Unit 2 Study Guide and Answers at Bottom: Direct Variation and Constant of Proportionality
1. A recent survey determined that 7 out of 9 students eat school lunch at least once a week. If 450 students were
surveyed, how many eat school lunch at least once a week?
.
2. The distance (d) in meters that an ant can travel varies directly with the amount of time (t) in hours it spends
walking. Assume that an ant’s constant of proportionality is 18.
a.
Write an equation representing the proportional relationship between d and t using the information
given.
.
b.
If an ant walks for 10 minutes, how far will it travel?
.
c.
If an ant traveled 22.5 meters, how long did it walk?
.
d.
What does the constant of proportionality represent in this problem?
.
3. Analyze the table, which shows the number of students in two grades who play on two or more sports teams
each year. Determine if the relationship is proportional.
Grade
Two or More Sports
Total Students
6
64
96
7
60
90
4. A recent survey on movies found that 1 out of 6 students like comedy films.
a.
Use the information to complete the table.
Total Number of Students
90
180
Students Who Like Action Films
270
360
b.
Graph the relationship.
5. Determine if the graph shows two quantities that vary directly. If possible, determine the constant of
proportionality. Explain your reasoning.
6. In the school orchestra, there are 5 woodwind instrument players for every 2 brass instrument players.
a.
If there are 12 brass players in the orchestra, how many woodwind players are in the orchestra? Write
a proportion and solve.
.
b.
.
If there are 20 woodwind players in the orchestra, how many brass players are in the orchestra? Write
a proportion and solve.
c.
Define variables for the quantities that are changing in this situation.
.
d.
Write an equation for the number of brass players based on the number of woodwind players.
.
Solve for the unknown value using the equation for the constant of proportionality,
7.
and
A recent survey determined that 5 out of 7 teens own portable music devices.
8. If 3500 teens were surveyed, how many owned portable music devices?
.
9. In a special survey group, 1510 teens owned portable music devices. How many teens were in this survey
group?
____ 10. A painter mixes cans of green and white paint to make the shade needed for painting a bedroom. The table
shows the proportions he can use for the color he wants.
Which is the direct variation equation for this relationship?
a.
Green Cans
6
10
16
18
White Cans
9
15
24
27
b.
c.
d.
____ 11. If two quantities vary directly, which must be true of a graph showing the relationship between them?
a.
b.
c.
d.
The graph is a curve.
The graph increases from left to right.
The points on the graph are connected.
The points of the graph form a straight line, and the line must pass through the origin.
Analyze each table shown. Determine if the relationship is proportional. If the relationship is proportional,
calculate the constant of proportionality to justify your answer.
12. 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
13. 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
Solve each using the equation for the constant of proportionality,
and x  21
14.
15. k  0.3 and y  18
16. 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.
b.
How many days did it take Chevrolet to produce 1190 Corvettes? Show your work.
c.
Complete the table.
Time (days)
7
21
Number of Corvettes
1190
952
d.
Use the data in the table to complete the graph.
e.
Is the data in the graph discrete or continuous? Explain your reasoning.
f.
Does it make sense to connect the points in this graph? If so, connect them. Explain your reasoning.
g.
What is the ratio of Corvette production to time?
h.
Does the number of Corvettes vary directly with time? Explain your reasoning.
.
17. 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
a.
1260
Define variables for the quantities that are changing in this problem situation.
.
b.
Analyze the table to determine if the relationship is proportional. Explain your reasoning and state a
constant of proportionality if possible.
.
c.
What does the constant of proportionality represent in this problem?
.
d.
Write an equation showing the relationship between the number of hours worked, the number of
bricks installed, and the constant of proportionality.
.
e.
Use your equation from part (d) to determine how many bricks Maya can install in 5.5 hours.Show
your work.
.
f.
Use your equation from part (d) to determine how many hours it will take Maya to install 840 bricks.
Show your work.
.
18. 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 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
a.
Graph each point from the table on the graph shown. Label the x- and y-axis and name your graph.
b.
What do you notice about the points you plotted?
.
c.
Determine if there is a constant of proportionality. Pick two points from the graph to form the ratio
Unit 2 Study Guide: Direct Variation and Constant of Proportionality
Answer Section
1. ANS:
Of the students surveyed, 350 eat school lunch at least once a week.
PTS: 2
TOP: Pre Test
2. ANS:
a.
d  18 t
REF: 2.3
NAT: 7.RP.2.b | 7.RP.2.c
KEY: constant of proportionality
b.
The ant will walk 3 meters in 10 minutes.
c.
The ant walked 1.25, or
d.
, hours.
The constant of proportionality represents how fast the ant can walk in meters per hour, or
PTS: 1
TOP: Pre Test
3. ANS:
REF: 2.6
.
NAT: 7.RP.2.a | 7.RP.2.b | 7.RP.2.c
Yes. The relationship is proportional because
, or , of the 6th graders play two or more sports and
, of the 7th graders play two or more sports. There is a constant of proportionality,
, or
PTS: 1
4. ANS:
a.
REF: 2.2
NAT: 7.RP.2.a
TOP: Post Test
Total Number of Students
Students Who Like Action Films
90
15
180
30
270
45
360
60
b.
PTS: 1
REF: 2.1
NAT: 7.RP.1 | 7.RP.2.a | 7.RP.2.d
TOP: Post Test
KEY: direct variation (direct proportional) | origin
5. ANS:
The graph does not show a direct variation because it does not go through the point (0, 0).
PTS: 1
TOP: Post Test
6. ANS:
a.
REF: 2.4
NAT: 7.RP.2.a | 7.RP.2.b | 7.RP.2.c | 7.RP.3
There are 30 woodwind players in the orchestra.
b.
There are 8 brass players in the orchestra.
c.
Let b equal the number of brass players in the orchestra, and let w equal the number of woodwind
players.
d.
PTS: 1
REF: 2.3
NAT: 7.RP.2.b | 7.RP.2.c
TOP: Mid Ch Test KEY: constant of proportionality
7. ANS:
PTS: 1
REF: 2.3
NAT: 7.RP.2.b | 7.RP.2.c
TOP: Mid Ch Test KEY: constant of proportionality
8. ANS:
Of the 3500 teens surveyed, 2500 owned portable music devices.
PTS: 1
TOP: End Ch Test
9. ANS:
REF: 2.3
NAT: 7.RP.2.b | 7.RP.2.c
KEY: constant of proportionality
There were 2114 teens in the special survey group.
PTS:
TOP:
10. ANS:
NAT:
11. ANS:
TOP:
12. ANS:
1
REF: 2.3
NAT: 7.RP.2.b | 7.RP.2.c
End Ch Test KEY: constant of proportionality
A
PTS: 1
REF: 2.7
7.RP.2.a | 7.RP.2.b | 7.RP.2.c | 7.RP.2.d
TOP: Standardized Test
D
PTS: 1
REF: 2.2
NAT: 7.RP.2.a
Standardized Test
No. This relationship is not proportional because
is not equal to
. The ratio of boys who 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 constant
of proportionality.
PTS: 1
13. ANS:
REF: 2.2
NAT: 7.RP.2.a
No. This relationship is not proportional. While
does equal
TOP: Assignment
, they do not equal
. 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 constant of proportionality.
PTS: 1
14. ANS:
REF: 2.2
NAT: 7.RP.2.a
TOP: Assignment
PTS: 1
REF: 2.2
NAT: 7.RP.2.a
TOP: Assignment
15. ANS:
PTS: 1
16. ANS:
a.
REF: 2.2
NAT: 7.RP.2.a
Chevrolet produced 238 cars in 7 days.
b.
It took Chevrolet 35 days to produce 1190 Corvettes.
c.
Time (days)
Number of Corvettes
7
238
21
714
17
35
1190
28
952
TOP: Assignment
d.
e.
The data is continuous because it is possible to work for only part of one day.
f.
Yes. It makes sense to connect the points because the data is continuous.
g.
The rate of Corvette production to time is
h.
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.
PTS: 1
REF: 2.1
NAT: 7.RP.1 | 7.RP.2.a | 7.RP.2.d
TOP: Assignment KEY: direct variation (direct proportional) | origin
17. ANS:
a.
Sample answer. Let h be the time in hours, and let b be the number of bricks.
b.
This relationship is proportional, because the ratio of bricks installed to hours worked is the constant
210. The constant of proportionality is 210.
c.
The constant of proportionality represents the number of bricks Maya can install in one hour.
d.
b  210h
e.
b  210h
b  210(5.5)
b  1155
Maya can install 1155 bricks in 5.5 hours.
f.
It will take Maya 4 hours to install 840 bricks.
PTS: 1
TOP: Assignment
18. ANS:
a.
REF: 2.3
NAT: 7.RP.2.b | 7.RP.2.c
KEY: constant of proportionality
b.
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).
c.
Answers will vary. Sample answer is given.
The two points I chose are (1,10) and (2,12).
Because the two ratios are not equivalent, there is no constant of proportionality. This is a nonproportional relationship.
PTS: 1
TOP: Assignment
REF: 2.5
NAT: 7.RP.2.a | 7.RP.2.b | 7.RP.2.c | 7.RP.2.d