Mixed Expressions and
Complex Fractions
Then
Now
Why?
You simplified
rational expressions.
1
2
1
A Top Fuel dragster can cover _
mile
(Lesson 11-3)
Simplify mixed
expressions.
4
2
seconds. The average speed in
in 4_
5
Simplify complex
fractions.
miles per second can be described
by the expression below. It is called
a complex fraction.
_1 mile
4
_
2
4_
seconds
5
NewVocabulary
mixed expression
complex fraction
1
4
Simplify Mixed Expressions An expression like 2 + _
is called a mixed
x+1
expression because it contains the sum of a monomial, 2, and a rational
4
expression, _
. You can use the LCD to change a mixed expression to a rational
expression.
x+1
Example 1 Change Mixed Expression to Rational Expressions
4
_
as a rational expression.
Virginia
i S
SO
SOL
OL
Write 2 +
Preparation for AII/T.1.b
The student will, given
rational, radical, or polynomial
expressions, add, subtract,
multiply, divide, and simplify
radical expressions containing
rational numbers and
variables, and expressions
containing rational exponents.
2(x - 1)
4
4
2+_
=_+_
x-1
x-1
x-1
x-1
2(x - 1) + 4
_
=
x-1
2x - 2 + 4
_
=
x-1
2x
+2
=_
x-1
The LCD is x - 1.
Add the numerators.
Distributive Property
Simplify.
GuidedPractice
Write each mixed expression as a rational expression.
6y
1B. _ + 5y
5
1A. 2 + _
x
4y + 8
2 Simplify Complex Fractions
A complex fraction has one or more fractions in
the numerator or denominator. You can simplify by using division.
numerical complex fraction
algebraic complex fraction
_2
_a
_3 = _2 ÷ _5
_b = _a ÷ _c
8
d
_5
3
8
8
2
×_
=_
3
16
=_
15
5
_c
b
d
d
= _a × _
b
c
ad
=_
bc
To simplify a complex fraction, write it as a division expression. Then find the
reciprocal of the second expression and multiply.
714 | Lesson 11-7
Real-World Example 2 Use Complex Fractions to Solve Problems
RACING Refer to the application at the beginning of the lesson. Find the average
speed of the Top Fuel dragster in miles per minute.
_1 mile
_1 mile
2
4_
seconds
2
4_
seconds
60 seconds
4
4
_
=_
×_
5
Real-WorldLink
5
=
A Jr. Dragster is a half-scale
verson of a Top Fuel dragster.
1
mile
This car, which can go _
=
8
9
seconds, is designed
in 7_
10
to be driven by kids ages
8–17 in the NHRA Jr. Drag
Racing League.
_1 × 60
4
_
2
4_
5
60
_
4
_
22
_
5
1 minute
Convert seconds to minutes. Divide by
common units.
Simplify.
Express each term as an improper fraction.
_a
_
15
_ _
60 × 5
=_
Use the rule bc = ad .
4 × 22
75
9
=_
or 3_
22
bc
d
1
Source: NHRA
22
Simplify.
9
So, the average speed of the Top Fuel dragster is 3_
miles per minute.
22
GuidedPractice
2. RACING Refer to the information about the Jr. Dragster at the left. What is the
average speed of the car in feet per second?
To simplify complex fractions, you can either use the rule as in Example 2, or you
can rewrite the fraction as a division expression, as shown below.
Example 3 Complex Fractions Involving Monomials
8t
_
v
Simplify _
.
2
4t
_
v3
8t 2
_
8t 2
v
4t
_
=_
÷_
4t
_
v
v3
v3
8t
v
=_
×_
2
v
3
4t
2t
2
2
3
v
4t
8t
v
=_
×_
or 2tv 2
1
Write as a division expression.
To divide, multiply by the reciprocal.
Divide by the common factors 4t and v and simplify.
1
GuidedPractice
Simplify each expression.
3
g h
_
b
3A. _
3
gh
_
b2
-24m 3t 5
_
2
p h
3B. _
2
16pm
_
t 4h
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715
Complex fractions may also involve polynomials.
Example 4 Complex Fractions Involving Polynomials
Simplify each expression.
2
_
5
_
y+3
_
a.
y2 - 9
2
_
y+3
5
2
_
=_
÷_
2
5
_
y+3
2
y -9
Write as a division expression.
y -9
y2 - 9
5
To divide, multiply by the reciprocal.
(y - 3)(y + 3)
5
Factor y 2 - 9.
2
=_
×_
Real-WorldCareer
y+3
Lab Technician
Lab technicians work
with scientists, running
experiments, conducting
research projects, and
running routine diagnostic
samples. Lab technicians in
any field need at least a twoyear associate degree.
2
=_
× __
y+3
1
(y - 3)(y + 3)
2
=_
× __
5
y+3
Divide by the GCF, y + 3.
1
2(y - 3)
=_
5
Simplify.
n + 7n - 18
__
n - 2n + 1
_
b.
n - 81
_
2
2
2
n-1
2
StudyTip
Factoring When simplifying
fractions involving
polynomials, factor the
numerator and the
denominator of each
expression if possible.
n__
+ 7n - 18
2
n
- 2n + 1
_
2
n
- 81
_
n-1
2
n 2 + 7n - 18
n
- 81
_
= __
÷
2
n - 2n + 1
n-1
n 2 + 7n - 18
n - 2n + 1
n - 81
Write as a division expression.
n-1
= __
×_
2
2
(n - 2)(n + 9)
(n - 1)(n - 1)
Multiply by the reciprocal.
n-1
= __ × __
1
Factor the polynomials.
(n - 9)(n + 9)
1
(n - 2)(n + 9)
n-1
= __× __
(n - 1)(n - 1)
1
n-2
= __
(n - 1)(n - 9)
GuidedPractice
a+7
_
4
4A. _
2
a - 49
_
10
c-d
_
4C.
(n - 9)(n + 9)
Divide out the common factors.
1
Simplify.
x+4
_
x-1
4B. _
2
x + 6x + 8
_
2x - 2
n 2 + 4n - 21
__
j+p
_
d2
n 2 - 9n + 18
4D. __
2
j2 - p2
n 2 - 10n + 24
c _
2
716 | Lesson 11-7 | Mixed Expressions and Complex Fractions
n + 3n - 28
__
Check Your Understanding
Example 1
= Step-by-Step Solutions begin on page R12.
Write each mixed expression as a rational expression.
2
1. _
n +4
1
2. r + _
5
3. 6 + _
x+7
4. _ - 5x
3r
t+1
Example 2
2x
1
1
5. ROWING Rico rowed a canoe 2_
miles in _
hour.
2
3
a. Write an expression to represent his speed in miles per hour.
b. Simplify the expression to find his average speed.
Examples 3–4 Simplify each expression.
1
2_
3
6. _
5
7. _
2
1_
5
8. _
5
b
_
2
2
2
2
q -4
12. _
p +p-6
13. _
2
q 2 - 6q + 8
p 2 + 6p + 9
q+4
_
(r + s)
_
x2 - x - 2
2x 2
p+3
_
2+q
_
x -y
11. _2
3
_
xy
_
a
r+s
_
x-2
10. _
x2
9. _
2
b3
2
6_
3
6
_
4
y
_
a2
_
_4
x-y
p + 4p + 3
_
Practice and Problem Solving
Example 1
Extra Practice begins on page 815.
Write each mixed expression as a rational expression.
6
14. 10 + _
7
15 p - _
2a
16. 5a - _
17. 3h + _
18. t + _
v-w
n-1
19. n 2 + _
k-1
20. (k + 2) + _
21. (d - 6) + _
2p
f
1+h
h
v+w
n+4
h-3
22. _
- (h + 2)
d+1
d-7
k–2
Example 2
b
h+5
3
23. READING Ebony reads 6_
pages of a book in 9 minutes. What is her average
4
reading rate in pages per minute?
3
1
mile in about _
minute. What is the horse’s
24. HORSES A thoroughbred can run _
2
4
speed in miles per hour?
Examples 3–4 Simplify each expression.
2
2_
9
25. _
5
26. _
3
7
1
3_
_2
a
29. _
1
_
a+6
B
5n 4
_
2
g
_
3
5_
3
p
28. _
h
27. _
5
1
2_
6n
_
g
_
5p
h2
2
j - 16
__
t+5
_
x-3
_
2
9
30. _
2
j + 10j + 16
31. _
15
x 2 + 3x + 2
32. _
2
_
t - t - 30
_
x -9
_
j+8
12
x+1
1
33. COOKING The Centralville High School Cooking Club has 12_
pounds of flour with
2
3
which to make tortillas. There are 3_
cups of flour in a pound, and it takes about
4
_1 cup of flour per tortilla. How many tortillas can they make?
3
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717
34. SCOOTER The speed v of an object spinning in a circle equals the circumference of
the circle divided by the time T it takes the object to complete one revolution.
a. Use the variables v, r (the radius of the circle), and T to write a formula
describing the speed of a spinning object.
1
b. A scooter has tires with a radius of 3_
inches. The tires make one revolution
2
1
every _
second. Find the speed in miles per hour. Round to the nearest tenth.
10
m
35 SCIENCE The density of an object equals _
Metal
Density (kg/m 3)
v , where
m is the mass of the object and V is the volume. The
copper
8900
densities of four metals are shown in the table.
gold
19,300
Identify the metal of each ball described below.
4 3
_
iron
7800
(Hint: The volume of a sphere is V = πr .)
3
lead
11,300
a. A metal ball has a mass of 15.6 kilograms and
a radius of 0.0748 meter.
b. A metal ball has a mass of 285.3 kilograms and a radius of 0.1819 meter.
36. SIRENS As an ambulance approaches, the siren sounds different than if it were
sitting still. If the ambulance is moving toward you at v miles per hour and
blowing the siren at a frequency of f, then you hear the siren as if it were blowing
f
at a frequency h. This can be described by the equation h = _
v , where s is the
1-_
s
speed of sound, approximately 760 miles per hour.
a. Simplify the complex fraction in the formula.
b. Suppose a siren blows at 45 cycles per minute and is moving toward you at
65 miles per hour. Find the frequency of the siren as you hear it.
C
Simplify each expression.
17x + 5
37. 15 - _
5x + 10
40.
38.
12
y-_
y-4
_
_
y - 18
y-3
H.O.T. Problems
41.
b
_
+2
2c - 6c - 10
1 + __
2
b+3
_
c+7
39. __
x 2 - 4x - 32
__
r 2 - 9r
_
2c + 1
b 2 - 2b - 8
x+1
_
2
x + 6x + 8
_
x2 - 1
r 2 + 7r + 10
42. _
2
r + 5r
_
r2 + r - 2
Use Higher-Order Thinking Skills
43. REASONING Describe the first step to simplify the expression below.
(x y)
_
_y - _x
x+y
_
xy
44. REASONING Is
n
_
5
1-_
p
+
n
_
_5 - 1
sometimes, always, or never equal to 0? Explain.
p
45. CHALLENGE Simplify the rational expression below.
1
1
_
+_
t-1
t+1
__
1
_1 - _
t
t2
1
46. OPEN ENDED Write a complex fraction that, when simplified, results in _
x.
47. WRITING IN MATH Explain how complex fractions can be used to solve a problem
involving distance, rate, and time. Give an example.
718 | Lesson 11-7 | Mixed Expressions and Complex Fractions
Virginia SOL Practice
A.4.f, AII/T.1.b
48. A number is between 44 squared and
45 squared. 5 squared is one of its factors, and
it is a multiple of 13. Find the number.
A
B
C
D
1950
2000
2025
2050
50. GEOMETRY Angela wanted a round rug to fit
her room that is 16 feet wide. The rug should
just meet the edges. What is the area of the
rug rounded to the nearest tenth?
F 50.3 ft 2
G 100.5 ft 2
H 152.2 ft 2
J 201.1 ft 2
10
51. Simplify 7x + _
.
49. SHORT RESPONSE Bernard is reading a 445page book. He has already read 157 pages. If
he reads 24 pages a day, how long will it take
him to finish the book?
7x + 10
A _
2xy
17x
C _
2xy
2xy
2
7x y + 5
B _
xy
7xy + 5
D _
2
x y
Spiral Review
Find each sum or difference. (Lesson 11-6)
5+x
6
52. _
-_
d
4
53. _
+_
q-5
3q + 2
54. _ + _
1
2
-_
55. _
3
5m
10
-3
56. _
-_
3g
6
b
57. _
+_
7x
7x
15m
2q + 1
2q + 1
1-d
d-1
b-2
b+3
4h
Find each quotient. Use long division. (Lesson 11-5)
58. (x 2 - 2x - 30) ÷ (x + 7)
59. (a 2 + 4a - 22) ÷ (a - 3)
60. (3q 2 + 20q + 11) ÷ (q + 6)
61. (3y 3 + 8y 2 + y - 7) ÷ (y + 2)
62. (6t 3 - 9t 2 + 6) ÷ (2t - 3)
63. (9h 3 + 5h - 8) ÷ (3h - 2)
64. GEOMETRY Triangle ABC has vertices A(7, -4), B(-1, 2), and C(5, -6). Determine
whether the triangle has three, two, or no sides that are equal in length. (Lesson 10-6)
Graph each function. Determine the domain and range. (Lesson 10-1)
65. y = 2 √"
x
1√
67. y = _
x
"
66. y = -3 √"
x
4
Factor each polynomial. If the polynomial cannot be factored, write prime. (Lesson 8-5)
68. x 2 - 81
69. a 2 - 121
70. n 2 + 100
71. -25 + 4y 2
72. p 4 - 16
73. 4t 4 - 4
74. PARKS A youth group traveling in two vans visited
Mammoth Cave in Kentucky. The number of people
in each van and the total cost of the cave are shown.
Find the adult price and the student price of the tour.
(Lesson 6-3)
Van
Number
of Adults
Number of
Students
Total Cost
A
2
5
$77
B
2
7
$95
Skills Review
Solve each equation. (Lessons 2-2 and 2-3)
75. 6x = 24
76. 5y - 1 = 19
77. 2t + 7 = 21
p
78. _ = -4.2
2m + 1
79. _ = -5.5
3
1
80. _
g=_
3
4
4
2
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