Complex Fractions
I.
Fractions
a. Definition: A complex fraction is a fraction where either the
numerator, denominator, or both contain fractions.
b. Examples:
1.
a
b
c
2.
1
4
x
3
3.
a c

b d
x
4
y
II.
To Simplify A Complex Fraction
a. Find the least common denominator of all fractions within the
fraction.
b. Multiply the numerator and denominator of the complex fraction
by the LCD found in part a.
c. Simplify the resulting fraction.
III.
Mixed Numbers & Improper Fractions
a.
3
3
is the only fraction within the complex
x
fraction. The LCD is x .
x
2
3 x

x 1
x
2
1
Answer:
Multiply the top and bottom by x .
Simplify as the last step.
3
2x
www.rit.edu/asc
Page 1 of 5
b.
3
2
2
3
and
are the only fractions within the
x
y
complex fraction. The LCD is xy .
x
y
3 xy

x 1
2 xy

y 1
Answer:
c.
Multiply the top and bottom by xy .
Simplify as the last step.
3y
2x
3
1
x
2 3

y 4
3 
  14 xy
x 
 2 3
  4 xy
 y 4
3 2
3
,
and
are the only fractions within
x y
4
the complex fraction. The LCD is 4 xy .
Multiply every term in the top and bottom
by 4 xy . Simplify as the last step.
Answer: 12 y  4 xy
8 x  3xy
www.rit.edu/asc
Page 2 of 5
Practice Problems:
1
1.
1
1
2.
1
a
5.
1 1

x y
1 1

y z
6.
1
1

x2 x
2
7.
1
1
 2
2
x
y
1
2 2
x y
8.
1
1
 2
x  3 x  2x  3
1
x 1
a2
a
b
1
3.
4.
x
y
1 1

x y
xy
www.rit.edu/asc
Page 3 of 5
9.
1 1

a b
a2  b2
10.
1
x
2x 1
11.
1 1

3 a
1
b
12.
1
1

a b a b
2b
2
a  b2
13.
1
1

x y y
1
2
x  y2
14.
1
1

2x 6 y
1
1

3y 4x
2
www.rit.edu/asc
Page 4 of 5
Answers to Complex Fractions:
1.
a
8.
x2
x 3
2.
b
a
9.
1
ab(a  b)
3.
1
xy
10.
1
x
4.
x y
x2 y2
11.
ab  3b
3a
5.
yz  xz
xz  xy
12.
1
13.
xy  x 2
y
14.
6 y  2x
4x  3 y
6.
7.
1
x( x  2)
y2  x2
www.rit.edu/asc
Page 5 of 5