LESSON
9.1
Name
Inverse Variation
9.1
Class
Date
Inverse Variation
Essential Question: What does it mean for one variable to vary inversely as another variable?
Texas Math Standards
A2.6.L Formulate and solve equations involving inverse variation.
Explore
A2.6.L
You know that the area A of a rectangle with length ℓ and width w is given by
the formula A = ℓw. If you hold the width constant and allow the length to
vary, the area will vary also. For instance, if w = 2, then the equation relating
A and ℓ is A = 2ℓ, A is said to vary directly as ℓ and 2 is called the constant of
variation. In general, any equation of the form y = ax, where a is a nonzero
constant, is a direct variation, and its graph is a line that passes through the
origin. For instance, the graph of A = 2ℓ is shown. The graph is a ray rather
than a line because length and area are nonnegative quantities.
Formulate and solve equations involving inverse variation.
Mathematical Processes
A2.1.A
Apply mathematics to problems arising in everyday life, society, and
the workplace.
Resource
Locker
Investigating Inverse Variation
Area
The student is expected to:
0
In this Explore, you will consider what happens to the length and width
of a rectangle if you hold the area constant.
Language Objective
8
7
6
5
4
3
2
1
A
ℓ
1 2 3 4 5 6 7 8
Length
2.C.4, 2.I.4, 2.I.5 3.G.3, 4.G
Explain to a partner how to tell the difference between inverse variation
and direct variation.

Let the area of a rectangle be 12 square units. Complete the table, which lists
possible positive-integer lengths and widths for the rectangle. Note that it
doesn’t matter whether the width is less than, equal to, or greater than the
length.
Length
Width
1
12
2
6
3
4
3
2
1
4
6
ENGAGE
If y varies inversely as x, then the relationship
between the variables is given by the equation
a where a is a nonzero constant. For positive
y = __
x
real-world quantities x and y, the value of y
approaches 0 as the value of x increases without
bound, and the value of y increases without bound
as the value of x approaches 0.
PREVIEW: LESSON
PERFORMANCE TASK
© Houghton Mifflin Harcourt Publishing Company
Essential Question: What does it mean
for one variable to vary inversely as
another variable?
12
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A2_MTXESE353947_U4M09L1.indd 455
for one variab
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7
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Area
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Turn to these pages to
find this lesson in the
hardcover student
edition.
7 8
4 5 6
1 2 3
Length
Length
1
Width
12
6
2
3
4
6
12
4
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2
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© Houghto
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Publishin
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View the Engage section online. Discuss the photo
and ask students to describe the quantities they can
vary in order to fit all their songs on the MP3 player.
Then preview the Lesson Performance Task.
Class
Lesson 1
455
Module 9
L1.indd
7_U4M09
SE35394
A2_MTXE
455
Lesson 9.1
455
2/21/14
2:56 AM
2/21/14 2:56 AM
C
D
On the coordinate plane shown, draw each of the
rectangles having the lengths and widths in the table from
Step A. Each rectangle should have its lower-left corner at
the origin. The first rectangle, having a length of 1 and a
width of 12, is already drawn for you.
Using the coordinate plane in Step B, draw a smooth
curve through the upper-right corners of the rectangles.
See graph in part B.
Write an equation that gives w in terms of ℓ.
12
w = __
ℓ
Width
B
12
11
10
9
8
7
6
5
4
3
2
1
w
0
EXPLORE
Investigating Inverse Variation
FOCUS ON MATH CONNECTIONS
Have students consider the coordinates of the
upper-right corner of each rectangle as they make a
table of these values and each area. Elicit that the area
is constant and that the width and length of each
rectangle are related.
ℓ
1 2 3 4 5 6 7 8 9 10 11 12
Length
Reflect
1.
In the direct variation equation A = 2ℓ, the value of A increases as the value of ℓ increases. For the
equation that you wrote in Step D, what can you say about the value of w as the value of ℓ increases?
As the value of ℓ increases, the value of w decreases.
\
Explain 1
Formulating and Solving Inverse Variation Equations
An inverse variation is a relationship between two variables x and y that can
be written in the form y = __ax where a ≠ 0. In this relationship, y is said to vary
inversely as x, and a is called the constant of variation.
QUESTIONING STRATEGIES
Suppose you continued making thinner and
thinner rectangles with a constant area. Could
there ever be a rectangle with a height or width of 0?
Will the graph ever actually touch or cross the line
x = 0? Will the graph ever actually touch or cross the
line y = 0? Why or why not? No; the product of the
length and width is constant. If x or y is 0, then the
product must be 0.
y
If x and y represent positive real-world quantities, then the graph of y = __ax is
a Quadrant I curve that passes through points of the form (x, __ax ) and has the
following end behavior:
a
x, x
• As x → 0, y → +∞.
• As x → +∞, y → 0.
x
a to solve problems, you may find it helpful
To use the equation y = _
x
to rewrite it as xy = a.
© Houghton Mifflin Harcourt Publishing Company
EXPLAIN 1
Formulating and Solving Inverse
Variation Equations
AVOID COMMON ERRORS
Module 9
456
Some students may connect the points to points on
the axis. Ask students if they can explain these points.
Remind them that dividing a positive number by a
positive number will never result in a value of 0.
Lesson 1
PROFESSIONAL DEVELOPMENT
A2_MTXESE353947_U4M09L1.indd 456
Integrate Mathematical Processes
2/21/14 12:31 AM
This lesson provides an opportunity to address Mathematical Process
TEKS A2.1.A, which calls for students to “apply mathematics to problems arising
in everyday life, society, and the workplace.” Students learn to recognize
relationships that can be modeled by using inverse variation equations. Students
then use their models to solve problems.
Inverse Variation 456
Example 1
QUESTIONING STRATEGIES

How can you use a data pair to create the
inverse variation equation? Explain. The
product of the x- and y-values is k, the constant of
variation. Write the equation xy = k, and then solve
for y.
Write an equation relating the variables and use it to answer the question.
The time t (in days) that it takes a theater crew to set up a stage for a musical varies
inversely as the number of workers w. If 30 workers can set up the stage in 4 days, how
many days would it take if only 24 workers are available?
Write an equation relating w and t by finding the constant of variation, a.
a
t = __
w
Write the general equation.
a
Substitute known values. 4 = __
30
120
So, an equation is t = ___
w .
120 = a
Solve for a.
Find the amount of time needed to set up the stage with 24 workers.
120
120
___
Using the equation t = ___
w , substitute 24 for w to get t = 24 = 5.
So, 24 workers can set up the stage in 5 days.

The time t (in hours) that it takes a group of volunteers to clean up a city
park varies inversely as the number of volunteers v. If 10 volunteers can
clean up the park in 6 hours, how many volunteers would be needed to
clean up the park in 4 hours?
Write an equation relating v and t by finding the constant of variation, a.
_
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Blend
Images/Alamy
a
Write the general equation. t = v
a
Substitute known values. 10 = _
6
Solve for a. 60 = a
60
So, an equation is t = _
v .
Find the number of volunteers needed to clean up the park in 4 hours.
Using the equivalent equation vt = 60 , substitute 4 for t and solve for v to
get v = 15 .
So,
15
volunteers are needed to clean up the park in 4 hours.
Reflect
2.
Discussion Using the equivalent equation xy = a for an inverse variation, explain why
the point (q, p) must be on the graph of the equation if the point (p, q) is.
If the point (p, q) is on the graph of xy = a, then x = p, y = q, and pq = a. Using the
commutative property of multiplication to rewrite pq = a as qp = a shows that when
x = q and y = p, the point (q, p) is also on the graph of xy = a.
3.
Discussion What does the fact that for every point (p, q) on the graph of y = __ax the
point (q, p) is also on the graph tell you about the symmetry of the graph?
The graph is symmetric across the line y = x.
Module 9
457
Lesson 1
COLLABORATIVE LEARNING
A2_MTXESE353947_U4M09L1 457
Peer-to-Peer Activity
Have students work in pairs and give each pair two number cubes. Have one
student in each pair first toss the number cubes and determine the product k, then
instruct the other student in graphing xy = k in the first quadrant. Have students
switch roles and repeat the exercise for a different value of k on the same grid. The
pairs should then agree on how the two graphs are related and why. One graph
will be inside the other.
457
Lesson 9.1
1/11/15 9:21 AM
Your Turn
EXPLAIN 2
Write an equation relating the variables and use it to answer the question.
4.
Kim lives in a suburb and drives to work in a city. The time t (in hours) it takes her to get
to work varies inversely with her average driving speed s (in miles per hour). When she
averages 20 miles per hour in heavy traffic, it takes her 1.5 hours to get to work. How long
would the trip take if her average driving speed is 50 miles per hour in light traffic?
Distinguishing between Inverse
Variation and Direct Variation
_
_
a
t= s
a
1.5 =
20
30 = a
30
So, an equation relating s and t is t = __
s.
CONNECT VOCABULARY
Relate inverse to opposite and inverse direction to
opposite direction. As one variable increases, the other
decreases, so their behaviors are opposites.
30
30
__
_3
Using the equation t = __
s , substitute 50 for s and get t = 50 = 5 .
So, if Kim’s average driving speed is 50 miles per hour, she can drive to work in
_3 hour, or 36 minutes.
5
5.
QUESTIONING STRATEGIES
Boyle’s law says that the volume V of a gas held in a container at a constant temperature
varies inversely with the pressure P on the gas. The volume of a particular gas is 8 liters at a
pressure of 3 atmospheres. What is the pressure when the volume is 6 liters?
a
V=
P
a
8=
3
24 = a
Consider direct variation equations and
inverse variation equations. How are they
alike? How are they different? They are alike in that
both have two variables and a constant of variation.
They are different in that the variables in a direct
variation equation have a constant proportion, and
the variables in an inverse variation equation have a
constant product.
_
_
24
So, an equation relating V and P is V = __
.
P
Using the equivalent equation PV = 24, substitute 6 for P and solve for V to get V = 4.
So, when the volume of the gas is 6 liters, the pressure is 4 atmospheres.
Explain 2
As you know, you can rewrite the inverse variation equation y = __ax as xy = a. The alternative form of the equation
gives you a way to check for inverse variation in a table of data: If the products of the paired values of the variables are
constant (or nearly constant), then inverse variation exists.
y
A direct variation equation has the form y = ax, which you can rewrite as __x = a. So, to check for direct
variation in a table of data, see if the ratios of the paired values of the variables are constant.
Module 9
458
If the point (a, b) is on the graph of an inverse
variation equation, what is another point?
How do you know? Another point is (b, a). The
product of a and b is ab, which is constant. So when
ab = a.
x = b, y = ___
b
© Houghton Mifflin Harcourt Publishing Company
Distinguishing between Inverse Variation and
Direct Variation
Lesson 1
DIFFERENTIATE INSTRUCTION
A2_MTXESE353947_U4M09L1 458
Visual Cues
1/11/15 9:28 AM
Visual learners may benefit from finding the value of g(x) = __2x as x increases from
2 ___
, 2 , __2 , __2 , __2 .
-4 to 4: -__24 , -__22 , -__21 , -___
0.5 0.5 1 2 4
As they simplify each fraction in the sequence, have them follow the path of the
corresponding points on the graph of g(x) = __2x , paying close attention to what
happens as x increases from –0.5 to 0.5.
0.4
0.4
___
Repeat the process for g(x) = ___
x as the values of x increase from -2 to 2: - 2 ,
0.4 ___
0.4 ___
0.4
0.4
0.4
0.4 ___
, -___
, -___
, -___
, 0.4 , ___
, 0.4 , ___
, 0.4
. Have them pay close attention to what
-___
2
1
0.4
0.2
0.1 0.1 0.2 0.4 1
happens as x increases from -0.1 to 0.1.
Inverse Variation 458
Example 2
INTEGRATE TECHNOLOGY
A graphing calculator can be used to check
irrational solutions to an equation. Graph the
corresponding function and compare its zeros to the
approximate decimal equivalent of the solutions.

Determine whether the two variables vary inversely or directly. Then write
an equation and use it to answer the question.
The table gives the total cost (in dollars) of various numbers of tickets to a school play. What
is the cost of 12 tickets?
3
7
Tickets, t
C
Check both the products tC and the ratios __
t to see which
12
28
Total cost, C
are constant.
C
Because the ratios __
are constant, C varies directly as t.
t
COGNITIVE STRATEGIES
When contrasting direct variation with inverse
variation, think of direct variation as a pair of up
arrows, ⇈, and an inverse variation as a pair of up
and down arrows, ⇅.
Write the equation.
C = 4t
Find the cost for 12 tickets.
C = 4(12) = 48
So, 12 tickets cost $48.

Because the products st are constant, s varies inversely as t.
600
Write the equation.
t= s
_
© Houghton Mifflin Harcourt Publishing Company
Find the time needed to get to the destination when driving at an
average speed of 48 miles per hour.
36
t
C
tC
C
_
t
3
12
36
4
7
28
196
4
9
36
324
4
The table gives the time (in hours) needed to drive to a
destination at various average speeds (in miles
per hour). What is the time needed to get to the
Average Speed, s
destination when driving at an average speed of
Time, t
48 miles per hour?
Check both the products st and the ratios _st to see
which are constant.
9
12
15
20
50
40
30
s
t
st
_t
s
12
50
600
15
40
600
_
2.6
20
30
600
1.5
_
4.16
600
t = _ = 12.5
48
12.5 hours to get to the destination when driving at an average speed of 48 miles per hour.
So, it takes
Reflect
6.
State the real-world significance of the constant of variation in Part A and Part B.
In Part A, the constant of variation represents the cost per ticket ($4). In Part B,
the constant of variation represents the distance driven (600 miles).
Module 9
459
Lesson 1
LANGUAGE SUPPORT
A2_MTXESE353947_U4M09L1 459
Communicate Math
Have students work in pairs. Provide each pair with several sticky notes or index
cards, each containing an equation; some are inverse variation equations while
others are not. The first student picks up a card, and explains why it is or is not an
inverse variation equation. Encourage students to use the terms decrease, increase,
and varies inversely.
459
Lesson 9.1
1/13/15 7:48 AM
Your Turn
ELABORATE
Determine whether the two variables vary inversely or directly. Then write an equation
and use it to answer the question.
7.
The table gives the number of pages read from a book for various amounts of time (in hours) spent
reading. How many pages can be read in 6 hours?
p
Because the ratios _ are 20, p varies
t
directly as t, and p = 20t.
When t = 6, p = 20(6) = 120, so the
number of pages read in 6 hours is 120.
8.
Time, t
2
3
5
Pages Read, p
40
60
100
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Technology
Discuss with students how to use a graphing
calculator to find a solution to an inverse-variation
equation. They can find the solution by using the
graphing capabilities with the trace function or by
generating a table and finding the appropriate value.
The table gives the amount (in dollars) that each person owes when a group of people buys a birthday
present for a friend. How much does each person owe when 8 people chip in for the present?
Because the products pA are 80, A varies
80
inversely as p, and A = __
p.
80
When p = 8 , A = __
= 10, so when a group
8
of 8 people chips in to buy the present,
each person owes $10.
Number of People, p
2
4
5
Amount each Owes, A
40
20
16
AVOID COMMON ERRORS
When graphing a function that represents a
real-world inverse variation situation, students may
graph both branches of the function even though the
context of the problem requires that only positive
values be used. Have students consider whether the
graph also consists of the branch in Quadrant III.
Have them identify several points on that branch and
analyze whether those points represent solutions that
make sense in the context of the situation. Encourage
students to read every real-world problem carefully,
identifying practical restrictions on values for any
variables and solutions.
Elaborate
9.
How do direct variation and inverse variation differ?
When y varies directly as x, the equation that relates y to x is y = ax for some nonzero
a
constant a. When y varies inversely as x, the equation that relates y to x is y = _
x for some
nonzero constant a. When x and y represent nonnegative real-world quantities (for direct
variation) or positive real-world quantities (for inverse variation), y varying directly as
x means that y increases as x increases, whereas y varying inversely as x means that y
decreases as x increases.
the value of x when the value of y is known. It is also helpful when determining whether
a set of paired data represents inverse variation, since inverse variation exists when the
products of the paired numbers are constant (or nearly constant).
11. Essential Question Check-In If x and y represent positive real-world quantities that vary inversely,
what happens to y as x approaches 0? What happens to y as x increases without bound?
As x approaches 0, y increases without bound. As x increases without bound,
© Houghton Mifflin Harcourt Publishing Company
10. What is another way to write the inverse variation equation y = __ax ? How is this equation helpful?
The alternative form of the inverse variation equation is xy = a. It is helpful in finding
QUESTIONING STRATEGIES
k
What is the domain of the function y = __
x,
where k is a positive real number? {x | x ≠ 0}
y approaches 0.
SUMMARIZE THE LESSON
Module 9
A2_MTXESE353947_U4M09L1.indd 460
460
Lesson 1
2/21/14 12:30 AM
How can you determine whether a situation
can be modeled by inverse variation and then
determine the model? Explain. If the variables are
related by having a constant product, they are
related by inverse variation. The constant product is
k.
k, so the equation is y = __
x
Inverse Variation 460
Evaluate: Homework and Practice
EVALUATE
ASSIGNMENT GUIDE
Practice
Explore Activity
Investigating Inverse Variation
Exercises 1–2
Example 1
Formulating and Solving Inverse
Variation Equations
Exercises 3–12
Example 2
Distinguishing between Inverse
Variation and Direct Variation
Exercises 13–23
Let the area of a rectangle be 16 square units. On the coordinate plane shown, draw
all rectangles having positive-integer lengths and widths. (Note that it doesn’t matter
whether the width is less than, equal to, or greater than the length.) Each rectangle
should have its lower-left corner at the origin. The first rectangle, having a length of
1 and a width of 16, is already drawn for you. After drawing the rectangles, draw a
smooth curve through their upper-right corners.
2.
Write an equation of the curve that you drew in
Evaluate 1. The equation should give the width
w of a rectangle in terms of the length ℓ.
w=
16
_
ℓ
Width
Concept and Skills
1.
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
• Online Homework
• Hints and Help
• Extra Practice
w
ℓ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Length
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
3.
© Houghton Mifflin Harcourt Publishing Company
Have students consider an inverse variation in which
the length of a rectangle is based on the area of the
rectangle in square feet being a constant product of
the length and width in feet, but then all units are
changed to inches. An inverse variation model can
still be used, but the change in units will result in a
change in the inverse variation constant.
Write an equation relating the variables and use it to answer the question.
Given that y varies inversely as x, and y = 8 when x = 3, what is the value of y
when x = 12?
Find the value of the constant of variation, a.
a
a
y= x 8=
24 = a
3
24
So, an equation relating x and y is y = x .
24
When x = 12, y =
= 2.
12
Given that y varies inversely as x, and y = 10 when x = 4, what is the value of x when
y = 8?
_
_
_
4.
_
Find the value of the constant of variation, a.
a
a
y = x 10 =
40 = a
4
40
So, an equation relating x and y is y = x . Using the equivalent equation
xy = 40, substitute 8 for y and solve for x to get x = 5.
_
_
_
Module 9
A2_MTXESE353947_U4M09L1.indd 461
Exercise
461
Lesson 9.1
Lesson 1
461
Depth of Knowledge (D.O.K.)
Mathematical Processes
1–2
1 Recall of Information
1.E Create and use representations
3–4
2 Skills/Concepts
1.E Create and use representations
5–12
2 Skills/Concepts
1.A Everyday life
13–22
2 Skills/Concepts
1.E Create and use representations
23
2 Skills/Concepts
1.A Everyday life
24
3 Strategic Thinking
1.G Explain and justify arguments
2/21/14 12:30 AM
5.
The time t (in hours) that it takes a pump to empty a tank of water varies inversely
with the pumping rate r (in gallons per hour). If it takes 3 hours to empty a tank of
water when the pumping rate is 80 gallons per hour, how long does it take to empty
the tank when the pumping rate is 60 gallons per hour?
AVOID COMMON ERRORS
When graphing an inverse variation function without
real-world restrictions, students may graph only the
branch of the function in the first quadrant. Remind
students that negative numbers are included in the
domain, and that k can be represented by the product
of two negative numbers. For example, for
6 , (-2, -3) is a solution.
y = __
x
Find the value of the constant of variation, a.
a
a
240 = a
t= r 3=
80
240
240
So, an equation relating t and r is t = r . When r = 60, t =
= 4.
60
So, it takes 4 hours to empty the tank when the pumping rate is 60 gallons per hour.
_
6.
_
_
_
The number of flowers f that a gardener can plant along a border of a garden varies
inversely with the distance d (in inches) between the flowers. If the gardener can fill
the border with 30 flowers planted 12 inches apart, how far apart should the gardener
plant 36 flowers?
a
a
f=
30 =
360 = a
12
d
360
So, an equation relating f and d is f =
. Using the equivalent equation fd = 360,
d
substitute 36 for f and solve for d to get d = 10.
_
_
_
So, the gardener can plant 36 flowers 10 inches apart.
7.
The number of presents p that Tim can afford to buy varies inversely with their
average cost C (in dollars). If Tim can afford 5 presents when their average cost is $12,
what average cost would 3 presents have?
a
a
p=
60 = a
5=
12
C
60
So, an equation relating p and C is p =
.
C
Using the equivalent equation Cp = 60, substitute 3 for p and solve for C to get C = 20.
_
_
_
So, 3 presents would have an average cost of $20.
8.
_
_
© Houghton Mifflin Harcourt Publishing Company
A club rents a bus for a trip. The cost C (in dollars) that each person pays to cover the
cost of the bus varies inversely with the number of people p who go on the trip. It will
cost $30 per person if 50 people go on the trip. How much will it cost per person if
40 people go on the trip?
a
a
C = p 30 =
1500 = a
50
1500
1500
So, an equation relating C and p is C = p . When p = 40, C =
= 37.5.
40
So, it will cost $37.50 per person if 40 people go on the trip.
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Exercise
Depth of Knowledge (D.O.K.)
Mathematical Processes
25
3 Strategic Thinking
1.F Analyze relationships
26
3 Strategic Thinking
1.A Everyday life
1/11/15 9:32 AM
Inverse Variation 462
9.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Critical Thinking
For a fundraiser, members of the booster club wash cars by hand. The time t (in
minutes) it takes to wash a car varies inversely as the number of people p who are
washing the car. If 2 people can wash a car in 20 minutes, how many people would
be needed to wash a car in 8 minutes?
a
a
40 = a
t = p 20 =
2
40
So, an equation relating t and p is t = p . Using the equivalent equation
pt = 40, substitute 8 for t and solve for p to get p = 5.
So, 5 people are needed to wash a car in 8 minutes.
_
When determining whether an inverse variation can
be used to model a situation, students should
recognize that a value of 0 for either the dependent or
independent variable indicates that an inverse
variation is not an appropriate model.
_
_
10. A gear with 32 teeth meshes with a gear with 40 teeth so that when one gear revolves,
the other one does as well. The number of revolutions r that each gear makes varies
inversely with the gear’s number of teeth t. When the gear with 32 teeth makes
5 revolutions, how many revolutions does the gear with 40 teeth make?
a
a
5=
160 = a
r=
t
32
160
160
So, an equation relating r and t is r = ___
. When t = 40, r = ___
= 4.
t
40
_
_
So, the gear with 40 teeth makes 4 revolutions.
11. Music The frequency f (in hertz) of a vibrating guitar string varies inversely as
its length ℓ (in centimeters). If a guitar string 65 centimeters long vibrates with a
frequency of 110 hertz, at what frequency would the guitar string vibrate when the
guitarist reduces the string’s length to 22 centimeters?
a
a
f=
7150 = a
110 =
65
ℓ
7150
7150
So, an equation relating f and ℓ is f =
. When ℓ = 22, f =
= 325.
22
ℓ
So, the string vibrates with a frequency of 325 hertz when the string length is 22 centimeters.
© Houghton Mifflin Harcourt Publishing Company
_
_
_a
_
a
100 = a
50 =
2
d
100
100
So, an equation relating m and d is m =
. When d = 5, m =
= 20.
d
d
So, a force of 20 pounds applied at a distance of 5 feet from the fulcrum counterbalances
the force of 50 pounds applied at a distance of 2 feet from the fulcrum.
m=
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Lesson 9.1
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12. Physics When a lever is placed on a fulcrum and a force is applied at each end, the
lever will be in balance as long as the magnitude m (in pounds) of each force and
the distance d (in feet) of each force from the fulcrum satisfy an inverse variation
relationship. If a force of 50 pounds is applied to one end of a lever at a distance of
2 feet from the fulcrum, what force must be applied to other end, which is 5 feet from
the fulcrum, to bring the lever into balance?
Module 9
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463
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Lesson 1
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Determine whether the two variables vary inversely or directly.
Then write an equation and use it to answer the question.
13. Given the table of data, what is y when x = 15?
Because the products xy are 30, y varies
30
inversely as x, and y = __
x.
x
1
3
4
y
30
10
7.5
x
2
5
6
y
30
75
90
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Technology
When generating a table for an inverse variation, an
error should be expected when x = 0, since the
inverse variation function is undefined due to
division by 0. When tracing along the curve of an
inverse variation function, there should be no value
of the function when x = 0.
30
When x = 15, x = __
= 2.
15
14. Given the table of data, what is y when x = 10?
Because the ratios _x are 15, y varies directly
y
as x, and y = 15x.
When x = 10, y = 15(10) = 150.
15. The table gives the cost (in dollars) per person
when friends share in renting a mountain cabin
for a weekend. What is the cost per person
when 6 friends rent the cabin?
Number of People, p
2
3
5
Cost per Person, C
90
60
36
180
Because the products pC are 180, C varies inversely as p, and C = ___
p .
180
When p = 6, C = ___
= 30, so the cost per person when 6 friends rent the cabin is $30.
6
16. The table gives the amount of gas (in gallons) used
when driving a car various distances (in miles)
on highways. What amount of gas is used when
driving 336 miles on highways?
Distance Driven, d
112
140
224
Amount of Gas, g
4
5
8
d
1
are __
, g varies directly as d, and g = __
.
Because the ratios _
28
28
d
g
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Elisseeva/Shutterstock
336
When d = 336, g = ___
= 12, so 12 gallons of gas are used when driving 336 miles on
28
highways.
17. The table gives the cost (in dollars) of renting a rowboat at a
lake for various amounts of time (in hours). What is the cost of
renting a rowboat for 3.5 hours?
Time, t
2
2.5
3
Rental Cost, C
28
35
42
C
Because the ratios _
t are 14, C varies directly as t, and C = 14t.
When t = 3.5, C = 14(3.5) = 49, so the cost of renting a rowboat for 3.5 hours is $49.
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Lesson 1
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Inverse Variation 464
18. The table gives the total working time (in hours) that it takes a crew of painters to paint a house.
How much time does it take 6 painters to paint the house?
MODELING
Students should be aware that data is not always
modeled perfectly by a function, especially when it is
approximate or rounded. An inverse variation
function can still be used when the product of the
variables is not exactly constant. For example, when
the products are 37.488, 37.504, 37.496, and so on, an
inverse variation model with a value of k of 37.5 is
appropriate.
Number of Painters, p
2
3
5
Total Working Time, t
48
32
19.2
96
Because the products pt are 96, t varies inversely as p, and t = __
p.
96
When p = 6, t = __
= 16, so 6 painters can paint the house in 16 hours.
6
19. The table gives the speed of a bicycle (in miles per hour) when a cyclist pedals at various rates (in
revolutions per minute of the pedals) with the bicycle in a particular gear. What is the bicycle’s speed
when the cyclist pedals at a rate of 72 revolutions per minute?
Pedaling Rate, r
30
60
90
Bicycle’s Speed, s
5
10
15
1
r
Because the ratios _sr are _
, s varies directly as r, and s = _
.
6
6
72
__
When r = 72, s = = 12, so the bicycle’s speed is 12 miles per hour when the cyclist pedals
6
at a rate of 72 revolutions per minute.
20. The table gives the number of laptops sold in a month when a store sells a particular model of laptop at various
prices (in dollars). How many laptops would be sold in a month when the store sells the laptop for $500?
Price of Laptop, p
600
720
800
Number of Laptops Sold, ℓ
60
50
45
Because the products pℓ are 36,000, ℓ varies inversely as p, and ℓ = _____
p .
36,000
When p = 500, ℓ = _____
= 72, so 72 laptops would be sold in a month when the store sells
500
36,000
the laptop for $500.
© Houghton Mifflin Harcourt Publishing Company
21. The table gives the number of small figurines that can be placed on a display shelf for various distances
(in centimeters) between them. How many figurines can be placed on the shelf when they are
24 centimeters apart?
Distance between Figurines, s
10
15
20
Number of Figurines, f
12
8
6
120
Because the products sf are 120, f varies inversely as s, and f = ___
s .
120
When s = 24, f = ___
= 5, so 5 figurines can be placed on the shelf when they
24
are 24 centimeters apart.
22. The table gives the amount of water (in gallons) coming out of a garden hose for various amounts of time
(in minutes). How much water comes out of the garden hose in 20 minutes?
Time, t
4
9
16
Amount of Water, w
72
162
288
w
Because the ratios __
t are 18, w varies directly as t, and w = 18t.
When t = 20, w = 18(20) = 360, so 360 gallons of water come out of the garden
hose in 20 minutes.
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Lesson 1
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23. Determine whether each of the following situations represents direct variation or
inverse variation. Select the correct answer for each lettered part.
CONNECT VOCABULARY
a. When traveling a
fixed distance, the
travel time is a function
of average speed.
Direct variation
Inverse variation
b. For items that are priced
the same, the total cost is
a function of the number
of items purchased.
Direct variation
Inverse variation
Direct variation
Inverse variation
Direct variation
Inverse variation
c.
When traveling at a
fixed speed, the distance
traveled is a function
of the travel time.
d. For a fixed amount of
money, the number of
identical items that can be
purchased is a function of
the cost per item.
In English, the word variation is a noun form of the
verb vary. The variables in a direct variation or
inverse variation vary, meaning they are different
from another, or from each other. In a direct
variation, they vary together in the same way, both
increasing proportionally. In an inverse variation,
they vary in opposition, one decreasing in proportion
to an increase in the other.
H.O.T. Focus on Higher Order Thinking
24. Justify Reasoning If y varies directly as x, what happens to the value of y when the value of x is
doubled? If y varies inversely as x, what happens to the value of y when the value of x is doubled? Use the
general equations for direct and inverse variation to justify your reasoning.
If y varies directly as x, then doubling the value of x doubles the value of y.
Justification: Since y = ax, replacing x with 2x gives this result: a(2x) = 2(ax) = 2y,
so the value of y is doubled.
If y varies inversely as x, then doubling the value of x halves the value of y.
25. Explain the Error Boyle’s law says that the volume V of a gas held in a container at a constant
temperature varies inversely with the pressure P on the gas. The volume of a particular gas is 6 liters at a
pressure of 2 atmospheres. Jerry says that if the pressure is changed to 3 atmospheres, the volume will be
9 liters. Explain and correct his error.
Jerry used the equation V = 3P to obtain his result, but this equation represents
a direct variation between V and P. Since V varies inversely as P, the correct
12
equation is V = __
p , so when P = 3 atmospheres, V = 4 liters.
Module 9
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© Houghton Mifflin Harcourt Publishing Company
()
a
a
__
_1 _a _1
Justification: Since y = _
x , replacing x with 2x gives this result: 2x = 2 x = 2 y, so the
value of y is halved.
Lesson 1
2/21/14 12:30 AM
Inverse Variation 466
26. Represent Real-World Situations The volume V of a gas varies inversely as the pressure P and directly
as the temperature T. A particular gas has a volume of 10 liters, a temperature of 300 kelvins, and a
pressure of 1.5 atmospheres. If the gas is compressed to a volume of 7.5 liters and is heated to a temperature
of 360 kelvins, what will the pressure be? Write an equation and use it to answer the question.
GRAPHIC ORGANIZERS
Have students create a graphic organizer with
examples of equations and graphs of inverse
variations, direct variations, and neither.
aT
The general equation is V = __
. Find the value of the constant of variation, a.
P
aT
V = __
P
a(300)
10 = _____
1.5
CRITICAL THINKING
0.05 = a
0.05T
So, the equation is V = ____
.
P
a if x = y? How do you
What is a solution to y = __
x
a , so x 2 = a, and x = a is
know? When x = y, x = __
x,
a solution.
―
Find the value of P when V = 7.5 and T = 360.
7.5 =
0.05(360)
_
7.5P = 18
P
P = 2.4
JOURNAL
So, the pressure will be 2.4 atmospheres when the volume is 7.5 liters and the temperature
is 360 kelvins.
© Houghton Mifflin Harcourt Publishing Company
Have students explain how to create an equation for a
situation that can be modeled by inverse variation,
and solve a problem by using the model, if they are
given a table of values.
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Lesson 1
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Lesson Performance Task
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
You have a collection of CDs whose songs you want to transfer to your new MP3 player.
The MP3 player has 32,000 megabytes (MB) of storage. An average song lasts 4 minutes and
requires 40 MB of storage on a CD.
Ask students to discuss the effect of a in the function
a , where a is the storage in MB on a player.
S(s) = __
s
Have them determine the value for a in this problem
and discuss any general limitations on a in similar
problems. Ask them if it makes sense for a to be
negative in this real-world context.
a. Write a function that gives the number S of songs that your MP3 player can store
if the average file size of a song is s MB.
b. If you were to transfer the songs from your CDs in their current size, how many
songs could you expect to store on your MP3 player?
c. The MP3 file format compresses songs without much loss in the quality of the
sound. Typically, the MP3 format compresses a CD file to a tenth of its size. How
many songs from your CDs can you expect to store as MP3 files on your MP3
player?
QUESTIONING STRATEGIES
d. In general, if you could use a file format that compresses the songs on your CDs
by a factor of __1f where f > 1, how many songs in that format can you expect to
store on your MP3 player?
In what situation can you use an inverse
variation function? When one variable
increases as the other decreases so that their
product is constant.
a. The number S of songs that can be stored on the MP3 player is the
available amount of storage, 32,000 MB, divided by the average file size s
(in megabytes). So, S(s) = _____
s .
32,000
b. S(40) = _____
= 800, so you can expect to store 800 songs in their current
40
32,000
What determines the shape of the graph for an
a ? The
inverse variation function S(s) = __
s
parameter a determines the curvature of the graph.
The smaller that a is, the more that the graph bends
in toward the origin.
size of 40 MB.
c. A CD file that is 40 MB becomes an MP3 file of 4 MB. S(4) = _____
= 8000,
4
32,000
so you can expect to store 8000 songs as MP3 files on your MP3 player.
40
d. A CD file that is 40 MB becomes a file in the new format of __
MB.
(f)
f
f
f
© Houghton Mifflin Harcourt Publishing Company
32,000
32,000f
32,000f
40
S __
= _____
= _____
= _____
= 800f, so you could expect to
40
40
__
__
40
·f
store 800f songs in the new format on your MP3 player.
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Lesson 1
EXTENSION ACTIVITY
A2_MTXESE353947_U4M09L1 468
Have students research typical memory capacities for a flash drive, a tablet device,
and a desktop computer. For each device, have students write the function
a for the number S of songs of file size s that can be stored on a device with
S(s) = __
s
memory capacity a. Have students graph all three functions on the same graph
together with the function for the MP3 player from the Performance Task, and ask
them to compare the devices.
1/11/15 9:41 AM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Inverse Variation 468