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Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 1
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 2
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 3
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 4
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 5
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 6
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 7
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 8
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 9
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 10
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 11
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 12
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 13
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 14
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 15
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 16
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 17
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 18
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 19
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 20
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 21
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 22
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 23
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 24
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 25
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 26
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 27
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 28
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 29
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 30
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 31
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 32
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 33
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 34
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 35
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 36
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 37
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 38
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 39
11-7 Mixed Expressions and Complex Fractions
Write each mixed expression as a rational expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
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: a. Use the formula d = r × t to write an expression for Rico's speed, r.
b. Simplify the complex fraction to find his average speed.
So, Rico's average speed is .
Simplify each expression.
6.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
7.
SOLUTION: To simplify complex fractions, rewrite as a division expression, then rewrite as a multiplication expression.
8.
SOLUTION:
9.
SOLUTION:
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each mixed expression as a rational expression.
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
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.
Her average reading rate is page/minute.
24. HORSES A thoroughbred can run mile in about minute. What is the horse’s speed in miles per hour?
SOLUTION:
Use the formula d = rt or to represent the horse's speed in miles per minute and convert minutes to hours.
The horse’s speed is 40 miles/hour.
Simplify each expression.
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33. CCSS MODELING The Centralville High School Cooking Club has pounds of flour with which to make
tortillas. There are cups of flour in a pound, and it takes about 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 about cup of flour. To find out how many tortilla they can make,
divide the total cups of flour by .
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 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.
The speed of the scooter is about 12.5 mi/hr.
35. SCIENCE The density of an object equals , 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 .)
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.
Metal Density (kg/m3)
copper 8900 gold 19,300 iron 7800 lead 11,300
SOLUTION: a.
The density is about 8900 kg/m3. Therefore, the metal ball is made of copper.
b.
The density is about 11,317 kg/m3. 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 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 frequencyof the siren as you hear it.
SOLUTION: a.
b. Substitute 65 for v, and 45 for f in the equation .
The frequency of the siren as you hear it is 49.21 cycles/min.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
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.
44. REASONING Is sometimes, always, or never equal to 0? Explain.
SOLUTION:
This expression is always equal to 0.
45. CCSS PERSEVERANCE Simplify the rational expression shown.
SOLUTION:
46. OPEN ENDED Write a complex fraction that, when simplified, results in .
SOLUTION:
Find two fractions that have the same denominators and when you divide the numerators, you get
Consider the fraction . It simplifies to .
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 walks mile in 10 minutes;
the time in miles per minute is , which simplifies to mi/min.
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.
The only number that has 52 as one of its factors and is a multiple of 13 is 1950. So, the correct choice is A.
Number Prime Factorization
Exponential Form
1950 2 • 3 • 5 • 5 • 13 2 • 3 • 52 • 13
2000 2 • 2 • 2 • 2 • 5 • 5 •5
24 • 5
3
1975 5 •5•79 52 • 79
49. SHORT RESPONSE Bernard is reading a 445-page book. He has already read 157 pages. If he reads 24 pages aday, how long will it take him to finish the book?
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 daysto 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
G 804.2 ft2
H 50.3 ft
J 201.1 ft2
SOLUTION:
The area of a circle is A = πr2. 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.
The area of the rug is about 201.1 square feet. So the correct choice is J.
51. Simplify .
A
B
C
D
SOLUTION:
The correct choice is B.
Find each sum or difference.
52.
SOLUTION:
53.
SOLUTION:
54.
SOLUTION:
55.
SOLUTION:
Find the LCD of 5m and 15m3. The LCD is 15m
3.
56.
SOLUTION: Find the LCD of 3g and 4h. The LCD is 12gh.
57.
SOLUTION: Find the LCD of (b + 3) and (b – 2). The LCD is (b + 3)(b – 2).
Find each quotient. Use long division.
58. (x2 − 2x − 30) ÷ (x + 7)
SOLUTION:
The quotient is .
59. (a2 + 4a − 22) ÷ (a − 3)
SOLUTION:
The quotient is .
60. (3q2 + 20q + 11) ÷ (q + 6)
SOLUTION:
The quotient is .
61. (3y3 + 8y
2 + y − 7) ÷ (y + 2)
SOLUTION:
The quotient is .
62. (6t3 − 9t
2 + 6) ÷ (2t − 3)
SOLUTION:
The quotient is .
63. (9h3 + 5h − 8) ÷ (3h − 2)
SOLUTION:
The quotient is .
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.
Graph each function. Determine the domain and range.
65.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≥ 0}.
x y
0 0
1 2
4 4
9 6
66.
SOLUTION: Use perfect squares for the x-values of the table.
The domain is {x|x ≥ 0}, and the range is {y |y ≤ 0}.
x y
0 0
1 –3
4 –6
9 –9
67.
SOLUTION: Use perfect squares for x-values of the table.
The domain is {x| x ≥ 0}, and the range is {y | y ≥ 0}.
x y
0 0
1 0.25
4 0.5
9 0.75
Factor each polynomial. If the polynomial cannot be factored, write prime .
68. x2 − 81
SOLUTION:
69. a2 − 121
SOLUTION:
70. n2 + 100
SOLUTION: This is the sum of two squares, not the difference. Therefore it cannot be factored and is a prime polynomial.
71. −25 + 4y2
SOLUTION:
72. p4 − 16
SOLUTION:
73. 4t4 − 4
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.
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.
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
SOLUTION:
76. 5y − 1 = 19
SOLUTION:
77. 2t + 7 = 21
SOLUTION:
78.
SOLUTION:
79.
SOLUTION:
80.
SOLUTION:
eSolutions Manual - Powered by Cognero Page 40
11-7 Mixed Expressions and Complex Fractions
