16 the multiplier method for simplifying complex fractions

Complex Fractions
Frank Ma © 2011

Complex Fractions
A fraction with its numerator or the denominator containing
fraction(s) is called a complex fraction.

Complex Fractions
In other words, a complex fraction is a fractional expression
divided by another fractional expression.

Complex Fractions
We want to simplify complex fractions to regular fractions.

Complex Fractions
The “easy” complex fractions are just regular divisions of
of a fraction by another fraction.

Complex Fractions
of a fraction by another fraction. In this case just flip the
denominator then simplify the multiplication,

Complex Fractions
denominator then simplify the multiplication, i.e.
A
B
C
D

Complex Fractions
A
B
C
D
A
B C
D*
flip

Complex Fractions
A
B
C
D
A
B C
D*
flip
Example A. Simplify
4x2y
9
xy2
6

Complex Fractions
A
B
C
D
A
B C
D*
flip
Example A. Simplify
4x2y
9
xy2
6
=
flip
4x2y
9 xy2
6

Complex Fractions
A
B
C
D
A
B C
D*
flip
Example A. Simplify
4x2y
9
xy2
6
=
flip
4x2y
9 xy2
6
3
2

Complex Fractions
A
B
C
D
A
B C
D*
flip
Example A. Simplify
4x2y
9
xy2
6
=
flip
4x2y
9 xy2
6
3
2x

Complex Fractions
A
B
C
D
A
B C
D*
flip
Example A. Simplify
4x2y
9
xy2
6
=
flip
4x2y
9 xy2
6
3
2x
y

Complex Fractions
A
B
C
D
A
B C
D*
flip
Example A. Simplify
4x2y
9
xy2
6
=
flip
4x2y
9 xy2
6
=
3
2x
y
8x
3y

Complex Fractions
We give two methods for simplifying general complex
fractions.

Complex Fractions
fractions. The first method is to reduce the problem to an
“easy” problem.

Complex Fractions
“easy” problem. We do this by combining the numerator
into one fraction and the denominator into one fraction.

Complex Fractions
We can use cross multiplication for combining two fractions.

Example B. Simplify
Complex Fractions
x
x + 1
+ 1
1 – x
x – 1

Example B. Simplify
Complex Fractions
x
x + 1
+ 1
1 – x
x – 1x
x + 1
+ 1
1 –
x
x – 1

Example B. Simplify
Complex Fractions
x
x + 1
+ 1
1 – x
x – 1x
x + 1
+ 1
1 –
x
x – 1
=
x
x + 1 +
–
x
x – 1
1
1
1
1

Example B. Simplify
Complex Fractions
x
x + 1
+ 1
1 – x
x – 1x
x + 1
+ 1
1 –
x
x – 1
=
x
x + 1 +
–
x
x – 1
1
1
1
1
=
x + (x + 1)
x + 1

Example B. Simplify
Complex Fractions
x
x + 1
+ 1
1 – x
x – 1x
x + 1
+ 1
1 –
x
x – 1
=
x
x + 1 +
–
x
x – 1
1
1
1
1
=
x + (x + 1)
x + 1
(x – 1) – x
x – 1

Example B. Simplify
Complex Fractions
x
x + 1
+ 1
1 – x
x – 1x
x + 1
+ 1
1 –
x
x – 1
=
x
x + 1 +
–
x
x – 1
1
1
1
1
=
x + (x + 1)
x + 1
(x – 1) – x
x – 1
=
2x + 1
x + 1
–1
x – 1

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
2x + 1
x + 1
–1
x – 1
Therefore,

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
2x + 1
x + 1
–1
x – 1
Therefore,
=
(2x + 1)
(x + 1)
(x – 1)
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
2x + 1
x + 1
–1
x – 1
Therefore,
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
The second method is to multiply the LCD of all the terms to
the numerator and the denominator of the complex fraction.
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
The LCD of
2
3
3
2
5
6
3
4, , , is 12.
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
The LCD of
2
3
3
2
5
6
3
4, , , is 12.
Multiply 12 to the top and bottom of the complex fraction.
3
2
+ 1
1 –
–
5
6
2
3
3
4
+
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1) (–1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
The LCD of
2
3
3
2
5
6
3
4, , , is 12.
3
2
+ 1
1 –
–
5
6
2
3
3
4
+(
( )
)
12
12
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1) (–1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
The LCD of
2
3
3
2
5
6
3
4, , , is 12.
3
2
+ 1
1 –
–
5
6
2
3
3
4
+(
( )
)
12
12
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
This is OK since is 1.
12
121
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
The LCD of
2
3
3
2
5
6
3
4, , , is 12.
3
2
+ 1
1 –
–
5
6
2
3
3
4
+(
( )
)
12
12
6
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
Distribute the multiplication.
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
The LCD of
2
3
3
2
5
6
3
4, , , is 12.
3
2
+ 1
1 –
–
5
6
2
3
3
4
+(
( )
)
12
12
6 12
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
The LCD of
2
3
3
2
5
6
3
4, , , is 12.
3
2
+ 1
1 –
–
5
6
2
3
3
4
+(
( )
)
12
12
6 12 2
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
The LCD of
2
3
3
2
5
6
3
4, , , is 12.
3
2
+ 1
1 –
–
5
6
2
3
3
4
+(
( )
)
12
12
6 12 2
12 4 3
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
The LCD of
2
3
3
2
5
6
3
4, , , is 12.
3
2
+ 1
1 –
–
5
6
2
3
3
4
+(
( )
)
12
12
=
6 12 2
12 4 3
18 + 12 – 10
12 – 8 + 9
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
(–1)

Complex Fractions
x
x + 1
+ 1
1 – x
x – 1
=
Therefore,
Example C. Simplify
3
2
+ 1
1 –
– 5
6
2
3
3
4
+
The LCD of
2
3
3
2
5
6
3
4, , , is 12.
3
2
+ 1
1 –
–
5
6
2
3
3
4
+(
( )
)
12
12
=
6 12 2
12 4 3
18 + 12 – 10
12 – 8 + 9 =
20
13
2x + 1
x + 1
–1
x – 1
=
(2x + 1)
(x + 1)
(x – 1)
= –
(2x + 1)
(x + 1)
(x – 1)
(–1)

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
x
y
y2

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
The LCD of
x
y
is y2.
y2
x2 x
yy2 ,

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
The LCD of
x
y
is y2.
Multiply the LCD to top and bottom of the complex fraction.
y2
x2 x
yy2 ,
x2
+ 1
– 1
x
y
y2

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
The LCD of
x
y
is y2.
y2
x2 x
yy2 ,
x2
+ 1
– 1
x
y
y2
(
(
)
) y2
y2

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
The LCD of
x
y
is y2.
y2
x2 x
yy2 ,
x2
+ 1
– 1
x
y
y2
(
(
)
) y2
y2
1

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
The LCD of
x
y
is y2.
y2
x2 x
yy2 ,
x2
+ 1
– 1
x
y
y2
(
(
)
) y2
y2
y21

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
The LCD of
x
y
is y2.
y2
x2 x
yy2 ,
x2
+ 1
– 1
x
y
y2
(
(
)
) y2
y2
y2
y2y
1

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
The LCD of
x
y
is y2.
y2
x2 x
yy2 ,
x2
+ 1
– 1
x
y
y2
(
(
)
) y2
y2
=
y2
y2y
1
x2 – y2
xy + y2

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
The LCD of
x
y
is y2.
y2
x2 x
yy2 ,
x2
+ 1
– 1
x
y
y2
(
(
)
) y2
y2
=
y2
y2y
1
x2 – y2
xy + y2
= (x – y)(x + y)
y(x + y)
To simplify this, put it in the factored form .

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
The LCD of
x
y
is y2.
y2
x2 x
yy2 ,
x2
+ 1
– 1
x
y
y2
(
(
)
) y2
y2
=
y2
y2y
1
x2 – y2
xy + y2 To simplify this, put it in the factored form .
= (x – y)(x + y)
y(x + y)

Complex Fractions
x2
+ 1
– 1
Example D. Simplify
The LCD of
x
y
is y2.
y2
x2 x
yy2 ,
x2
+ 1
– 1
x
y
y2
(
(
)
) y2
y2
=
y2
y2y
1
x2 – y2
xy + y2 To simplify this, put it in the factored form .
= (x – y)(x + y)
y(x + y)
= x – y
y

Complex Fractions
x
x – 2
–
+
3
x + 2
Example E. Simplify
3
x + 2
x
x – 2

Complex Fractions
x
x – 2
–
+
3
x + 2
Example E. Simplify
The LCD of
3
x + 2
x
x – 2
x
x – 2
x
x – 2
3
x + 2
3
x + 2, , , is (x – 2)(x + 2).

Complex Fractions
x
x – 2
–
+
3
x + 2
Example E. Simplify
The LCD of
3
x + 2
x
x – 2
x
x – 2
x
x – 2
3
x + 2
3
x + 2, , , is (x – 2)(x + 2).
x
x – 2
–
+
3
x + 2
3
x + 2
x
x – 2

Complex Fractions
x
x – 2
–
+
3
x + 2
Example E. Simplify
The LCD of
3
x + 2
x
x – 2
x
x – 2
x
x – 2
3
x + 2
3
x + 2, , , is (x – 2)(x + 2).
x
x – 2
–
+
3
x + 2
3
x + 2
x
x – 2
(
( )
)
(x – 2)(x + 2)
(x – 2)(x + 2)

Complex Fractions
x
x – 2
–
+
3
x + 2
Example E. Simplify
The LCD of
3
x + 2
x
x – 2
x
x – 2
x
x – 2
3
x + 2
3
x + 2, , , is (x – 2)(x + 2).
x
x – 2
–
+
3
x + 2
3
x + 2
x
x – 2
(
( )
)
(x – 2)(x + 2)
(x – 2)(x + 2)
(x + 2)

Complex Fractions
x
x – 2
–
+
3
x + 2
Example E. Simplify
The LCD of
3
x + 2
x
x – 2
x
x – 2
x
x – 2
3
x + 2
3
x + 2, , , is (x – 2)(x + 2).
x
x – 2
–
+
3
x + 2
3
x + 2
x
x – 2
(
( )
)
(x – 2)(x + 2)
(x – 2)(x + 2)
(x + 2) (x – 2)

Complex Fractions
x
x – 2
–
+
3
x + 2
Example E. Simplify
The LCD of
3
x + 2
x
x – 2
x
x – 2
x
x – 2
3
x + 2
3
x + 2, , , is (x – 2)(x + 2).
x
x – 2
–
+
3
x + 2
3
x + 2
x
x – 2
(
( )
)
(x – 2)(x + 2)
(x – 2)(x + 2)
=
(x + 2)
(x + 2)
(x – 2)
(x – 2)
x(x + 2) + 3(x – 2)
x(x + 2) – 3(x – 2) Expand and put the fraction
in the factored form.

Complex Fractions
x
x – 2
–
+
3
x + 2
Example E. Simplify
The LCD of
3
x + 2
x
x – 2
x
x – 2
x
x – 2
3
x + 2
3
x + 2, , , is (x – 2)(x + 2).
x
x – 2
–
+
3
x + 2
3
x + 2
x
x – 2
(
( )
)
(x – 2)(x + 2)
(x – 2)(x + 2)
=
(x + 2)
(x + 2)
(x – 2)
(x – 2)
x(x + 2) + 3(x – 2)
=
x2 + 2x + 3x – 6
x2 + 2x – 3x + 6

Complex Fractions
x
x – 2
–
+
3
x + 2
Example E. Simplify
The LCD of
3
x + 2
x
x – 2
x
x – 2
x
x – 2
3
x + 2
3
x + 2, , , is (x – 2)(x + 2).
x
x – 2
–
+
3
x + 2
3
x + 2
x
x – 2
(
( )
)
(x – 2)(x + 2)
(x – 2)(x + 2)
=
(x + 2)
(x + 2)
(x – 2)
(x – 2)
x(x + 2) + 3(x – 2)
=
x2 + 2x + 3x – 6
x2 + 2x – 3x + 6
=
x2 + 5x – 6
x2 – x + 6

Complex Fractions
x
x – 2
–
+
3
x + 2
Example E. Simplify
The LCD of
3
x + 2
x
x – 2
x
x – 2
x
x – 2
3
x + 2
3
x + 2, , , is (x – 2)(x + 2).
x
x – 2
–
+
3
x + 2
3
x + 2
x
x – 2
(
( )
)
(x – 2)(x + 2)
(x – 2)(x + 2)
=
(x + 2)
(x + 2)
(x – 2)
(x – 2)
x(x + 2) + 3(x – 2)
=
x2 + 2x + 3x – 6
x2 + 2x – 3x + 6
=
x2 + 5x – 6
x2 – x + 6
This is the simplified.=
(x + 6)(x – 1)
x2 – x + 6

Complex Fractions
2
x – 1
–
+
3
x + 2
Example F. Simplify
3
x + 2
1
x – 2

Complex Fractions
2
x – 1
–
+
3
x + 2
Example F. Simplify
3
x + 2
1
x – 2
Use the crossing method.

Complex Fractions
2
x – 1
–
+
3
x + 2
Example F. Simplify
3
x + 2
1
x – 2
2
x – 1
–
+
3
x + 2
3
x + 2
1
x – 2

Complex Fractions
2
x – 1
–
+
3
x + 2
Example F. Simplify
3
x + 2
1
x – 2
2
x – 1
–
+
3
x + 2
3
x + 2
1
x – 2
=
2(x + 2) + 3(x – 1)
(x – 1)(x + 2)
2(x + 3) – 3(x – 2)
(x – 2)(x + 2)

Complex Fractions
2
x – 1
–
+
3
x + 2
Example F. Simplify
3
x + 2
1
x – 2
2
x – 1
–
+
3
x + 2
3
x + 2
1
x – 2
=
2(x + 2) + 3(x – 1)
(x – 1)(x + 2)
2(x + 3) – 3(x – 2)
(x – 2)(x + 2)
=
5x +1
(x – 1)(x + 2)
– x + 12
(x – 2)(x + 2)

Complex Fractions
2
x – 1
–
+
3
x + 2
Example F. Simplify
3
x + 2
1
x – 2
2
x – 1
–
+
3
x + 2
3
x + 2
1
x – 2
=
2(x + 2) + 3(x – 1)
(x – 1)(x + 2)
2(x + 3) – 3(x – 2)
(x – 2)(x + 2)
flip the bottom
fraction
=
5x +1
(x – 1)(x + 2)
– x + 12
(x – 2)(x + 2)

Complex Fractions
2
x – 1
–
+
3
x + 2
Example F. Simplify
3
x + 2
1
x – 2
2
x – 1
–
+
3
x + 2
3
x + 2
1
x – 2
=
2(x + 2) + 3(x – 1)
(x – 1)(x + 2)
2(x + 3) – 3(x – 2)
(x – 2)(x + 2)
flip the bottom
fraction
=
5x +1
(x – 1)(x + 2)
– x + 12
(x – 2)(x + 2)
=
(5x +1)
(x – 1)(x + 2) (– x + 12)
(x – 2)(x + 2)

Complex Fractions
2
x – 1
–
+
3
x + 2
Example F. Simplify
3
x + 2
1
x – 2
2
x – 1
–
+
3
x + 2
3
x + 2
1
x – 2
=
2(x + 2) + 3(x – 1)
(x – 1)(x + 2)
2(x + 3) – 3(x – 2)
(x – 2)(x + 2)
flip the bottom
fraction
=
5x +1
(x – 1)(x + 2)
– x + 12
(x – 2)(x + 2)
=
(5x +1)
(x – 1)(x + 2) (– x + 12)
(x – 2)(x + 2)
=
(5x + 1)(x – 2)
(x – 1)(–x + 12)

Complex Fractions
Ex. A. Simplify.
1.
2x2y
5
4y2 2.
3
4x2
3x
2 3.
4x2
2x
3 4.
12x2
5y
4xy
15
3
+ 1
–
1
3
4
5.
2
3
5
–
–
2
3
6
6.
4
3
1
4
4
+
–
1
6
5
7.
11
15
7
12
1
+ 1
– 2
2
y
x8.
1
+ 2
1 –
1
x2
xy9.
1
+
–
1
x
y10.
1
x
1
y
2
–
–
4
x2
y11.
2
x
4
y2
2
–
–
4
x2
y12.
2
x
4
y2
2
x + 1 + 1
1 –
1
x – 1
13.

Complex Fractions
1
2x + 1
– 2
3 –
1
x + 2
14.
2
x – 3
– 1
2 –
1
2x + 1
15.
2
x + 3
–
+
1
x + 3
16.
3
x + 2
3
x + 2
–2
2x + 1
–
+
3
x + 4
17.
1
x + 4
2
2x + 1
2
x + 2
–
+ 2
x + 5
18.
1
3x – 1
3
x + 2
4
2x + 3
–
+
3
x + 4
19.
3
3x – 2
5
3x – 2
–5
2x + 5
–
+ 3
–x + 4
20.
2
2x – 3
6
2x – 3
2
3
+ 2
2 –
–
1
6
2
3
1
2
+
21.
1
2
– +
5
6
2
3
1
4
–
22.
3
4
3
2
+

Complex Fractions
23.
2
x – 1
–
+
3
x + 3
x
x + 3
x
x – 1
24.
3
x + 2
–
+
3
x + 2
x
x – 2
x
x – 2
25.
2
x + h
–
2
x
h
26.
3
x – h
–
3
x
h
27.
2
x + h
– 2
x – h
h
28.
3
x + h
–
h
3
x – h

16 the multiplier method for simplifying complex fractions

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16 the multiplier method for simplifying complex fractions