes multiplying
4
5
3
2
10
1
10
1
10
by 10 , as
, such a good idea is that 10 will cancel with each
denominator in the complex fraction; here it is again:
4
5·
3
2·
10
1
10
1
40
5
30
2
=
8
1
15
1
which simplifies to
8
= 15 , which is no longer a complex fraction.
15
Multiplying by a different value of 1, such as 15 , would be helpful in reducing only the numerator fraction:
4
5·
3
2·
15
1
15
1
=
60
5
45
2
which simplifies to
12
1
45
2
12
= 45 , but this is still a complex fraction.
2
Let’s put Method 2 into practice. (More steps are shown, below, than are actually necessary.)
Example 2:
Simplify each simpler complex fraction by using Method 2.
a)
Procedure:
a)
b)
c)
d)
5
3
7
8
b)
3
x
x
2
c)
d)
x
2y
3
y2
First recognize the common denominator, then use it to multiply by 1.
5
3
7
8
The common denominator is 24:
The common denominator is 8:
The common denominator is 2x:
The common denominator is 2y2:
Complex Fractions
7
8
3
4
7
8
3
4
24
1
24
1
·
8
1
8
1
·
3
x
x
2
·
x
2y
3
y2
=
7
8
3
4
=
2x
1
2x
1
·
5 24
3· 1
7 24
8· 1
=
2y2
1
2y2
1
8
·1
8
1
·
=
3 2x
x· 1
x 2x
2· 1
=
5
1
7
1
=
7
1
3
1
8
·1
·
1
·1
·
=
x 2y2
2y · 1
3 2y2
y2 · 1
40
= 21
3
1
7
= 6
2
1
3
1
x
1
2
·1
·
=
6
= x2
x
1
x
1
3
1
y
·1
·
2
1
xy
= 6
page 6.6 - 5
Exercise 3
Simplify each simpler complex fraction by first finding the common denominator and using it
to multiply the complex fraction by 1. Simplify completely.
a)
2
3
7
5
b)
c)
4
9
8
3
d)
e)
x
y
3x
y2
f)
g)
4a2
b
2a3
Complex Fractions
h)
5
8
3
2
10
5
4
4c
8c2
3b
5xy2
4w
10x2y
3w2
page 6.6 - 6
SIMPLIFYING HIGHER LEVEL COMPLEX FRACTIONS, METHOD 1
Complex fractions with more operations than just division are being called higher level complex fraction.
We saw a couple of examples earlier:
2
3
5
4
5
+ 6
4
x
and
1
– 2
3
– 2x
5
1 – x2
Clearly, these complex fractions have more operations than just division. We cannot directly simplify these
higher level complex fractions in the same manner as demonstrated in Example 1 (using Method 1).
In order to simplify these higher level complex fractions using Method 1 we need to first find a way to write
the numerator as one fraction and (separately) the denominator as one fraction. That usually means getting a least
common denominator (LCD); in this case, it means getting the LCD of the numerator and (separately) getting the
LCD of the denominator.
2
3
5
4
For example,
5
+ 6
has a two LCD’s, one for the numerator and one for the denominator. Let’s express
1
– 2
LCD = 6
the LCD’s of each this way: LCD = 4 :
2
3
5
4
2 2
5
3·2 + 6
5
1 2
4 – 2·2
5
+ 6
–
=
1
2
=
4
6
5
4
5
+ 6
–
2
4
=
9
6
3
4
9
3
= 6 ÷ 4 =
9
4
36
6 · 3 = 18
= 2
9
Notice that we were able to use Method 1 only after we first simplified the numerator to a single fraction, 6 ,
3
and (separately) simplified the denominator to a single fraction, 4 .
Example 3:
Simplify this higher level complex fraction by first simplify both the numerator and
denominator.
4
x
3
– 2x
1 –
Procedure:
4 2
x·2
1 x2
1 · x2
LCD = 2x
The LCD’s are: LCD = x2
5
x2
Simplify the numerator separately from the denominator; then use Method 1 to simplify the
complex fraction.
3
– 2x
5
– x2
Complex Fractions
=
8
2x
x2
x2
3
– 2x
5
– x2
=
5
2x
x2 – 5
x2
5
x2 – 5
= 2x ÷ x2
5
x2
5
x
5x
= 2x · x2 – 5 = 2 · (x2 – 5) = 2(x2 – 5)
page 6.6 - 7
Exercise 4
Simplify each higher level complex fraction by first simplifying both the numerator and
denominator. Simplify completely.
a)
b)
c)
2
3
3
8
1
+ 6
=
1
2 – 4
a
4
1
2
1
+ a
=
1
– a
3
+ x2
=
9
1 – x2
1
2
e)
=
1
1 – 6
1
x
d)
1
– 2
3
4
– x + x2
2
1 – x
Complex Fractions
=
page 6.6 - 8
SIMPLIFYING HIGHER LEVEL COMPLEX FRACTIONS, METHOD 2
Clearly, these complex fractions have more operations than just division. We cannot directly simplify these
higher level complex fractions in the same manner as demonstrated in Example 1 (using Method 1). However,
we can find the common denominator of all of the fractions within and use Method 2 to simplify.
For example, the higher level complex fraction
2
3
5
4
5
+ 6
1
– 2
has four fractions within it. The
denominators of those fractions are 3, 6, 4 and 2. The least common denominator is 12. We’ll want to multiply
12
1
12
the complex fraction by 12 , better written as 12 .
1
When multiplying this higher level complex fraction by
12
1
12
1
, however, we do need to be a little more
12
careful, as shown here. We’ll need to distribute 1 to each of the fractions and cross-cancel.
2
3+
5
4–
5
6
1
2
·
12
1
12
1
2
=
5
(3 + 6) · 121
5 1
(4 – 2) · 121
=
2 12 5 12
3· 1 +6· 1
5 12 1 12
4· 1 –2· 1
=
2 4
1·1+
5 3
1·1–
5 2
1·1
1 6
1·1
=
8 + 10
15 – 6
=
18
9
= 2
Your work might look a little more like this:
Either way you look at it, it’s a bit “busy” and you’ll need to be careful step by step. As you practice, you
might find that you can reduce the number of steps even further.
Complex Fractions
page 6.6 - 9
Example 4:
Simplify each higher level complex fraction by first finding the least common denominator
and using it to multiply by 1.
4
x
a)
Procedure:
1
3
– 2x
b)
5
1 + x2
1
3
1 – c2
1
c
c)
1
+ c2
1
4
– 3x – x2
1
3
3
– x2
Make every term a fraction, if it isn’t already. That will help in the multiplication process.
a)
First, the common denominator is 2x2. Let’s go right to the distribution step where every
2x2
small fraction is multiplied by the common denominator 1
4 2x2
x· 1
1 2x2
1· 1
3
2x2
– 2x · 1
+
5 2x2
x2 · 1
=
4 2x
1· 1
1 2x2
1· 1
3 x
– 1·1
+
5 2
1·1
8x – 3x
= 2x2 + 10
5x
= 2x2 + 10
This fraction cannot reduce any further.
b)
1 c2
1· 1
1 c2
c· 1
c2
The common denominator is c2 . Let’s distribute 1 to every small fraction.
1
c2
1
c2
– c2 · 1
+ c2 · 1
=
1 c2
1· 1
1 c
1·1
1 1
– 1·1
+
1 1
1·1
c2 – 1
(c + 1)(c – 1)
= c+1 =
(c + 1)
This fraction can reduce further.
c)
3x2
=
= c – 1
 Factor it and cancel the common factor.
The common denominator is 3x2. Let’s distribute 1
1 3x2
1 3x2
4 3x2
3 · 1 – 3x · 1 – x2 · 1
1 3x2
3 3x2
3 · 1 – x2 · 1
to every small fraction.
1 x2
1 x
4 3
1· 1 – 1·1 – 1·1
1 x2
3 3
1· 1 – 1·1
=
This fraction can reduce further.
Complex Fractions

x2 – x – 12
(x + 3)(x – 4)
= (x – 3)(x + 3)
x2 – 9
(x – 4)
= (x – 3)
 Factor it and divide out the common factor.
page 6.6 - 10
Exercise 5
Simplify each higher level complex fraction by first finding the least common denominator
and using it to multiply by 1. Simplify completely.
a)
b)
c)
2
3
3
8
1
+ 6
=
1
2 – 4
a
4
1
2
1
+ a
=
1
– a
3
+ x2
=
9
1 – x2
1
2
e)
3
4
– x + x2
2
1 – x
1
6
f)
=
1
1 – 6
1
x
d)
1
– 2
4
=
2
+ 3x + x2
2
1 + x
Complex Fractions
=
page 6.6 - 11
Answers to each Exercise
Section 6.6
Exercise 1:
a)
2
7
2
5
10
3 ÷ 5 = 3 · 7 = 21
b)
5
3
8 ÷ 2
Exercise 2:
a)
1
6
b)
8
c)
e)
y
3
f)
3b
2c
a)
2
3
7
5
c)
1
6
d)
8
g)
2
ab
h)
3wy
8x
Exercise 3:
Exercise 4:
Exercise 5:
Complex Fractions
a)
2
3
15
1
· 15
1
1
– 2
1 –
=
1
6
=
2 5
1·1
7 3
1·1
=
10
21
2 2
1 3
3 · 2 – 2· 3
1
6
1· 6 – 6
=
4
6
6
6
5 2
= 8 ·3
5
= 12
3
8
d)
30
g)
2
ab
h)
3wy
8x
b)
5
12
e)
y
3
f)
3b
2c
3
– 6
–
1
6
=
1
6
5
6
1
= .... = 5
b)
13
42
c)
a2 + 4
2a – 4
d)
1
x – 3
e)
x2 – 6x + 8
2x2 – 4x
a)
1
5
b)
13
42
c)
a2 + 4
2(a – 2)
d)
1
x–3
e)
x–4
2x
f)
x+6
6x
or
a2 + 4
2(a – 2)
=
x – 4
2x
page 6.6 - 12
Section 6.6
1.
Focus Exercises
Simplify each simpler complex fraction using any method.
a)
c)
e)
g)
Complex Fractions
5
3
7
4
4
11
b)
8
d)
c2
p
5c
p3
f)
3a3
b2
6a4
h)
10
9
25
12
12
6
5
12x3
16x2
5y4
21x2y
10w
14xy2
15w2
page 6.6 - 13
2.
Simplify each higher level complex fraction using any method.
5
a)
1 – 2
2
1 + 3
b)
5
12
2
3
d)
1
6
1
3x
2
c)
1 + x
x
3
2
+ 3
1
+ 3
1
– 4
1
– 2x
1
– x2
4
e)
1 – x
2
8
1 – x – x2
1
3
f)
Complex Fractions
5
2
+ 3x + x2
1
3
3
– x2
page 6.6 - 14