9
MATHEMATICS
Guided Learning Activity Kit
Direct Variation and Inverse Variation
Quarter 2- Week 1
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Mathematics – Grade 9
Guided Learning Activity Kit
Direct Variation and Inverse Variation
Quarter 2- Week 1
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DIRECT VARIATION AND
INVERSE VARIATION
Introduction
One of the interesting topics in Mathematics with a variety of applications in real
life situations is variations. In this Guided Learning Activity Kit, different fields of
study will act as a mathematical model in the application of variation. Variation is
very useful in Science and Physics as well as in Business.
This week’s lessons deal with two specific types of relationships between variables
-- direct variation and inverse variation.
Learning Competency
-Illustrates situations that involve the following variations: (a) direct; (b)
inverse; (c) joint; (d) combined. (M9AL-lla-1)
-Translates into variation statement a relationship between two
quantities given by: (a) a table of values; (b) a mathematical equation; (c) a graph,
and vice versa. (M9AL-lla-b-1)
-Solves problems involving variation. (M9AL-lla-c-1)
Objectives
At the end of this Guided Learning Activity Kit, you are expected to:
1. define direct variation and inverse variation;
2. translate into variation statement a relationship between two quantities
given by: (a) a table of values; b) a mathematical equation; (c) a graph, and
vice versa; and
3. solve problems involving direct variation and inverse variation.
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Review
Let’s start the lesson by recalling the concept of evaluating algebraic
expressions. To evaluate an algebraic expression means to find the value of the
expression when the variable is replaced by a given number. To evaluate an
expression, we substitute the given number for the variable in the expression using
order of operations. For example, if we evaluate x + 6 when x = -5, substitute -5 for
x in the expression and simplify, then -5 + 6 = 1. When x =-5, the expression x + 6 =
has a value of 1.
Evaluate the expression 4y – 3, when y = 2. Again substitute 2 for y in the
expression and simplify the expression, we have 4(2) – 3 = 5. When y = 2, the expression
4y – 3 has a value of 5.
Recall also the process of solving linear equations.
For example: Find y when x = 5 in the equation y = 8x.
Solution:
y = 8x
y = 8(5)
y = 40
How did you get the value of y in the equation?
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Discussion
DIRECT VARIATION
A direct variation, also called direct proportion is a relationship between
two variables x and y that can be written as y = kx, where k ≠ 0. This situation occurs
when the ratio of two variables is constant.
When y = kx, we say that y varies directly with x or y is directly proportional to x.
When z = kt, we say that z varies directly with t or z is directly proportional to t.
k is called the constant of variation.
A simple problem situation will show clearly the idea of direct relationship
between quantities.
A motorist drives along the Expressway at a constant speed of 80 km per hour.
How far will he go in two hours? In 5 hours? In 3.5 hours? In t hours?
The table below shows the relation between the distance travelled by the motorist
and the period of time spent.
Time
hours
in
Distance
in km
1
2
3
4
5
t
80
160
240
320
400
80t
The longer the time, the farther the distance traveled, that is, the distance varies
directly as the time t or distance and time are in direct variation.
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑖𝑚𝑒
=
80
160
240
80𝑡
=
=
=
= 80
1
2
3
𝑡
This means that the ratio of any specific pair of corresponding values of
distance to time is constant. Ratios of successive pairs of values may be equated to
form a proportion. Thus, two variables in direct variation are sometimes said to be
in direct proportion or to vary proportionally.
Mathematical statements that describe relationships can be translated into
equations, where k represents the constant of variation or constant of
proportionality.
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Illustrative Examples
1. Translate the statements into formulas expressing direct variation. Use k as
the constant of variation.
a. P varies directly as Q.
Solution:
P = kQ or k =
𝑃
𝑄
b. The cost c of meat varies directly as the number n of kilos bought.
Solution:
c = kn
or k =
𝑐
𝑛
c. The weight (w) of an athlete is directly proportional to his height (h).
Solution:
w = kh
or k =
𝑤
ℎ
Concept
If the ratio between two quantities is constant, it means that one
quantity varies directly as the other quantity, or the two quantities are
in direct variation.
In symbols, y = kx or k =
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𝒚
𝒙
2. The table below shows that the distance d varies directly as the time t. Find
the constant of variation and the equation that describes the relation.
Time
hours
in
Distance
in km
1
2
3
4
5
6
10
20
30
40
50
60
Solution:
d = kt
Using one pair of value from the table, (2, 20), then
20 = 2k
1
1
( ) (20) = (2k)( )
2
2
k = 10
Therefore, the constant of variation is 10.
Form the mathematical equation by substituting 10 in the statement d = kt.
d = 10t
3. The graph that shows the relation d = 10t is a line which also describes a
direct variation in the form y = kx where k = 10 and the constant of variation.
60
Distance (d)
50
40
30
20
10
0
1
2
3
4
5
Time (t)
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4. If y varies directly as x and y = 48 and x = 6, find the variation constant and
the equation of variation.
Solution:
a. Find k or the variation constant.
y = kx
48 = 6k
1
1
6
6
( )(48) = (6k)(
)
k=8
b. Find the equation of variation:
y = kx
y = 8x
5. The variable x varies directly with y, and y = 3 when x = 15. Find the constant
of variation. What is x when y = 15?
Solution:
a. Find k or the constant of variation:
x = ky
15 = 3k
1
1
( )(15) = (3k)( )
3
3
k=5
b. What is x when y = 15?
Method 1: Using the equation of variation
x = 5y
equation of the variation
x = 5(15) solving x when y = 15
x = 75
Method 2: Using Proportion
𝑥
Since k = , we can establish a proportion such that
𝑦
= 15, y₁ = 3 and y₂ = 15.
Substituting the value to the proportion
𝒙₁
𝒚₁
15
3
=
=
𝒙₂
𝒚₂
𝑥₂
15
3x₂ = (15)(15)
3x₂ = 225
x₂ = 75
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𝒙₁
= 𝒙₂
where x₁
𝒚₁
𝒚₂
6. Solve the given problems:
a. The weight (E) of an object on Earth varies directly as the weight of the
same object (m) on the moon. A 300-pound object would weigh 48 pounds
on the moon. How much would a 65-pound object weigh on the moon?
Solution:
Step 1:
y = kx → E = km
Step 2: 300 = 48k
Step 3:
k=
k=
300
48
25
4
Step 4: E = km
65 =
25
(
4
)m
260 = 25m
m=
260
25
m = 10.4 pounds
b. The circumference (c) of a circle varies directly with its diameter (d). If the
circumference of a 7-cm in diameter is 7𝜋, what is the circumference of
the circle whose diameter is 10cm? 15cm? 18cm?
(c = 𝜋𝑑)
Solution:
The circumference of a circle varies directly with its diameter
can be expressed as
𝑘=
Step 1: 𝑘 =
𝑘=
𝑐
𝑑
or c = kd
𝑐
𝑑
7𝜋
7
=𝜋
Step 2: c = 𝜋𝑑
c = 𝜋10
c = 10𝜋 𝑐𝑚, when d = 10 cm
Step 3: c = 𝜋𝑑
c = 𝜋15
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c = 15𝜋 𝑐𝑚, when d = 15 cm
Step 4: c = 𝜋𝑑
c = 𝜋18
c = 18𝜋 𝑐𝑚, when d = 18 cm
INVERSE VARIATION
Inverse variation refers to the relationship between two variables where as
one variable increases, the other decreases by the same factor. The relationship
between the two variables can be modeled by the equation y =
𝑘
𝑥
where k is a
constant of proportionality. The product of x and y will always be equal to k.
From the word inverse, the opposite is the result. An increase of one
quantity brings about the corresponding decrease in the other quantity and vice
versa.
Consider a rectangle whose area is 36 cm². If the width (w) is 2 cm, then its
length (l) is 18 cm. What happens when the width is increased by 3 cm? 4 cm? The
table below shows the relation between the length and the width of the rectangle.
Width (w)
2
3
4
5
Length (l)
18
12
9
7.2
Area (A)
36
36
36
36
When the width (w) increased, there is a corresponding decreased in the
length (l) and the Area (A) remains the same.
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How does the graph of an inverse variation look like?
The graph below shows an inverse variation.
Source: geogebra.org
The graph is neither a straight line nor a parabola but a half of a two-part curve
known as the hyperbola.
Concept
Whenever the product of corresponding values of two
quantities is a constant, then one quantity varies inversely
as the other.
In symbols, k = xy; y =
𝒌
𝒙
𝟏
or y = k( )
𝒙
This means that y is inversely proportional to x, or x is
inversely proportional to y.
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Illustrative examples
1. Translate the statements into formulas expressing direct variation. Use k as
the constant of variation.
a. The altitude (h) of a triangle with constant area varies inversely as its base (b).
Solution:
𝑘
h=
or k = bh
𝑏
b. The number of persons (n) needed to do a job varies inversely as the number
of days (d) to finish the job.
Solution:
𝑘
n=
or k = nd
𝑑
c. The acceleration (a) of a car is inversely proportional to its mass (m).
Solution:
𝑘
a=
𝑚
or k = am
2. Find the constant of variation and write the equation representing the
relationship between the quantities of the table of values.
x
1
2
3
4
y
2
1
2
3
1
2
Solution:
a. Find the constant of variation.
Since the value of x increases and the value of y decreases, it
shows the relationship of an inverse variation.
k = xy or y =
𝑘
𝑥
2
Using a pair of values from the table (3, ), then
3
2
k = 3( )
3
k=2
b. Write the equation of the variation.
Using y =
y=
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𝑘
𝑥
2
𝑥
and k = 2,
3. Find the constant of variation and write the equation representing the
relationship between the quantities of the graph.
Source: geogebra.org
Solution:
a. Find the constant of variation.
y=
𝑘
𝑥
or k = xy
k = 12 using (2, 6) as one of the points on the graph
b. Write the equation of the variation.
y=
y=
𝑘
𝑥
12
𝑥
4. Find the equation and solve for k: y varies inversely as x and y = 12 when x
= 5.
Solution:
The relation y varies inversely as x translates to y =
values to find k.
y=
12 =
𝑘
𝑥
. Substitute the
𝑘
𝑥
𝑘
5
k = (12)(5)
k = 60
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The equation of variation is y =
60
𝑥
.
5. If y varies inversely as x and y = 10 when x = 2, find y when x = 10. This
concerns two pairs of values of x and y which may be solved in two ways.
Solution:
Method 1:
a. Find k or the constant of variation:
k = xy
k = (10)(2)
k = 20
The equation of variation is y =
20
𝑥
b. What is y when x = 10?
Using the equation of variation.
y=
y=
20
𝑥
20
10
y=2
Method 2:
Since k = xy, then for any pairs x and y, x₁y₁ = x₂y₂
If we let x₁ = 2, y₁ = 10, and x₂ = 10, find y₂.
Substituting the values,
x₁y₁ = x₂y₂
(2)(10) = 10y₂
y₂ =
20
10
y₂ = 2
Hence, y = 2 when x = 10
6. Solve the given problems.
a. For a given gas of constant temperature, the pressure (P) varies inversely
as the volume (V). If P = 6 when V = 24, find P when V = 36.
Solution: Using Method 2
P₁V₁ = P₂V₂
(6)(24) = P₂(36)
144 = 36P₂
P₂ =
144
36
P₂ = 4
Hence, P = 4 when V = 36.
Similarly, P=4 when V=36 using method 1. Try it!
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b. Find the number of days the 10 workers can complete a job if 5 workers
can complete the same job in 7 days.
Solution: Using Method 1.
The number of days (d) a certain job can be completed varies
inversely as the number of workers (w) assigned to do the job.
Assuming that that each works at the same rate, then d =
a. Find the constant variation k
k = dw
k = (7)(5)
k = 35
d=
35
𝑤
𝑘
𝑤
or k = dw.
, the equation of variation
b. Find d if w = 10.
d=
35
10
1
d = 3.5 or 3
2
1
Hence, d = 3.5 or 3
1
2
when w = 10.
d = 3 days for 10 workers to complete the same job.
2
Similarly, d = 3.5 when w = 10 using method 2. Please try method 2 also!
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Activities
Guided Practice 1
A. Directions: Write an equation for each of the following sentences. Use k as the
constant of variation.
Example: The cost (c) of chicken varies directly as its weight (w).
Answer: c = kw
1. The electric current (I) passing through an electric component varies directly with
the applied voltage (V).
2. The kinetic energy (E) of a moving object is directly proportional to the square of
its speed (V).
3. The time (t) needed to empty a water tank varies inversely with the pumping rate
(r).
4. The amount of time (t) traveled when driving a car varies inversely with the car’s
speed (s).
5. The frequency (f) of a radio signal varies inversely as the wave length (w).
B. Directions: Determine if the tables and graphs below express a direct variation or
inverse variation between the variables. If they do, find the constant of variation and
an equation that defines the relation.
1.
x
1
2
3
4
y
-3
-6
-9
-12
x
1
2
3
4
y
2
1
2
3
1
2
x
-15
10
-20
25
y
-3
2
-4
5
2.
3.
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4.
Source: geogebra.org
5.
Source: geogebra.org
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Guided Practice 2
A. Write an equation for the following. Use k as the constant of variation:
1. y varies directly as x if y = 28 when x = 7
2. y varies directly as x if y = 0.8 when x = 0.5
3. y varies directly as x if y = 48 when x =6
4. a varies inversely b if a = 24 when b = 0.3
5. a varies inversely b if a = 5 when b =
2
5
B. Solve the following problems:
1. If m varies directly as n and m = 35 when n = 42, find m when n = 78.
2.
If y varies inversely as x, and y = 48 when x = 10, find y when x = 32.
3. If c is inversely proportional to d, and c = 30 when d = 9, find d when
c = 540.
4. The electric current I (in amperes) in an electric circuit varies directly as
the voltage V. When 21 volts are applied, the current is 7 amperes. Find
the current when 12 volts are applied.
5. The number of hours required to finish a certain job varies inversely as
the number of persons on the job. If 8 persons require 9 hours to finish
the job, how long will it take for 24 persons to finish the job?
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Independent Practice
Directions: Answer the following. Write your answer on a separate sheet of paper.
1. Beverly travels by car for 5 hours. The distance she covers varies directly as
the rate.
a. Assuming a uniform rate for the whole trip, make a table for the following
rates: 60km/hr; 80km/hr; 100km/hr; 110km/hr; 120km/hr.
rate (km/hr)
distance (d)
b. Write the equation of variation showing the relation of the distance to the
rate.
2. Make a graph of the variation expressed by the condition “if y varies directly
as x and y = 10 when x = 2, what is the constant of variation?
3. If y is directly proportional to x, and y = 40, when x = 16, find y when x = 30.
4. If p varies directly as q, and p = 25, when q = 20, find q when p = 140.
5. If y is inversely proportional to x, and y = 8 when x = 2, find y when x = 4.
6. Find x when y = 32, given that x varies inversely as y and x = 132 when y = 4.
7. The weight w of an object on Mars is directly proportional to its weight B on
earth. A man who weighs 118.75 kg on earth weighs 47.5 kg on Mars. Find
the weight of a man on Mars if he weighs 98 kg.
8. The bases of triangles having equal areas are inversely proportional to their
altitudes. The base b of a certain triangle is 12 cm and its altitude h is 15 cm.
Find the base of a triangle whose altitude is 20 cm.
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9. A photocopy machine can finish 500 pages in 1 minute. How many books of
250 pages can the machine copy in 20 minutes?
10.At 60 kilometers per hour it takes Loida 10 hours to travel from her house to
their house in the province. How long will it take her if she travels at 80 kilometers
an hour?
Assessment
Directions: Read each statement/question carefully. Write the letter of the correct
answer on your answer sheet.
1. What is the constant (k) in the inverse variation y =
400
𝑥
?
a. x
b. y
c. 400
d. Not enough information given
2. The cost c varies directly as the number n of pencils is written as
a. c = kn
b. k = cn
c. n =
𝑘
d. c =
𝑐
𝑘
𝑛
3. The speed r of a moving object is inversely proportional to the time t travelled
is written as
a. r = kt
b. r =
𝑘
c. t = kr
𝑡
d. t =
4. Which is an example of direct variation?
a. xy = 10 b. y =
2
c. y = 5x
𝑡
d.
2
𝑦
𝑘
𝑟
=x
5. y varies directly as x and y = 32 when x = 4. Find the constant of variation.
a. 128
b. 28
c. 36
d. 8
6. Which of the following describes an inverse variation?
a.
x
2
3
4
5
y
5
10
3
5
2
2
x
1
2
3
4
y
5
10
15
20
x
40
30
20
10
y
8
6
4
2
b.
c.
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d.
x
4
8
10
12
y
2
4
5
6
7. If y varies directly as x and y = 12 when x = 4, find y when x = 12.
a. 3
b. 4
c. 36
d. 48
8. What happens to T when h is doubled in the equation T = 4h?
a. T is halved
b. T is tripledc. T is doubled
d. T becomes zero
9. The time it takes to complete a trip varies inversely with the speed of the bus.
When Darcy’s school bus travels at 30 miles per hour, it gets from her home
to school in 12 minutes. What is the speed of Darcy’s bus if it completes the
same trip in 18 minutes?
a. 15 miles/hour
c. 25 miles/hour
b. 20 miles/hour
d. 30 miles/hour
10.The height of a wave in the ocean varies directly with the second that pass by.
At 4 seconds, the wave is 6 feet high. How many seconds will give you a wave that
is 10 feet high?
a. 8.67 seconds b. 7.70 seconds
c. 7.76 seconds
d. 6.67 seconds
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Reflection
Answer the following:
1. Determine some quantities in your home that show direct and
inverse relationships. Formulate a problem using these
quantities and solve it.
2. Write a journal about the importance of direct and inverse
variations in your daily life. Cite in your journal your
experiences wherein direct and inverse variations are applied.
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References
Bryant, Merden L., et al. Mathematics Grade 9 Learners Material. Vibal
Group Inc. 2014.
Dilao, Soledad Jose, et al. Algebra II Functional Approach. 1253 G. Araneta
Ave., Quezon City: Vibal Publishing House Inc. 2002.
Orince, Orlando A. and Mendoza, Marilyn O. Exploring Mathematics II.
Sampaloc, Manila: Rex Book Store Inc. 2003.
Solis, Reynaldo Escano. Exploration and Application Workbook (2 nd Year).
Sta. Ana, Manila: VICARISH Publishing & Trading, Inc. 2008.
Teaching Guide on the 2010 Secondary Education curriculum (Mathematics
II). Mady’ Printing (02)664-1158.
https://www.basic-mathematics.com/direct-variation.html
https://www.basic-mathematics.com/inverse-variation.html
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Independent Practice:
1.
a.
r(km/hr)
60
80
100
d(distance) 300 400
500
110
550
120
600
b. d = 5r
2. k = 5
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Activities:
Guided Practice 1:
A.
1.I = KV
2.E = KV²
𝑘
3.t =
𝑟
𝑘
4. t =
𝑠
𝑘
𝑤
5. f =
B.
1. k = -3
y=
y=
1
5
y=
2. k =2
3. k =
2
𝑥
𝑥
5
4. k = 20
5. k = 10
−3
𝑥
20
y=
𝑥
y = 10x
Guided Practice 2:
3. y = 12
4. q = 175
5. y = 4
6. x = 4
7. w = 24.5 kg
8. b = 9 cm
9. x = 40 books
10. 13 hours and 55 minutes
A.
1. k
2. k
3. k
4. k
5. k
Assessment:
=
=
=
=
=
4
1.6
8
1
5
2
y = 4x
y = 1.6x
y = 8x
1
𝑏
a = b or
5
5
a = 2b
B.
1. m = 65
2. y = 15
1
3. d =
2
4. 4 amperes
5. 3 hours
1. C
2. A
3. B
4. C
5. D
6. A
7. C
8. C
9. B
10.D
Key to Corrections
Acknowledgment
The Schools Division of Zambales would like to express its heartfelt gratitude
to the following, who in one way or another, have contributed to the successful
preparation, development, quality assurance, printing, and distribution of the
Quarter 2 Guided Learning Activity Kits (GLAKs) in all learning areas across grade
levels as a response to providing the learners with developmentally-appropriate,
contextualized and simplified learning resources with most essential learning
competencies (MELCs)-based activities anchored on the principles of guided learning
and explicit instruction:
First, the Learning Resources (LR) Development Team composed of the writers
and graphic artists for devoting much of their time and giving their best efforts to
produce these indispensable learning kits which will be used for the implementation
of learning delivery modalities.
Second, the content editors, language reviewers, and layout evaluators who
made up the Division Quality Assurance Team (DQAT) for having carefully evaluated
all GLAKs to ensure quality and compliance to DepEd standards;
Third, the Provincial Government of Zambales, for unceasingly extending its
financial assistance to augment the funds for the printing of these learning resources
to be used by learners and parents at home;
Fourth, the teacher-advisers and subject teachers, in close coordination with
the school heads, for their weekly distribution and retrieval of the GLAKs and for
their frequent monitoring of the learners’ progress through various means; and
Finally, the parents and other home learning facilitators for giving the learners
the necessary guidance and support for them to possibly accomplish the tasks and
for helping them become independent learners.
To deliver learning continuity in this challenging circumstance would not be
possible without your collective effort and strong commitment to serving our
Zambaleño learners.
Again, our sincerest gratitude towards you all!
The Management Team
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For inquiries or feedback, please write or call:
Schools Division of Zambales
Zone 6, Iba, Zambales
Tel./Fax No.: (047) 602 1391
E-mail Address: zambales@deped.gov.ph
Website: www.depedzambales.ph
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