11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational
expression.
4. 1. SOLUTION: SOLUTION: 2. 5. ROWING Rico rowed a canoe
miles in hour.
a. Write an expression to represent his speed in
miles per hour.
b. Simplify the expression to find his average speed.
SOLUTION: SOLUTION: a. Use the formula d = r × t to write an expression
for Rico's speed, r.
3. SOLUTION: b. Simplify the complex fraction to find his average
speed.
4. SOLUTION: So, Rico's average speed is
.
Simplify each expression.
eSolutions Manual - Powered by Cognero
6. Page 1
11-7So,
Mixed
Expressions
and
Rico's
average speed
is Complex Fractions
.
Simplify each expression.
8. 6. SOLUTION: SOLUTION: To simplify complex fractions, rewrite as a division
expression, then rewrite as a multiplication
expression.
9. SOLUTION: 7. SOLUTION: To simplify complex fractions, rewrite as a division
expression, then rewrite as a multiplication
expression.
10. SOLUTION: 8. SOLUTION: eSolutions Manual - Powered by Cognero
Page 2
11-7 Mixed Expressions and Complex Fractions
10. 12. SOLUTION: 11. SOLUTION: 13. SOLUTION: SOLUTION: 12. Write each mixed expression as a rational
expression.
SOLUTION: 14. SOLUTION: eSolutions Manual - Powered by Cognero
13. Page 3
15. 11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational
expression.
17. 14. SOLUTION: SOLUTION: 18. 15. SOLUTION: SOLUTION: 19. SOLUTION: 16. SOLUTION: 20. 17. SOLUTION: SOLUTION: eSolutions Manual - Powered by Cognero
Page 4
11-7 Mixed Expressions and Complex Fractions
20. SOLUTION: 23. READING Ebony reads
pages of a book in 9
minutes. What is her average reading rate in pages
per minute?
SOLUTION: The number of pages Ebony reads equals the product
of the rate she reads per minute and the time she
reads in minutes, or N = r × t.
21. SOLUTION: Her average reading rate is page/minute.
24. HORSES A thoroughbred can run
22. mile in about minute. What is the horse’s speed in miles per
SOLUTION: hour?
SOLUTION: Use the formula d = rt or to represent the
horse's speed in miles per minute and convert
minutes to hours.
23. READING Ebony reads
pages of a book in 9
minutes. What is her average reading rate in pages
per minute?
SOLUTION: The number of pages Ebony reads equals the product
eSolutions
Manual
Powered
Cognero
of the
rate -she
readsbyper
minute and the time she
reads in minutes, or N = r × t.
Page 5
11-7Her average reading rate is Mixed Expressions and Complex
Fractions
page/minute.
24. HORSES A thoroughbred can run
The horse’s speed is 40 miles/hour.
Simplify each expression.
mile in about minute. What is the horse’s speed in miles per
25. hour?
SOLUTION: SOLUTION: Use the formula d = rt or to represent the
horse's speed in miles per minute and convert
minutes to hours.
26. SOLUTION: The horse’s speed is 40 miles/hour.
Simplify each expression.
27. 25. SOLUTION: eSolutions Manual - Powered by Cognero
SOLUTION: Page 6
11-7 Mixed Expressions and Complex Fractions
29. 27. SOLUTION: SOLUTION: 30. SOLUTION: 28. SOLUTION: 31. SOLUTION: 29. SOLUTION: eSolutions Manual - Powered by Cognero
Page 7
11-7 Mixed Expressions and Complex Fractions
33. CCSS MODELING The Centralville High School
Cooking Club has
pounds of flour with which to
31. make tortillas. There are
and it takes about
SOLUTION: cups of flour in a pound,
cup of flour per tortilla. How many tortillas can they make?
SOLUTION: First find out how many total cups of flour the Club
has by multiplying the number of pounds of flour
times the amount of cups in a pound.
They have
cups of flour. Each tortilla needs
32. about
cup of flour. To find out how many tortilla
they can make, divide the total cups of flour by
SOLUTION: .
So, they can make about 140 tortillas.
33. CCSS MODELING The Centralville High School
Cooking Club has
pounds of flour with which to
make tortillas. There are
and it takes about
cups of flour in a pound,
cup of flour per tortilla. How many tortillas can they make?
SOLUTION: First find out how many total cups of flour the Club
has by multiplying the number of pounds of flour
times the amount of cups in a pound.
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.
b. A scooter has tires with a radius of
inches.
The tires make one revolution every
second. Find
the speed in miles per hour. Round to the nearest
tenth.
SOLUTION: a. The circumference of a circle is 2πr. So, the
eSolutions Manual - Powered by Cognero
velocity of an object is
.
Page 8
Metal
copper
gold
iron
lead
11-7 Mixed Expressions and Complex Fractions
So, they can make about 140 tortillas.
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.
b. A scooter has tires with a radius of
inches.
The tires make one revolution every
second. Find
3
Density (kg/m )
8900
19,300
7800
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.
SOLUTION: a.
the speed in miles per hour. Round to the nearest
tenth.
SOLUTION: a. The circumference of a circle is 2πr. So, the
velocity of an object is
.
b.
3
The density is about 8900 kg/m . Therefore, the
metal ball is made of copper.
b.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals
,
3
where m is the mass of the object and V is the
volume. The densities of four metals are shown in
the table. Identify the metal of each ball described
below. (Hint: The volume of a sphere is .)
Metal
copper
gold
iron
lead
3
Density (kg/m )
8900
19,300
7800
11,300
eSolutions Manual - Powered by Cognero
a. A metal ball has a mass of 15.6 kilograms and a
radius of 0.0748 meter.
The density is about 11,317 kg/m . Therefore, the
metal ball is made of lead.
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 at a frequency h.
This can be described by the equation
,
where s is the speed of sound, approximately 760
Page 9
miles per hour.
a. Simplify the complex fraction in the formula.
b. Suppose a siren blows at 45 cycles per minute
3
11-7The
Mixed
Expressions
and kg/m
Complex
Fractions
density
is about 11,317
. Therefore,
the
metal ball is made of lead.
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 at a frequency h.
This can be described by the equation
The frequency of the siren as you hear it is 49.21
cycles/min.
Simplify each expression.
37. SOLUTION: ,
where s is the 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.
SOLUTION: a.
38. SOLUTION: b. Substitute 65 for v, and 45 for f in the equation
.
39. The frequency of the siren as you hear it is 49.21
cycles/min.
SOLUTION: Simplify each expression.
37. SOLUTION: eSolutions Manual - Powered by Cognero
Page 10
11-7 Mixed Expressions and Complex Fractions
39. 41. SOLUTION: SOLUTION: 42. 40. SOLUTION: SOLUTION: 43. REASONING Describe the first step to simplify
the expression shown.
41. SOLUTION: SOLUTION: Find the lowest common denominator for the
fractions in the numerator. Then subtract and
simplify.
eSolutions Manual - Powered by Cognero
Page 11
11-7 Mixed Expressions and Complex Fractions
This expression is always equal to 0.
43. REASONING Describe the first step to simplify
the expression shown.
SOLUTION: Find the lowest common denominator for the
fractions in the numerator. Then subtract and
simplify.
45. CCSS PERSEVERANCE Simplify the rational
expression shown.
SOLUTION: 44. REASONING Is
sometimes,
46. OPEN ENDED Write a complex fraction that,
when simplified, results in
always, or never equal to 0? Explain.
SOLUTION: .
SOLUTION: Find two fractions that have the same denominators
and when you divide the numerators, you get
Consider the fraction
. It simplifies to
.
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational
expression shown.
SOLUTION: eSolutions Manual - Powered by Cognero
47. WRITING IN MATH Explain how complex
fractions can be used to solve a problem involving
distance, rate, and time. Give an example.
SOLUTION: Sample answer: Time equals distance divided by
rate
Page 12
or
. When the distance or the rate is given as a
11-7 Mixed Expressions and Complex Fractions
47. WRITING IN MATH Explain how complex
fractions can be used to solve a problem involving
distance, rate, and time. Give an example.
SOLUTION: Sample answer: Time equals distance divided by rate
or
. When the distance or the rate is given as a
fraction or mixed number, the expression
becomes a complex fraction. Example: Someone 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 1950
B 2000
C 2025
D 1975
SOLUTION: The value of 44 squared is 1936. The value of 45
squared is 2025. Therefore, the answer cannot be C.
walks
mile in 10
minutes; the time in miles per
Number
1950
minute is
, which simplifies to
mi/min.
2000
1975
Prime
Factorization
2 • 3 • 5 • 5 • 13
2 • 2 • 2 • 2 • 5 • 5 •5
5 •5•79
Exponential
Form
2
2 • 3 • 5 • 13
4
3
2 • 5
2
5 • 79
2
The only number that has 5 as one of its factors and
is a multiple of 13 is 1950. So, the correct choice is
A.
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?
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 1950
B 2000
C 2025
D 1975
SOLUTION: The value of 44 squared is 1936. The value of 45
squared is 2025. Therefore, the answer cannot be C.
SOLUTION: Bernard still needs to read 445 – 157 or 288 pages.
He will read 24 pages a day. It will take him 288 ÷ 24 or 12 days to finish the book.
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 100.5 ft
2
G 804.2 ft
H 50.3 ft
J 201.1 ft2
SOLUTION: Number
1950
2000
1975
Prime
Factorization
2 • 3 • 5 • 5 • 13
2 • 2 • 2 • 2 • 5 • 5 •5
5 •5•79
Exponential
Form
2
2 • 3 • 5 • 13
4
3
2 • 5
2
The area of a circle is A = πr . If the rug just meets
the edge of the room, then the diameter is 16 feet, so
the radius is 16 ÷ 2 or 8 feet.
2
5 • 79
2
eSolutions
- Powered
by Cognero
The Manual
only number
that
has 5 as one of its factors and
is a multiple of 13 is 1950. So, the correct choice is
A.
Page 13
SOLUTION: Bernard still needs to read 445 – 157 or 288 pages.
11-7He will read 24 pages a day. It will take him 288 ÷ Mixed Expressions and Complex Fractions
24 or 12 days to finish the book.
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 100.5 ft
2
G 804.2 ft
H 50.3 ft
J 201.1 ft2
The area of the rug is about 201.1 square feet. So
the correct choice is J.
51. Simplify
.
A B C SOLUTION: 2
The area of a circle is A = πr . If the rug just meets
the edge of the room, then the diameter is 16 feet, so
the radius is 16 ÷ 2 or 8 feet.
D SOLUTION: The area of the rug is about 201.1 square feet. So
the correct choice is J.
51. Simplify
.
A The correct choice is B.
Find each sum or difference.
B 52. C SOLUTION: D SOLUTION: 53. SOLUTION: eSolutions Manual - Powered by Cognero
The correct choice is B.
Find each sum or difference.
Page 14
11-7 Mixed Expressions and Complex Fractions
57. 53. SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b
+ 3)(b – 2).
SOLUTION: Find each quotient. Use long division.
2
58. (x − 2x − 30) ÷ (x + 7)
54. SOLUTION: SOLUTION: 55. The quotient is
.
SOLUTION: 3
3
Find the LCD of 5m and 15m . The LCD is 15m .
2
59. (a + 4a − 22) ÷ (a − 3)
SOLUTION: 56. SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
The quotient is
.
2
60. (3q + 20q + 11) ÷ (q + 6)
SOLUTION: 57. eSolutions Manual - Powered by Cognero
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b
Page 15
11-7The
Mixed
Expressions
and .Complex Fractions
quotient
is
2
The quotient is
3
63. (9h + 5h − 8) ÷ (3h − 2)
60. (3q + 20q + 11) ÷ (q + 6)
SOLUTION: SOLUTION: The quotient is
3
.
.
2
61. (3y + 8y + y − 7) ÷ (y + 2)
The quotient is
.
SOLUTION: 64. GEOMETRY A rectangle has a base of 8 meters
and a height of 14 meters. What is the length of the
diagonal?
SOLUTION: Use the Pythagorean theorem.
The quotient is
3
.
2
62. (6t − 9t + 6) ÷ (2t − 3)
SOLUTION: Graph each function. Determine the domain and
range.
65. SOLUTION: Use perfect squares for the x-values of the table.
x
y
The quotient is
.
SOLUTION: eSolutions Manual - Powered by Cognero
0
1
2
4
4
6
9
3
63. (9h + 5h − 8) ÷ (3h − 2)
0
Page 16
11-7 Mixed Expressions and Complex Fractions
Graph each function. Determine the domain and
range.
The domain is {x|x ≥ 0}, and the range is {y|y ≥ 0}.
66. SOLUTION: Use perfect squares for the x-values of the table.
65. SOLUTION: Use perfect squares for the x-values of the table.
x
y
0
0
1
2
4
4
6
9
x
y
0
0
1
4
9
–3
–6
–9
The domain is {x|x ≥ 0}, and the range is {y|y ≤ 0}.
The domain is {x|x ≥ 0}, and the range is {y|y ≥ 0}.
67. 66. SOLUTION: Use perfect squares for the x-values of the table.
x
y
0
0
1
4
9
–3
SOLUTION: Use perfect squares for x-values of the table.
–6
–9
x
y
0
0
1
0.25
4
0.5
9
0.75
The domain is {x| x ≥ 0}, and the range is {y| y ≥ 0}.
eSolutions Manual - Powered by Cognero
The domain is {x|x ≥ 0}, and the range is {y|y ≤ 0}.
Factor each polynomial. If the polynomial
cannot be factored, write prime .
2
68. x − 81
Page 17
11-7 Mixed Expressions and Complex Fractions
The domain is {x|x ≥ 0}, and the range is {y|y ≤ 0}.
SOLUTION: This is the sum of two squares, not the difference.
Therefore it cannot be factored and is a prime
polynomial.
71. −25 + 4y
67. 2
SOLUTION: SOLUTION: Use perfect squares for x-values of the table.
x
y
0
0
1
0.25
4
0.5
9
0.75
4
72. p − 16
SOLUTION: 4
73. 4t − 4
SOLUTION: The domain is {x| x ≥ 0}, and the range is {y| y ≥ 0}.
Factor each polynomial. If the polynomial
cannot be factored, write prime .
2
68. x − 81
SOLUTION: 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.
2
69. a − 121
SOLUTION: 2
70. n + 100
SOLUTION: This is the sum of two squares, not the difference.
Therefore it cannot be factored and is a prime
polynomial.
71. −25 + 4y
2
SOLUTION: Let x = the price of the adult ticket and y = the price
of the student ticket. In Van A, there were 2 adults
and 5 students, and the total cost was $77. In Van B, there were 2 adults and 7 students, and the total
cost was $95.
2x + 5y = 77
2x + 7y = 95
Because 2x and 2x have the same coefficient,
elimination using subtraction is the best method.
SOLUTION: eSolutions Manual - Powered by Cognero
Page 18
11-7 Mixed Expressions and Complex Fractions
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.
76. 5y − 1 = 19
SOLUTION: 77. 2t + 7 = 21
SOLUTION: Let x = the price of the adult ticket and y = the price
of the student ticket. In Van A, there were 2 adults
and 5 students, and the total cost was $77. In Van B, there were 2 adults and 7 students, and the total
cost was $95.
SOLUTION: 78. 2x + 5y = 77
2x + 7y = 95
SOLUTION: Because 2x and 2x have the same coefficient,
elimination using subtraction is the best method.
79. SOLUTION: Now, substitute 9 for y in either equation to find x.
So, the price of the adult ticket was $16 and the price
of the student ticket was $9.
Solve each equation.
75. 6x = 24
80. SOLUTION: SOLUTION: 76. 5y − 1 = 19
SOLUTION: eSolutions Manual - Powered by Cognero
Page 19