2-6 Ratios and Proportions
Determine whether each pair of ratios are
equivalent ratios. Write yes or no.
5. 1. SOLUTION: SOLUTION: No, the ratios are not equivalent.
6. 2. SOLUTION: SOLUTION: Yes, the ratios are equivalent.
3. SOLUTION: 7. RACE Jennie ran the first 6 miles of a marathon in
58 minutes. If she is able to maintain the same pace,
how long will it take her to finish the 26.2 miles?
SOLUTION: Let t represent the time it will take her to finish 26.2
miles. Write a proportion.
No, the ratios are not equivalent.
Solve each proportion. If necessary, round to
the nearest hundredth.
4. SOLUTION: It will take Jennie 253.3 minutes or 4 hours and 13.3
minutes.
8. MAPS On a map of North Carolina, Raleigh and
Asheville are about 8 inches apart. If the scale is 1
inch = 12 miles, how far apart are the cities?
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SOLUTION: SOLUTION: Let d represent the distance between the two cities.
Write a proportion.
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Yes, the ratios are equivalent.
2-6 Ratios
andJennie
Proportions
It will take
253.3 minutes or 4 hours and 13.3
minutes.
8. MAPS On a map of North Carolina, Raleigh and
Asheville are about 8 inches apart. If the scale is 1
inch = 12 miles, how far apart are the cities?
11. SOLUTION: Use cross products to check if the ratios are
equivalent.
SOLUTION: Let d represent the distance between the two cities.
Write a proportion.
No, the ratios are not equivalent.
Raleigh and Asheville are about 96 miles apart.
12. Determine whether each pair of ratios are
equivalent ratios. Write yes or no.
SOLUTION: Use cross products to check if the ratios are
equivalent.
9. SOLUTION: Yes, the ratios are equivalent.
No, the ratios are not equivalent.
13. 10. SOLUTION: Use cross products to check if the ratios are
equivalent.
SOLUTION: Yes, the ratios are equivalent.
11. SOLUTION: Use cross products to check if the ratios are
equivalent.
Yes, the ratios are equivalent.
14. SOLUTION: Use cross products to check if the ratios are
equivalent.
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No, the ratios are not equivalent.
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2-6 Ratios
and Proportions
Yes, the ratios are equivalent.
14. 17. SOLUTION: Use cross products to check if the ratios are
equivalent.
SOLUTION: Yes, the ratios are equivalent.
18. SOLUTION: Solve each proportion. If necessary, round to
the nearest hundredth.
15. SOLUTION: 19. SOLUTION: 16. SOLUTION: 20. SOLUTION: 17. SOLUTION: 21. SOLUTION: eSolutions Manual - Powered by Cognero
18. Page 3
2-6 Ratios and Proportions
21. 25. SOLUTION: SOLUTION: 26. 22. SOLUTION: SOLUTION: 23. SOLUTION: 27. SOLUTION: 24. SOLUTION: 28. SOLUTION: 25. SOLUTION: 29. SOLUTION: eSolutions Manual - Powered by Cognero
26. Page 4
2-6 Ratios and Proportions
29. SOLUTION: They will wash about 341 cars.
31. GEOGRAPHY On a map of Florida, the distance
between Jacksonville and Tallahassee is 2.6
centimeters. If 2 centimeters = 120 miles, what is the
distance between the two cities?
30. CAR WASH The B–Clean Car Wash washed 128
cars in 3 hours. At that rate, how many cars can they
wash in 8 hours?
SOLUTION: Let c represent the number of cars they wash in 8
hours. Write a proportion.
SOLUTION: Use the given scale to convert the map distance to
the real world distance.
32. CCSS PRECISION An artist used interlocking
building blocks to build a scale model of Kennedy
Space Center, Florida. In the model, 1 inch equals
1.67 feet of an actual space shuttle. The model is
110.3 inches tall. How tall is the actual space shuttle?
Round to the nearest tenth.
They will wash about 341 cars.
31. GEOGRAPHY On a map of Florida, the distance
between Jacksonville and Tallahassee is 2.6
centimeters. If 2 centimeters = 120 miles, what is the
distance between the two cities?
SOLUTION: Use the scale given to convert the model height to
the actual space shuttle height. 33. MENU On Monday, a restaurant made $545 from
selling 110 hamburgers. If they sold 53 hamburgers
on Tuesday, how much did they make?
SOLUTION: Let p represent their profit from lunch. Write a
proportion.
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Use the given scale to convert the map distance to
the real world distance.
Page 5
Their profit from lunch is about $262.59.
SOLUTION: Use the scale given to convert the model height to
the actual space shuttle height. 2-6 Ratios and Proportions
33. MENU On Monday, a restaurant made $545 from
selling 110 hamburgers. If they sold 53 hamburgers
on Tuesday, how much did they make?
36. SOLUTION: SOLUTION: Let p represent their profit from lunch. Write a
proportion.
37. SOLUTION: Their profit from lunch is about $262.59.
Solve each proportion. If necessary, round to
the nearest hundredth.
34. SOLUTION: 38. SOLUTION: 35. SOLUTION: 39. SOLUTION: 36. SOLUTION: eSolutions Manual - Powered by Cognero
Page 6
ATHLETES
2-6 Ratios and Proportions
So, 130 students in the ninth grade have braces.
42. PAINT Joel used a half gallon of paint to cover 84
square feet of wall. He has 932 square feet of wall
to paint. How many gallons of paint should he
purchase?
39. SOLUTION: SOLUTION: Let p represent the number of gallons of paint
needed. Write a proportion.
40. ATHLETES At Piedmont High School, 3 out of
every 8 students are athletes. If there are 1280
students at the school, how many are not athletes?
It will take about 5.55 gallons of paint, so Paul should
purchase 6 gallons of paint.
SOLUTION: Let s represent the number of students who are
athletes. If
of the students are athletes, then
of
43. MOVIE THEATERS the students are not athletes. Write a proportion.
So, 800 students are not athletes.
41. BRACES Two out of five students in the ninth
grade have braces. If there are 325 students in the
ninth grade, how many have braces?
SOLUTION: Let s represent the number of students that have
braces. Write a proportion.
a. Write a ratio of the number of indoor theaters to
the total number of theaters for each year.
b. Do any two of the ratios you wrote for part a form
a proportion? If so, explain the real-world meaning of
the proportion.
SOLUTION: a. For each year, divide the number of indoor
theaters by the number total number of theaters. 2003:
, 2004:
, 2007:
So, 130 students in the ninth grade have braces.
b. Compare the ratios. 42. PAINT Joel used a half gallon of paint to cover 84
square
feet- of
wall.by
HeCognero
has 932 square feet of wall
eSolutions
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to paint. How many gallons of paint should he
purchase?
Year
Indoor
2003 35,361
2004 36,012
2005 37,092
, 2005:
, 2008:
Total
35,995
36,652
37,740
, 2006:
, 2009:
Ratio
0.9823
0.9825
0.9828
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2009
2-6 Ratios
andabout
Proportions
It will take
5.55 gallons of paint, so Paul should
purchase 6 gallons of paint.
43. MOVIE THEATERS 38,605
39,233
0.9839
No, none of the ratios from part a form a proportion.
If two of these ratios formed a proportion, the indoor
theaters compared to the total number of theaters
would produce equivalent fractions.
44. DIARIES In a survey, 36% of the students said that
they kept an electronic diary. There were 900
students who kept an electronic diary. How many
students were in the survey?
SOLUTION: Let s represents the number of students in the
survey. 36% can be written as
. Now, write a
proportion.
a. Write a ratio of the number of indoor theaters to
the total number of theaters for each year.
b. Do any two of the ratios you wrote for part a form
a proportion? If so, explain the real-world meaning of
the proportion.
SOLUTION: a. For each year, divide the number of indoor
theaters by the number total number of theaters. There were 2500 students in the survey.
2003:
, 2004:
, 2007:
, 2005:
, 2008:
, 2006:
, 2009:
b. Compare the ratios. Year
Indoor
2003 35,361
2004 36,012
2005 37,092
2006 37,776
2007 38,159
2008 38,201
2009 38,605
Total
35,995
36,652
37,740
38,425
38,794
38,834
39,233
Ratio
0.9823
0.9825
0.9828
0.9832
0.9836
0.9836
0.9839
No, none of the ratios from part a form a proportion.
If two of these ratios formed a proportion, the indoor
theaters compared to the total number of theaters
would produce equivalent fractions.
45. MULTIPLE REPRESENTATIONS In this
problem, you will explore how changing the lengths
of the sides of a shape by a factor changes the
perimeter of that shape.
a. GEOMETRIC Draw a square ABCD. Measure
and label the sides. Draw a second square MNPQ
with sides twice as long as ABCD. Draw a third
square FGHJ with sides half as long as ABCD.
b. TABULAR Complete the table below using the
appropriate measures.
c. VERBAL Make a conjecture about the change in
the perimeter of a square if the side length is
increased or decreased by a factor.
SOLUTION: a.
44. DIARIES In a survey, 36% of the students said that
they kept an electronic diary. There were 900
students who kept an electronic diary. How many
students were in the survey?
SOLUTION: Let s represents the number of students in the
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survey. 36% can be written as
proportion.
Page 8
. Now, write a
b.
2-6 Ratios
and Proportions
There were 2500 students in the survey.
45. MULTIPLE REPRESENTATIONS In this
problem, you will explore how changing the lengths
of the sides of a shape by a factor changes the
perimeter of that shape.
a. GEOMETRIC Draw a square ABCD. Measure
and label the sides. Draw a second square MNPQ
with sides twice as long as ABCD. Draw a third
square FGHJ with sides half as long as ABCD.
b. TABULAR Complete the table below using the
appropriate measures.
c. VERBAL Make a conjecture about the change in
the perimeter of a square if the side length is
increased or decreased by a factor.
c. If the length of a side is increased by a factor, the
perimeter is also increased by that factor. If the
length of the sides are decreased by a factor, the
perimeter is also decreased by the same factor.
46. CCSS STRUCTURE In 2007, organic farms
occupied 2.6 million acres in the United States and
produced goods worth about $1.7 billion. Divide one
of these numbers by the other and explain the
meaning of the result.
SOLUTION: This the average amount of food produced (in dollar
amount) per acre of land used. SOLUTION: a.
0.0015 acres is the average land area used to
produce a dollars worth of goods.
47. REASONING Compare and contrast ratios and
rates.
SOLUTION: Ratios and rates each compare two numbers by
using division. However, rates compare two
measurements that involve different units of
b.
measure. An example of a ratio is
example of a rate is
c. If the length of a side is increased by a factor, the
perimeter is also increased by that factor. If the
length of the sides are decreased by a factor, the
perimeter is also decreased by the same factor.
46. CCSS STRUCTURE In 2007, organic farms
occupied 2.6 million acres in the United States and
produced goods worth about $1.7 billion. Divide one
of these numbers by the other and explain the
meaning of the result.
SOLUTION: .
and
48. CHALLENGE If value of
. An
, find the
. (Hint: Choose different values of a and
b for which the proportions are true and evaluate the
expression
.)
SOLUTION: The hint suggests that we choose different values of
a and b for which the proportions are true and
evaluate the expression
.
This the average amount of food produced (in dollar
amount) per acre of land used. eSolutions Manual - Powered by Cognero
How do we go about choosing these numbers? Let’s
look at the first equation.
Page 9
What values of a and b make this equation true? The
simplest values to use are a = 4 and b = 2, because 4
How do we go about choosing these numbers? Let’s
2-6 Ratios
Proportions
look at and
the first
equation.
What values of a and b make this equation true? The
simplest values to use are a = 4 and b = 2, because 4
+ 1 = 5 and 2 – 1 = 1, making the numerators and
denominators both equal.
If the values for a and b did not work for the second
equation, then we would have had to find two
different values that work in the first equation and try
them in the second equation until we found two that
worked for both.
49. WRITING IN MATH On a road trip, Marcus
reads a highway sign and then looks at his gas gauge.
Marcus’s gas tank holds 10 gallons and his car gets
32 miles per gallon at his current speed of 65 miles
per hour. If he maintains this speed, will he make it to
Atlanta without having to stop and get gas? Explain
your reasoning.
Now that we have two values that work for the first
equation, we can test them in the second equation.
As it happens, these values work in the second
equation as well.
the tank is about full, he has about
gal of gas left. Then determine the number of miles he can drive on
the
gal of gas. At 32 miles per gallon, he will be
able to travel 32 × or 180 miles. Since Atlanta is 200 miles away, he will run out of
gas about 20 miles before reaching the city if he
doesn’t stop to get gas.
and
So, SOLUTION: First, determine how much gas is left in the tank. If
.
If the values for a and b did not work for the second
equation, then we would have had to find two
different values that work in the first equation and try
them in the second equation until we found two that
worked for both.
49. WRITING IN MATH On a road trip, Marcus
reads a highway sign and then looks at his gas gauge.
50. WRITING IN MATH Describe how businesses
can use ratios. Write about a real-world situation in
which a business would use a ratio.
SOLUTION: For example, a business can use ratios to compare
how many of the potential customers in an area use
their business.
Also, using ratios, a pizza business can find the
number of potential customers in their area and
compare them to how many they have. They can do
the same with their
competitors.
Marcus’s gas tank holds 10 gallons and his car gets
32 miles per gallon at his current speed of 65 miles
per hour. If he maintains this speed, will he make it to
Atlanta without having to stop and get gas? Explain
your reasoning.
51. In the figure, x : y = 2 : 3 and y : z = 3 : 5. If x = 10,
find the value of z.
SOLUTION: First, determine how much gas is left in the tank. If
A 15
B 20
C 25
D 30
the tank is about
SOLUTION: eSolutions Manual - Powered by Cognero
full, he has about
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number of potential customers in their area and
compare them to how many they have. They can do
the same with their
2-6 Ratios and Proportions
competitors.
51. In the figure, x : y = 2 : 3 and y : z = 3 : 5. If x = 10,
find the value of z.
Choice C is correct.
52. GRIDDED RESPONSE A race car driver records
the finishing times for recent practice trials. What is
the mean time, in seconds, for the trials?
A 15
B 20
C 25
D 30
SOLUTION: First, find y.
SOLUTION: The mean time is 5.02 seconds.
53. GEOMETRY If
Now, find z.
is equal to
, what
is z?
Choice C is correct.
52. GRIDDED RESPONSE A race car driver records
the finishing times for recent practice trials. What is
the mean time, in seconds, for the trials?
F 240
G 140
H 120
J 70
SOLUTION: is equal to
SOLUTION: because they are vertical angles. The two right angles are equals and . Then since all three
is equal to
angles in
are equal to all three angles in
, . Similar triangles have
is similar to proportional sides. So, set up a proportion of the
sides.
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The mean time is 5.02 seconds.
Page 11
Choice G is correct.
2-6 Ratios and Proportions
The mean time is 5.02 seconds.
53. GEOMETRY If
is equal to
, what
is z?
54. Which equation below illustrates the Commutative
Property?
A (3x + 4y) +2z = 3x + (4y + 2z)
B 7(x + y) = 7x + 7y
C xyz = yxz
Dx+0=x
SOLUTION: Choice B represents the Distributive Property,
Choice A shows the Associative Property, and
Choice D represents the Additive Identity. The
Commutative Property switches the order of
multiplying or adding. Therefore, choice C is correct.
Solve each equation.
55. SOLUTION: F 240
G 140
H 120
J 70
SOLUTION: is equal to
because they are vertical angles. The two right angles are equals and . Then since all three
is equal to
angles in
are equal to all three angles in
, . Similar triangles have
is similar to proportional sides. So, set up a proportion of the
means the distance between x and –5 is
–8. Since distance cannot be negative, the solution is
the empty set .
56. SOLUTION: Case 1:
Case 2:
sides.
The solution set is (–7,–11).
57. Choice G is correct.
54. Which equation below illustrates the Commutative
Property?
A (3x + 4y) +2z = 3x + (4y + 2z)
B 7(x + y) = 7x + 7y
C xyz = yxz
Dx+0=x
SOLUTION: Choice B represents the Distributive Property,
Choice A shows the Associative Property, and
Choice
D represents
Additive Identity. The
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Cognero
Commutative Property switches the order of
multiplying or adding. Therefore, choice C is correct.
SOLUTION: Case 1:
Case 2:
Page 12
Case 2:
2-6 Ratios and Proportions
The solution set is (–7,–11).
The solution set is
.
59. HEALTH When exercising, a person’s pulse rate
should not exceed a certain limit. This maximum rate
is represented by the expression 0.8(220 – a), where
a is age in years. Find the age of a person whose
maximum pulse rate is 152.
57. SOLUTION: Case 1:
SOLUTION: Case 2:
A person whose maximum pulse rate is 152 is 30
years old.
Solve each equation. Check your solution.
60. 15 = 4a – 5
The solution set is {10, –7}.
SOLUTION: 58. SOLUTION: Case 1:
Check:
Case 2:
61. 7g – 14 = –63
SOLUTION: The solution set is
.
59. HEALTH When exercising, a person’s pulse rate
should not exceed a certain limit. This maximum rate
is represented by the expression 0.8(220 – a), where
a is age in years. Find the age of a person whose
maximum pulse rate is 152.
Check:
SOLUTION: eSolutions Manual - Powered by Cognero
Page 13
62. 2-6 Ratios and Proportions
61. 7g – 14 = –63
63. SOLUTION: SOLUTION: Check:
Check:
62. SOLUTION: ABC if each small
triangle has a base of 5.2 inches and a height of 4.5
inches.
64. GEOMETRY Find the area of
Check:
SOLUTION: The area of one small triangle is:
63. SOLUTION: There are 4 small triangles. Multiply the area of the
small triangle by 4 to determine the area of the large
triangle. So, the area of
is 4 • 11.7 or 46.8
square inches.
Evaluate each expression.
Check:
65. SOLUTION: eSolutions Manual - Powered by Cognero
Page 14
66. There are 4 small triangles. Multiply the area of the
small triangle by 4 to determine the area of the large
2-6 Ratios
triangle.and
So,Proportions
the area of
is 4 • 11.7 or 46.8
square inches.
Evaluate each expression.
65. SOLUTION: 66. SOLUTION: Solve each equation.
67. 4p = 22
SOLUTION: 68. 5h = 33
SOLUTION: 69. 1.25y = 4.375
SOLUTION: 70. 9.8m = 30.87
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