2. Mathematics – Grade 9
Quarter 2 – Self-Learning Module 3: Solving Direct Variations
First Edition, 2020
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Published by the Department of Education Division of Pasig City
Printed in the Philippines by Department of Education – Schools Division of
Pasig City
Development Team of the Self-Learning Module
Writer: Rochelle B. Laranang
Editor: Cristina DC. Prado
Reviewer (Language): Ma. Cynthia P. Badana, Ma. Victoria Peñalosa
(Technical): Glady O. Dela Cruz
Illustrator: Edison P. Clet
Layout Artist: Anthony G. Fijo , Clifchard D. Valente
Management Team: Ma. Evalou Concepcion A. Agustin
OIC-Schools Division Superintendent
Carolina T. Rivera
OIC-Assistant Schools Division Superintendent
Victor M. Javeña EdD
Chief, School Governance and Operations Division and
OIC-Chief, Curriculum Implementation Division
Education Program Supervisors
Librada L. Agon EdD (EPP/TLE/TVL/TVE)
Liza A. Alvarez (Science/STEM/SSP)
Bernard R. Balitao (AP/HUMSS)
Joselito E. Calios (English/SPFL/GAS)
Norlyn D. Conde EdD (MAPEH/SPA/SPS/HOPE/A&D/Sports)
Wilma Q. Del Rosario (LRMS /ADM)
Ma. Teresita E. Herrera EdD (Filipino/GAS/Piling Larang)
Perlita M. Ignacio PhD (EsP)
Dulce O. Santos PhD (Kindergarten/MTB-MLE)
Teresita P. Tagulao EdD (Mathematics/ABM)
4. Introductory Message
For the facilitator:
Welcome to the Mathematics 9 Self-Learning Module on Solving Direct
Variations!
This Self-Learning Module was collaboratively designed, developed and
reviewed by educators from the Schools Division Office of Pasig City headed by its
Officer-in-Charge Schools Division Superintendent, Ma. Evalou Concepcion A.
Agustin, in partnership with the City Government of Pasig through its mayor,
Honorable Victor Ma. Regis N. Sotto. The writers utilized the standards set by the K
to 12 Curriculum using the Most Essential Learning Competencies (MELC) in
developing this instructional resource.
This learning material hopes to engage the learners in guided and independent
learning activities at their own pace and time. Further, this also aims to help learners
acquire the needed 21st century skills especially the 5 Cs, namely: Communication,
Collaboration, Creativity, Critical Thinking, and Character while taking into
consideration their needs and circumstances.
In addition to the material in the main text, you will also see this box in the
body of the module:
As a facilitator you are expected to orient the learners on how to use this
module. You also need to keep track of the learners' progress while allowing them to
manage their own learning. Moreover, you are expected to encourage and assist the
learners as they do the tasks included in this module.
Notes to the Teacher
This contains helpful tips or strategies
that will help you in guiding the learners.
5. For the Learner:
Welcome to the Mathematics 9 Self-Learning Module on Solving Direct
Variations!
This module was designed to provide you with fun and meaningful
opportunities for guided and independent learning at your own pace and time. You
will be enabled to process the contents of the learning material while being an active
learner.
This module has the following parts and corresponding icons:
Expectations - This points to the set of knowledge and skills
that you will learn after completing the module.
Pretest - This measures your prior knowledge about the lesson
at hand.
Recap - This part of the module provides a review of concepts
and skills that you already know about a previous lesson.
Lesson - This section discusses the topic in the module.
Activities - This is a set of activities that you need to perform.
Wrap-Up - This section summarizes the concepts and
application of the lesson.
Valuing - This part integrates a desirable moral value in the
lesson.
Posttest - This measures how much you have learned from the
entire module.
6. 1. Translate statements involving direct variations to mathematical equations
2. Solve problems involving direct variations
Directions: Read each question carefully and choose the letter that corresponds to
the correct answer.
1) Which of the following situations represents direct variation?
A. The number of hours to do a job as to the number of people doing the job
B. The gas consumed as to the length of cooking time.
C. The time traveled by a car as to its speed.
D. The atmospheric pressure as to the altitude.
2) If y varies directly as x and y is 15 when x is 3, what is the value of the constant
of variation?
A. k = 3 C. k = 5
B. k = 4 D. k = 6
3) Which of the following equations illustrates direct variation?
A. y = 0.25x C. y = xz + 5
B. y =
3
𝑥
D. y = √6𝑥𝑧
4) If a varies directly as b and b is 108 when a is 12, find b when a is 7.
A. 1296 C. 63
B. 84 D. 9
5) A recipe for salad requires 250 mL of cream for every 300 mL of condensed milk.
How many liters of cream would be needed for 1.5 liters of milk?
A. 7.5 liters C. 2.25 liters
B. 5 liters D. 1.25 liters
EXPECTATIONS
PRETEST
7. Directions: Find the value of the unknown of each of the following proportions:
1) 6 : a = 4 : 10
2) b : 6 = 8 : 24
3) 4 : 20 = c : 15
4)
2
3
=
8
𝑑
5)
𝑒
20
=
3
7
Direct Variation
Direct variation exists whenever the ratio
between two quantities is a nonzero constant. The
statement “y varies directly as x”, “y is directly
proportional to x” and “y is proportional to x” may be
written as y = kx where k is the constant of variation.
This means, as one quantity increases, the other
quantity also increases. Similarly, as one quantity
decreases, the other quantity also decreases.
Example 1: Write the equation for the statement “the circumference (C) of a circle
varies directly with its radius r”.
Solution:
Using the given variables, the corresponding equation will be C = kr where k
is the constant of variation.
Example 2: Write in symbols: “r varies directly with the square root of s” with k as
the constant of variation.
Solution:
Using the given variables, the corresponding equation will be 𝑟 = 𝑘√𝑠.
LESSON
RECAP
(References: Wikimedia Commons. Accessed June 13, 2020.
https://commons.wikimedia.org/wiki/File:HK_Central_結志
街_Gage_Street_market_雞蛋_Chicken_n_鴨蛋
_Duck_Eggs_on_sale_March-2012.jpg; Free Images & Free
Stock Photos. Accessed June 13, 2020.
https://pxhere.com/en/photo/661869.
8. Example 3: If y varies directly as x and y is 32 when x is 4, find the variation constant
and the equation of the variation.
Solution:
y = kx Translate “y varies directly as x” into an equation
32 = k(4) Substitute the given values in the equation
32
4
= k Solve for k
8 = k
Therefore, the constant of variation is 8 and the equation is y = 8x.
Example 4: If a varies directly as b and a is 54 when b is 9, what is the value of
a when b is 12?
Solution:
a = kb Translate “a varies directly as b” into an equation
54 = k(9) Substitute the first given set of values in the equation
54
9
= k Solve for k
6 = k
Therefore, the constant of variation is 6 and the equation is a = 6b.
a = 6b
a = 6(12) Solve for a when b is 12
a = 72
Hence, a = 72 when b is 12.
Example 5: If p varies directly as the square of r and p is 324 when r is 9, what is
the value of r when p is 100?
Solution:
p = k𝑟2
Translate “p varies directly as the square of r” into an
equation
324 = k(9)2
Substitute the first given set of values in the equation
324 = k(81) Solve for k
324
81
= k
4 = k
Thus, the constant of variation is 4 and the equation is p = 4𝑟2
p = 4𝑟2
100 = 4(𝑟)2
Solve for r when p is 100
9. 100
4
= 𝑟2
√25 = √𝑟2 Get the square root of both sides
± 5 = r
Then, r = ± 5 when p is 100.
Example 6: Anna is going to make a leche flan for dessert. She knows that 8 egg
yolks are needed for 300 mL of condensed milk, but she is planning to use all the
450 mL condensed milk that she has. How many egg yolks does she need to maintain
the proportion of the recipe?
Method 1 Solution:
Let e be the number of egg yolks needed
m be the milliliter for the condensed milk
e = km working equation for the variation
8 = k(300) substitute the first given set of values to the equation
𝑘 =
8
300
solve for k
𝑘 =
2
75
Therefore, the constant of variation is
2
75
and the equation is e =
2
75
𝑚
e =
2
75
𝑚
e =
2
75
(450) solve for e when m is 450
e = 12
Hence, there are 12 egg yolks needed for 450 ml of milk.
Method 2 Solution:
Since the constant of variation is k =
𝑦
𝑥
, we can establish the proportion that
𝑦1
𝑥1
=
𝑦2
𝑥2
where x will be the number of egg yolks and y will be the milliliters of milk
needed.
Let 𝑥1 = 8 egg yolks
𝑦1 = 300 ml condensed milk
𝑦2 = 450 ml condensed milk
𝑥2 = number of egg yolks needed
𝑦1
𝑥1
=
𝑦2
𝑥2
establish the proportion needed
300
8
=
450
𝑥2
substitute the given information
10. 300𝑥2 = 450(8) apply cross product property of proportions
𝑥2 =
450 (8)
300
solve for 𝑥2
𝑥2 = 12
Thus, there are 12 egg yolks needed for 450 ml of milk.
ACTIVITY 1: LET’S PRACTICE!
Directions: Translate the following direct variation statements into equations.
1) m varies directly as n.
2) d varies directly as the square of c.
3) j varies directly as the cube root of h
4) The fare (f) cost is directly proportional to the distance (d) traveled.
5) The pressure (p) at the bottom of the sea is directly proportional to the
depth (d) reached.
ACTIVITY 2: KEEP PRACTICING!
Directions: Determine the constant of variation of the following statements:
1) y varies directly as x. If y is 36 then x is 9.
2) R varies directly as S. If R is 56 then S is 8.
3) Q is directly proportional to P. If Q is 150 then P is 10.
4) c is directly proportional to the square of b. If c is 16 then b is 3.
5) m is proportional to the square root of n. If m is 125 then n is 25.
ACTIVITY 3: TEST YOURSELF!
Directions: Write each of the following statements into direct variation equation and
then solve for the unknown.
1) If a varies directly as b and a is 15 when b is 5, find a when b is 8.
2) If e is directly proportional to f and e is 40 when f is 16, find e when f is 30.
3) If r varies directly as the square of t and r is 20 when t is 2, find r when t is 7.
ACTIVITIES
11. 4) If g is proportional to the cube root of h and g is 24 when h is 27, find h when
g is 16.
5) If w varies directly as v + 4 and w is 42 when v is 2, find v when w is 63.
6) A recipe for cake requires 3 teaspoonfuls of yeast for 5 cupfuls of flour. How
much yeast would be needed for 18 cupfuls of flour?
7) Liza, a professional typist, can type 70 words per minute. How many minutes
will it take her to finish a manuscript of 1610 words?
8) At a local market, a kilo of pork costs 240 pesos. If you plan to buy 2
1
2
kilos,
how much will you have to pay?
9) The weight on the moon varies directly with the weight on Earth. A person
that weighs 70 kg on Earth weighs 12 kg on the moon. How much will a person
who weigh 120 kg on Earth weighs on the moon?
10) The distance that a body falls from rest varies directly as the square of the
time it falls. If a ball falls 180 feet in two seconds, how far will the ball fall in
five seconds?
How are you going to identify whether the given equation or situation
illustrates direct variation? How do we solve problems involving direct variation?
In a family, the amount of money that can be spent varies directly with the
household income. With the recent problem in the COVID-19 pandemic such as the
implementation of lockdown, a lot of families lose their source of income. Many
families do not have enough savings to be used during emergency needs. How will
you describe your family’s economic situation during the lockdown? What valuable
experiences have you learned from this situation? As a young person, how can you
help in managing your family expenses? Write your answer in your notebook.
WRAP-UP
VALUING
12. Directions: Read each question carefully and choose the letter that corresponds to
the correct answer.
1) Which of the following situations DOES NOT represent direct variation?
A. The area of a circle as to the length of its radius.
B. The salary as to the number of hours worked.
C. The time to reach the destination as to the speed of the car.
D. The cost of fare as to the distance traveled.
2) If y varies directly as x and y is 36 when x is 9, what is the value of the constant
of variation?
A. k = 3 C. k = 5
B. k = 4 D. k = 6
3) Which of the following equations illustrates direct variation?
A. z = √6𝑥𝑦 C. f = 3gh
B. p =
7
𝑟
D. a =
3𝑏
4
4) If m varies directly as n and m is 60 when n is 5, find n when m is 108.
A. 15 C. 9
B. 12 D. 5
5) Phoebe deposits P 5,000 in her savings account every 3 months. How many years
will it take her to have a savings of P 100,000?
A. 5 years C. 25 years
B. 15 years D. 60 years
POSTTEST
14. References
BOOKS:
Bryant, Merden L., Bulalayao, Leonides E., Callanta, Melvin M., Cruz, Jerry D., De
Vera, Richard F., Garcia, Gilda T. and Javier, Sonia E., et. al. Mathematics
Grade 9 Learner’s Material. First Edition. Pasig City: Department of
Education, 2014.
Diaz, Zenaida B., Mojica, Maharlika P., Suzara, Josephine L., Mercado, Jesus P.,
Esparrago, Mirla S. and Reyes, Nestor Jr. V. Next Century Mathematics 9.
Quezon City: Phoenix Publishing House, Inc., 2014.
Dilao, Soledad Jose and Bernabe, Julieta G. Intermediate Algebra Textbook for
Second Year. Pilot Edition. Quezon City: JTW Corporation, 2002.
Oronce, Orlando A. and Mendoza, Marilyn O. E-Math 9. Revised Edition. Manila: Rex
Book Store Inc., 2015.
ONLINE RESOURCES:
Free Images & Free Stock Photos - PxHere. 2020,
https://pxhere.com/en/photo/661869. (Accessed June 13, 2020).
Pike, Scott. Welcome to MAT 120/121/122 Intermediate Algebra. Mesa Community
College, 2020. https://www.mesacc.edu/~scotz47781/mat120/notes/
variation/direct/ direct.html. (Accessed June 17, 2020).
Practical Algebra Lessons │ Purplemath, 2020, https://www.purplemath.com/
modules/variatn.html. (Accessed June 17, 2020).
Wikimedia Commons. 2020, https://commons.wikimedia.org/wiki/File:HK_Central
_結志_Gage_Street_market_雞蛋_Chicken_n_鴨蛋
_Duck_Eggs_on_sale_March-2012.jpg. (Accessed June 13, 2020).
