Complex fractions
Objective
• Simplify complex fractions
• Lets Review fraction rules first…………..
Multiplying Fractions
a c a c
 
, b  0 and d  0
b d bd
-5
Multiply
21
·
3
.
4
1
-5
21
·
3
4
=
-5
21
7
·
3
4
=
-5
7
·
1
4
=
-5
28
Multiplying Rational Expressions
1. Factor all numerators and denominators
completely.
2. Divide out common factors.
3. Multiply numerators together and multiply
denominators together.
2
3
18x
y
22z
Multiply
 2 5.
11z x y
2
36z
- 18x y 22z
- 18x y 22z
 2 5 
 2 5  4
y
11z x y
11z x y
2
3
2
3
Dividing Two Fractions
a c a d ad
   
, b  0 , d  0 and c  0
b d b c bc
-2
Divide
9
-2
9

5
9
=
-2
9

5
.
9
·
9
5
=
-2
9
1
1
·
9
5
=
-2
5
Dividing Rational Expressions
Invert the divisor (the second fraction)
and multiply
1
1
Divide x 2  7x  12  x 2 - 17  30 .
1
1
1
x 2 - 17  30
 2
 2


2
x  7x  18 x - 17  30
x  7x  18
1
1
(x  2)(x  15)
(x  15)


(x  9)(x  2)
1
(x  9)
Adding/Subtracting Fractions
a b a b
 
, c0
c c
c
Add 5 + 2 .
12 12
5
2
+
12 12
=
7
12
a b a b
 
, c0
c c
c
Common Denominators
1. Add or subtract the numerators.
2. Place the sum or difference of the
numerators found in step 1 over the
common denominator.
3. Simplify the fraction if possible.
Subtract 2x - 7  6 .
5
5
2x - 7 6
2x - 7 - 6 2x - 13
 

5
5
5
5
Common Denominators
Example:
a.)
3w  5
- 2w - 4
 2
.
Add 2
w  2w  1 w  2w  1
3w  5
- 2w - 4
3w  5 - 2w - 4
 2


2
2
w  2w  1 w  2w  1
w  2w  1
w 1
1
3w  5 - 2w - 4


2
2
(w  1)
(w  1)
w  2w  1
Common Denominators
Example:
b.)
4x 2  5 x 2 - x  29
Subtract 2

.
2
9x  64 9x  64
4x 2  5 x 2 - x  29 4x 2  5 - (x2 - x  29)


2
2
9x  64 9x  64
9x 2  64
4x 2  5 - x 2  x  29 3x 2  x  24


2
2
9x  64
9x  64
(x  3)(3x  8)
(x  3)

(3x  8)(3x  8) (3x  8)
Unlike Denominators
1. Determine the LCD.
2. Rewrite each fraction as an equivalent
fraction with the LCD.
3. Add or subtract the numerators while
maintaining the LCD.
4. When possible, factor the remaining
numerator and simplify the fraction.
Unlike Denominators
Example:
3
5

a.)
w 2 w
The LCD is w(w+2).
3
w 5 w 2
3w
5(w  2)
  



w 2 w w w 2
w (w 2) w (w 2)
3w
5w  10
8w  10


w (w 2) w (w 2) w (w 2)
2(4w  5)
8w  10
and 2
are also acceptable answers.
w (w 2)
w  2w
Unlike Denominators
Example:
x
1
x
1



The LCD is 12x(x – 1).
b.)
4x - 4 3x 4(x - 1) 3x
x
3x 1 4(x - 1)




4(x - 1) 3x 3x 4(x - 1)
3x 2
4(x - 1)
3x 2  4x  4



12x(x - 1) 12x(x - 1)
12x(x - 1)
This cannot be factored
any further.
Complex Fractions
Simplifying Complex Fractions
A complex fraction is one that has a fraction
in its numerator or its denominator or in
both the numerator and denominator.
Example:
4
5
4
x3
x
x3
a
b
a-9
a b
So how can we simplify them?
• Remember, fractions are just division problems.
• We can rewrite the complex fraction as a division
problem with two fractions.
• This division problem then changes to multiplication
by the reciprocal.
5
6
2
3
5 2
 
6 3
5 3
 
6 2
5

4
Simplifying Complex Fractions Rule
• Any complex fraction
a
b
c
d
Where b ≠ 0, c ≠ 0, and d ≠
0, may be expressed as:
ad
bc
What if we have mixed numbers in the
complex fraction?
• If we have mixed numbers, we treat it as an
addition problem with unlike denominators.
• We want to be working with two fractions, so
make sure the numerator is one fraction, and
the denominator is one fraction
• Now we can rewrite the complex fraction as a
division of two fractions
Example
1
2
2
5
Try on your own…
4
1
1
3
What about complex rational
expression?
• Treat the complex rational expression as a
division problem
• Add any rational expressions to form rational
expressions in the numerator and
denominator
• Factor
• Simplify
• “Bad” values
Ex. 2: Simplify
1 1
y
x


x y xy xy

1 1
y
x


x y xy xy
yx
xy

yx
xy
1

x
1

x
1
y
.
1
y
← The LCD is xy for both the numerator and the
denominator.
← Add to simplify the numerator and subtract to
simplify the denominator.
y  x xy


xy y  x
← Multiply the numerator by the reciprocal of the
denominator.
Ex. 2: Simplify
y  x xy


xy y  x
yx

yx
1

x
1

x
1
y
.
1
y
← Eliminate common factors.
Example
1
x
x
x 1


1
 (x  )  (x 1)
x
x 2 1
1
(
)
x
x 1
x 1
(x 1)(x 1)
1

, x  0,1


x
x
x 1
Example
x2
5 6
1  2
x x
Try on your own
2
3x
1
x
One more for you
16
x
x
2
x  8x  16
Ex. 3: Simplify
( x  4)(x  4)  1
x4
( x  11)(x  3)  48
x 3
1
x4
x4
48
x  11
x 3
← The LCD of the numerator is x +
4, and the LCD of the denominator
is x – 3.
Ex. 3: Simplify
x  8 x  16  1
x4
2
x  8 x  33  48
x 3
1
x4
x4
48
x  11
x 3
2
← FOIL the top and don’t forget to
subtract the 1 and add the 48 on
the bottom.
Ex. 3: Simplify
x  8 x  15
x4
2
x  8 x  15
x 3
1
x4
x4
48
x  11
x 3
2
← Simplify by subtracting the 1 in
the numerator and adding the 48 in
the denominator.
Ex. 3: Simplify
1
x4
x4
48
x  11
x 3
x  8 x  15
x 3
 2
x4
x  8 x  15
2
← Multiply by the reciprocal.
x2 + 8x +15 is a common factor that
can be eliminated.
1
x4
x4
48
x  11
x 3
Ex. 3: Simplify
x3
x4
← Simplify
Model Problems
1)
5
3x
1
2x
3)
24
x3
36
x3
5)
k
k

2 6
k
k

2
3
2)
y 1
1
y
y
4)
x y
x
1
1

x
y
6)
7
1
y2
3
1
y2
Homework
• Practice Sheet