Lesson 2.1 Assignment
Name_________________________________________________________ Date__________________________
What Makes You Tap Your Feet?
Introduction to Direct Variation
1. In 2010, Chevrolet produced 12,194 Corvettes. That means they made Corvettes at a rate of about
34 cars per day.
a. How many Corvettes did Chevrolet produce in 7 days? Show your work.
7 ? 34 5 238
Chevrolet produced 238 cars in 7 days.
b. How many days did it take Chevrolet to produce 1190 Corvettes? Show your work.
1190 4 34 5 35
It took Chevrolet 35 days to produce 1190 Corvettes.
© 2011 Carnegie Learning
c. Complete the table.
Time (days)
Number of Corvettes
7
238
21
714
1
​ __  ​ 
2
17
35
1190
28
952
Chapter 2 Assignments • 27
Lesson 2.1 Assignment
page 2
d. Use the data in the table to complete the graph.
Corvette Production
y
1080
Number of Corvettes
960
840
720
600
480
360
240
120
4
8
12
16 20 24
Time (days)
28
32
36
x
e. Is the data in the graph discrete or continuous? Explain your reasoning.
The data is continuous because it is possible to work for only part of one day.
f. Does it make sense to connect the points in this graph? If so, connect them.
Explain your reasoning.
Yes. It makes sense to connect the points because the data is continuous.
g. What is the ratio of Corvette production to time?
34 Corvettes
  
 ​. 
The rate of Corvette production to time is ​ ____________
1 day
h. Does the number of Corvettes vary directly with time? Explain your reasoning.
Yes. As time increases by 1 day, the number of Corvettes increases by a constant of 34 cars.
The number of Corvettes varies directly with time.
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Lesson 2.2 Assignment
Name_________________________________________________________ Date__________________________
Building Bird Feeders Is for the Birds!
Determining Equivalent Ratios
The student council at Camp Creek Middle School determine that 3 out of 4 students prefer that all
school assemblies be held on Friday afternoon.
1. If 200 students are surveyed, how many will prefer that school assemblies be held on Friday
afternoon? Explain how you determined your answer.
3 ​ 5 ​ 
x   ​ 
​ __
  ____
4 200
4x 5 3(200)
4x 5 600
x 5 150
I set up a proportion using the ratio 3 : 4 and then used the means and extremes method to
determine that 150 students will prefer that school assemblies be held on Friday afternoon.
2. If 747 students prefer that school assemblies be held on Friday afternoon, how many students
© 2011 Carnegie Learning
were surveyed? Explain how you determined your answer.
3  ​ 5 ​ ____
747
 ​
​ __
x   
4
3x 5 4(747)
3x 5 2988
x 5 996
I set up a proportion using the ratio 3 : 4 and then used the means and extremes method to
determine that 996 students were surveyed.
Chapter 2 Assignments • 29
Lesson 2.2 Assignment
page 2
Analyze each table shown. Determine if the relationship is proportional. If the relationship is
proportional, calculate the equivalent ratio to justify your answer.
3. Of the 75 boys in the 7th grade class, 25 participate in at least one sport. Of the 120 girls in the
7th grade class, 30 participate in at least one sport.
7th Grade Class
Plays Sports
Total
Boys
25
75
Girls
30
120
25  ​ is not equal to ​ ____
30  ​. The ratio of boys who
No. This relationship is not proportional because ​ ___
75
120
participate in at least one sport is 1 : 3, while the ratio of girls who participate in at least one
sport is 1 : 4. There is no equivalent ratio.
4. Of the 210 boys in the 8th grade, 190 have a cell phone. Of the 168 girls in the 8th grade,
8th Grade Class
Cell Phones
Total
Boys
190
210
Girls
152
168
190 
152 
Yes. This relationship is proportional because ​ ____
 ​ is equal to ​ ____
 ​. The ratio of boys who have
210
168
cell phones is 19 : 21 and the ratio of girls who have cell phones is also 19 : 21. The equivalent
19 ​ .
ratio is ​ ___
21
© 2011 Carnegie Learning
152 have a cell phone.
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Lesson 2.2 Assignment
page 3
Name_________________________________________________________ Date__________________________
5. Of the 250 sixth graders, 75 said they went to the movies over the weekend. Of the 180 seventh
graders, 54 said they went to the movies over the weekend. Of the 200 eighth graders, 65 said
they went to the movies over the weekend.
Class
Attended a Movie
Total
6th Graders
75
250
7th Graders
54
180
8th Graders
65
200
75  ​ does
54  ​, they do not equal ​ ____
65  ​. 
No. This relationship is not proportional. While ​ ____
 
equal ​ ____
250
180
200
The ratio of 6th graders who went to the movies over the weekend is 3 : 10, and the ratio of 7th graders who went to the movies over the weekend is 3 : 10, but the ratio of 8th graders
who went to the movies this weekend is 13 : 40. There is no equivalent ratio.
6. In the town of Clover, 3 out of 5 citizens who are eligible to vote did so in the spring election.
a. Write an equation showing the relationship between the citizens who voted (v) and the citizens
who are eligible to vote (e) and the equivalent ratio.
3  ​ 5 __
​ v  ​
​ __
5 e
b. Use the equation from part (a) to determine the number of citizens that voted in the spring
© 2011 Carnegie Learning
election if 400 citizens were eligible to vote.
3 ​ 5 ​ 
v   ​ 
​ __
  ____
5 400
5v 5 3(400)
5v 5 1200
v 5 240
Of the 400 citizens eligible to vote, 240 citizens actually voted.
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Lesson 2.2 Assignment
page 4
c. Use the equation from part (a) to determine the number of citizens that were eligible to vote in
the spring election if 180 actually voted.
180   
3  ​ 5 ​ ____
 ​
​ __
e
5
3e 5 5(180)
3e 5 900
e 5 300
There were 300 citizens eligible to vote in the spring election.
© 2011 Carnegie Learning
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Lesson 2.3 Assignment
Name_________________________________________________________ Date__________________________
Kids Just Wanna Have Fun!
Determining and Applying the Constant of Proportionality
1. Shilo is riding her bicycle across the state of Georgia to raise money for her favorite charity. The
distance (d ) in miles that she can travel varies directly with the length of time (t) in hours she
spends riding.
Assume that her constant of proportionality is 18.
a. Write an equation showing the proportional relationship between d and t using the
information given.
d 5 18t
b. If Shilo rides for 3 hours, how far will she travel? Use your equation to show the
© 2011 Carnegie Learning
relationship between distance and time.
d 5 18t
d 5 18(3)
d 5 54
Shilo will travel 54 miles in 3 hours.
c. How long does it take Shilo to ride 6 miles? Use your equation to show the relationship
between distance and time.
d 5 18t
6 5 18t
6  ​ 5
  t
​ ___
18
1 ​ 5
​ __
  t
3
1 ​ of
It takes Shilo ​ __
  an hour, or 20 minutes, to ride 6 miles.
3
Chapter 2 Assignments • 33
Lesson 2.3 Assignment
page 2
d. What does the constant of proportionality represent in this problem?
miles ​ 
.
The constant of proportionality represents how fast Shilo can ride in miles per hour, or ​ _____
hour
2. Maya’s construction company builds brick houses. The table shows the relationship between the
time Maya works and the number of bricks she can install.
Hours Worked
Bricks Installed
8
1680
7
1470
6
1260
a. Define variables for the quantities that are changing in this problem situation.
Sample answer. Let h be the time in hours, and let b be the number of bricks.
b. Analyze the table to determine if the relationship is proportional. Explain your reasoning and
state a constant of proportionality if possible.
This relationship is proportional, because the ratio of bricks installed to hours worked is the
constant 210. The constant of proportionality is 210.
c. What does the constant of proportionality represent in this problem?
The constant of proportionality represents the number of bricks Maya can install in one hour.
d. Write an equation showing the relationship between the number of hours worked, the number
of bricks installed, and the constant of proportionality.
b 5 210h
34 • Chapter 2 Assignments
© 2011 Carnegie Learning
Lesson 2.3 Assignment
page 3
Name_________________________________________________________ Date__________________________
e. Use your equation from part (d) to determine how many bricks Maya can install in 5.5 hours.
Show your work.
b 5 210h
b 5 210(5.5)
b 5 1155
Maya can install 1155 bricks in 5.5 hours.
f. Use your equation from part (d) to determine how many hours it will take Maya to install
840 bricks. Show your work.
b 5 210h
840 5 210h
45h
It will take Maya 4 hours to install 840 bricks.
© 2011 Carnegie Learning
Chapter 2 Assignments • 35
Lesson 2.3 Assignment
page 4
y
Solve each using the equation for the constant of proportionality, __
​ x ​5 k.
4. k 5 0.3 and y 5 18
3. k 5 __
​ 2 ​ and x 5 21
7
y
y
__
​ x ​5 k
__
​ x ​5 k
y
___
​ 2 ​
​    ​ 5 __
21
7
7y 5 2(21)
7y 5 42
y 5 6
___
​ 18 ​ 5 0.3
x
18 ​ 5 ___
​ ___
​  3  ​ 
x
10
3x 5 10(18)
3x 5 180
© 2011 Carnegie Learning
x 5 60
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Lesson 2.4 Assignment
Name_________________________________________________________ Date__________________________
Stop that Speeding Snail?
Using the Constant of Proportionality to Solve Proportions
1. Dudley and Bob monitored the distance their pet turtle could walk in a certain amount of time.
Their results are shown in the table. The table of values represents a proportional relationship.
Time (minutes)
Distance (inches)
 5
14.5
14
40.8
19
55.1
25
72.5
a. Define variables for the quantities that are changing in this problem situation.
Let t equal the time in minutes, and let d equal the distance in inches.
b. How can you determine the constant of proportionality using the values in the table?
I can set up one rate from one value from the time column and a corresponding value from
the temperature column. Next, I can define the quantities that are changing. Finally, I can set
© 2011 Carnegie Learning
up a proportion with the ratio to determine the constant of proportionality.
c. What is the constant of proportionality?
The constant of proportionality is 2.9.
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Lesson 2.4 Assignment
page 2
2. Analyze each table or problem situation to determine if the relationship is proportional. State a
constant of proportionality if possible. Show your work.
a.
Girls
Boys
7
14
9
21
11
22
7  ​ 5 ​ 
1 ​ 
​ ___
  __
14 2
9  ​ 5 ​ 
3
  __ ​ 
​ ___
21 7
11 ​ 5 ​ 
1 ​ 
  __
​ ___
22 2
3 ​ . The third ratio is ​ __
1 ​ . The second ratio is ​ __
1 ​ .
The first ratio is ​ __
7
2
2
No. This relationship is not proportional. b. A baby blue whale weighed 5520 pounds at birth. After two days, the baby weighed 5710. After
14  ​ 5 ​ 
7   ​ 
  _____
​ _____
8180 4090
1   ​ 
7   ​ 
The first ratio is ​ _____
. The second ratio is ​ _____
.
2855
4090
No. This relationship is not proportional. 38 • Chapter 2 Assignments
© 2011 Carnegie Learning
14 days, the baby weighed 8180 pounds. Is there a constant of proportionality?
1   ​ 
2   ​ 5 ​ 
  _____
​ _____
5710 2855
Lesson 2.5 Assignment
Name_________________________________________________________ Date__________________________
The Man Who Ran from Marathon to Athens
Graphing Direct Proportions
1. Tony is going into sixth grade in the fall. During the summer, he is required to read at least two
books from an approved reading list and to complete a mini-project to accompany each book.
Suppose the number of pages ( p) that Tony can read varies directly with the amount of time
(t) in hours he spends reading. Suppose the constant of proportionality relating the variables
p and t is 30.
a. Write an equation showing the relationship between the number of pages read and the time
spent reading. Assume that Tony always maintains the same rate of reading.
p 5 30t
b. What does the constant of proportionality represent in this problem?
The constant of proportionality means that Tony is able to read 30 pages in one hour.
c. Complete the table showing the amount of time spent reading and the number of pages read
© 2011 Carnegie Learning
using your equation from part (a).
Time (in hours)
Number of Pages Read
t
p
0.5
15
1.25
37.5
2
60
2.5
75
3
90
Chapter 2 Assignments • 39
Lesson 2.5 Assignment
page 2
d. Graph the values in the table from part (c) on the coordinate plane shown. Graph the values of t
on the x-axis and the values of p on the y-axis.
p
90
Number of Pages Read
80
70
60
50
40
30
20
10
1
2
3
4
t
Time (in hours)
e. Explain how the graph verifies that the number of pages Tony reads is directly proportional to
the amount of time he spends reading.
The graph is a straight line going through the origin (0, 0), which shows that there is a direct variation.
f. Interpret the meaning of the point (2.5, 75) on the graph.
The point (2.5, 75) means that Tony reads 75 pages in 2.5 hours.
© 2011 Carnegie Learning
40 • Chapter 2 Assignments
Lesson 2.5 Assignment
page 3
Name_________________________________________________________ Date__________________________
y-coordinate
g. For each of the following points on the graph, form the ratio ____________
​ 
   ​in the table.
x-coordinate
Then, simplify the ratio. What do you notice? What is significant about what you noticed?
x-coordinate
y-coordinate
y-coordinate
​ _____________
   ​
x-coordinate
0.5
15
15
 ​____  ​ 
5 30
0.5
1.25
37.5
37.5
_____
 ​ 
5 30
​ 
2
60
60
___
 5 30
​   ​ 
2.5
75
75
____
3
75
90
___
 5 30
​   ​ 
1.25
2
​   ​  
5 30
2.5
3
I notice that all of the ratios are the same. It is significant that the ratio of 30 is the same as
the constant of proportionality given in this problem.
2. Mariah is trying to improve her math skills over the summer. When she reads her math book, the
number of pages she can read depends on the difficulty of the concept being studied. Mariah
© 2011 Carnegie Learning
keeps track of the amount of time and the number of pages read during several study sessions
in the table.
Time (in hours)
Number of Pages Read
0.5
5
0.75
7
1
10
1.5
8
2
12
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Lesson 2.5 Assignment
page 4
a. Graph each point from the table on the graph shown. Label the x- and y-axis and name
your graph.
Reading in Math
18
Number of Pages Read
16
14
12
10
8
6
4
2
1
2
3
4
Time (in hours)
b. What do you notice about the points you plotted?
The points don’t appear to make a straight line. Also, if the plotted points were to lie in a
straight line, the line would not pass through the origin (0, 0).
y-coordinate
   ​.
the ratio ​ ____________
x-coordinate
Answers will vary. Sample answer is given.
The two points I chose are (1, 10) and (2, 12).
10 ​ 5
  10
​ ___
1
___
  6
​ 12 ​ 5
Because the two ratios are not equivalent, there is no constant of proportionality. 2
This is a non-proportional relationship.
42 • Chapter 2 Assignments
© 2011 Carnegie Learning
c. Determine if there is a constant of proportionality. Pick two points from the graph to form
Lesson 2.6 Assignment
Name_________________________________________________________ Date__________________________
Racing to the Finish Line!
Using Direct Proportions
1. Jameer works in a men’s clothing store. The amount of money (m) he makes is directly proportional
to the number of hours (h) he works.
a. Complete the table to show the amount of money Jameer earns for the different amounts of
time that he works.
Hours (h)
Money (m)
(in dollars)
3
$37.50
4.5
56.25
6
75.00
7.5
93.75
b. Determine the constant of proportionality and interpret it in the context of the problem.
37.50
 
 ​ 
, or 12.50. This tells me that Jameer earns
The constant of proportionality is k 5 ​ ______
3
$12.50 per hour.
© 2011 Carnegie Learning
c. Write an equation showing the relationship between m and h.
m 5 12.50h
d. How many hours does Jameer have to work to make $100?
m 5 12.50h
100 5 12.50h
100  ​ 5
  h
​ ______
12.50
8 5 h
Jameer has to work 8 hours to make $100.
Chapter 2 Assignments • 43
Lesson 2.6 Assignment
page 2
e. How much money does Jameer earn if he works 40 hours?
m 5 12.50h
m 5 12.50(40)
m 5 500
Jameer earns $500 if he works 40 hours.
2. Henry knows that whenever he buys something he is charged a sales tax. This sales tax (t)
varies directly with the amount of money (m) he spends on the item.
a. Write an equation representing the direct proportional relationship between the sales tax and
the amount of money spent. Use the variable k as the constant of proportionality.
t 5 km
b. If Henry pays $1.76 in sales tax on an item that costs $22, what is the value of the constant
of proportionality?
t 5 km
1.76 5 k(22)
1.76 ​ 5
 
  k
​ _____
22
0.08 5 k
The constant of proportionality is 0.08.
c. What does the constant of proportionality represent in this problem?
The constant of proportionality represents the percent of sales tax applied to the cost of
each item purchased.
44 • Chapter 2 Assignments
© 2011 Carnegie Learning
Lesson 2.6 Assignment
page 3
Name_________________________________________________________ Date__________________________
d. If Henry buys an item that costs $76, how much is the sales tax? Use the constant of
proportionality you calculated in part (b).
t 5 km
t 5 0.08m
t 5 0.08(76)
t 5 6.08
The sales tax is $6.08.
e. How much does an item cost if the sales tax is $1.04? Use the constant of proportionality
© 2011 Carnegie Learning
you calculated in part (b).
t 5 km
t 5 0.08m
1.04 5 0.08m
1.04 ​ 5
  m
​ _____
0.08
13 5 m
The item costs $13 without the sales tax.
Chapter 2 Assignments • 45
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Lesson 2.7 Assignment
Connecting Representations of Proportional Relationships
Interpreting Multiple Representations of Direct Proportions
1. Millie is cutting out stars to decorate the gym for the school dance. The number of stars (s) she can
cut out varies directly with the time (t) in minutes she spends cutting out the stars.
a. Write an equation showing the relationship between s and t.
s 5 kt
b. Complete the table to show the number of stars Millie is able to cut out for various amounts of
time. Explain how you completed the table.
Time (t)
(in minutes)
Number of Stars (s)
0
0
------
12
6
___
​  6   ​5 __
​ 1 ​
30
15
15  ​5 __
​ ___
​ 1 ​
30 2
44
22
22  ​5 __
​ ___
​ 1 ​
44 2
50
25
25  ​5 __
​ ___
​ 1 ​
50 2
s
__
​   ​5 k
t
12
2
© 2011 Carnegie Learning
• Because s varies directly with t, I knew that when t is 0, s would have to be 0.
s ​ 5 ​ 
6  ​ 5 ​ 
1 ​ , simplifying gives me the
  __
• I saw that for t 5 12, s 5 6. So by setting up the ratio ​ __
  ___
t
12 2
1  ​.
constant of proportionality, k, to be equal to ​ __
2
1  ​, I could then determine the unknown values using the equation
• Because I determined k 5 ​ __
2
1  ​t and solving for s or for t.
s 5 ​ __
2
Chapter 2 Assignments • 47
Lesson 2.7 Assignment
page 2
c. Write the equation representing the relationship between s and t using the value of k you
determined from the table.
1  ​t
s 5 ​ __
2
d. Graph the data. Label the x- and y-axes and name your graph.
Millie’s Stars
y
27
Number of Stars
24
21
18
15
12
9
6
3
0
10
20
30
40
50
60
70
80
90 x
Time (in minutes)
e. Did the graph turn out as you expected? Explain.
Yes. Because s varies directly with t, I expected the points to be on a line that passes
through the origin (0, 0).
f. Determine the constant of proportionality in two different ways by using the graph.
One way to determine the constant of proportionality is to set up the ratio between s and t
using any of the points graphed. For instance, if I choose the point (30, 15) and set up the
15 ​ , I get k 5 ​ __
1 ​ .
ratio ​ ___
30
2
Another way to determine the constant of proportionality is to set up a ratio between the
vertical distance and the horizontal distance between any two points on the graph. Once I calculate the vertical and horizontal distance, I can set up a ratio to determine the constant
of proportionality. For instance, if I choose the points (12, 6) and (30, 15), the vertical 9  ​ , or ​ __
1 ​ , is the
distance would be 9 and the horizontal distance would be 18. The ratio ​ ___
18
2
constant of proportionality.
48 • Chapter 2 Assignments
© 2011 Carnegie Learning