C
\
Second Quarter Module 1
Week 1
SDO TAGUIG CITY AND PATEROS
Introductory Message
For the facilitator:
This module was collaboratively designed, developed and evaluated by the Development and
Quality Assurance Teams of SDO TAPAT to assist you in helping the learners meet the
standards set by the K to 12 Curriculum while overcoming their personal, social, and economic
constraints in schooling.
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.
Furthermore, you are expected to encourage and assist the learners as they do the tasks
included in the module.
For the learner:
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 helped to process the
contents of the learning resource while being an active learner.
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part of the module.
Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer Let’s Try before moving on to the other activities included in the
module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and in checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not hesitate to
consult your teacher or facilitator. Always bear in mind that you are not alone.
We hope that through this material, you will experience meaningful learning and gain deep
understanding of the relevant competencies. You can do it!
Let’s Learn
This module was designed and written with you in mind. It is here to help you master Variation.
The scope of this module permits it to be used in many different learning situations. The
language used recognizes the diverse vocabulary level of students. The lessons are arranged
to follow the standard sequence of the course. But the order in which you read them can be
changed to correspond with the textbook you are now using.
The module is divided into two lessons, namely:
•
•
Lesson 1 – Direct Variation
Lesson 2 – Inverse Variation
After going through this module, you are expected to:
1. illustrate situations that involve the following variations:
(a) direct and
(b) inverse;
2. translate into variation statement a relationship between two quantities given (a) a table
of values;
(b) a mathematical equation;
(c) a graph, and vice versa and
3. solve problems involving variation.
Let’s Try
Directions: Let’s find out how much you already know about this topic. Write only the letter
of the choice that you think is the correct answers. Please answer all items.
1. The cost c varies directly as the number n of pencils is written as
𝑘
A. 𝑐 = 𝑘𝑛
B. 𝑘 = 𝑐𝑛
C. 𝑛 =
𝑐
D. 𝑐 =
𝑘
𝑛
2. The speed r of a moving object is inversely proportional to the time t travelled is written as
𝑘
𝑟
A. 𝑟 = 𝑘𝑡
B. 𝑟 =
C. 𝑡 = 𝑘𝑟
D. = 𝑟
𝑡
3. Which is an example of a direct variation?
2
A. 𝑥𝑦 = 10
B. 𝑦 =
𝑥
𝑘
C. 𝑦 = 5𝑥
D.
4. If 𝑦 varies directly as 𝑥 and 𝑦 = 32 when 𝑥 = 4. Find the constant of variation.
A. 8
B. 36
C. 28
D. 128
2
=𝑥
𝑦
5. Which of the following describes an inverse variation?
A.
C.
B.
D.
6. 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
7. If 𝑦 varies inversely as 𝑥 and 𝑦 =
−2
A. 3
2
B.3
1
3
when 𝑥 = 8, find y when 𝑥 = −4.
C.
−32
3
32
D. 3
8. If 3 men can do a portion of a job in 8 days, how many men can do the same job in 6
days?
A. 7
B. 6
C. 5
D. 4
9. If (𝑥 – 4) varies inversely as (𝑦 + 3) and 𝑥 = 8 when 𝑦 = 2, find 𝑥 when 𝑦 = −1.
A. 20
B. 18
C. 16
D. 14
10. Francisco’s income varies directly as the number of days that he works. If he earns ₱
8,000.00 in 20 days, how much will he earn if he worked 3 times as long?
A. ₱ 26,000
B. ₱ 24,000
C. ₱ 20,000
D. ₱ 16,000
https://forms.gle/W1FHXKhmQEs6JPkZ8L6
Lesson
Direct Variation and
Inverse Variation
1
Let’s Recall
I. Solve for the value of 𝑥.
1.
3.
4.
3 𝑥
=
2 4
6 1
=
𝑥 4
2.
5 𝑥
=
8 2
5.
12: 5 = 8: 𝑥
7: 8 = 𝑥: 4
II. Write the following in mathematical proportion form and solve.
1. If 3 cookies cost ₱3.50, how much will 10 cookies cost?
2. A 5-m high post casts a shadow of 8 meters when the building casts a shadow of
14-m. How high is the building?
3. A team won 5 times in 8 games. At this rate, how many games must the team play to
win 9 games?
4. A certain recipe calls for 2 cups of sugar for every 5 glasses of water. How many
glasses of water are needed for 9 cups of sugar?
Let’s Explore
Occasionally, changes in the values of two variable quantities can be related. When a
change in the value of one quantity links to an expected change in the value of the other
quantity, then we say that the two quantities are related.
Let us begin by investigating some related quantities in very similar situation.
Related Quantities
Which of the following quantities are related?
Socio-economic status and number of friends
Speed and distance covered by a car in an hour
Nature of activity and heartbeat
hard work and success
For sample quantities above, except for socio-economic status and number of friends’ the
other pairs of quantities or variables are related. Can you explain how the quantities are
related?
Let’s Explain, Analyze & Solve
Direct Variation
In previous years, you have learned how to form and solve proportion both direct and
inverse. Variation is the same as proportion. It relates the value of one variable to the value of
the other variable, so that, if a variable 𝑦 varies directly as another variable 𝑥, then there exists
a constant k such that 𝑦 = 𝑘𝑥.
In this case, 𝑦 is said to be directly proportional to 𝑥, and 𝑘 is called the constant of
variation.
In a direct variation, for 𝑘 > 0, as the value of 𝑥 increases the value of 𝑦 also increases.
Similarly, as 𝑥 decreases, 𝑦 also decreases.
Let us consider this relationship:
Example 1. A car travels at 60 kilometers per hour.
How far has it travel after
(a) 1 hour?
(b) 2 hours? (c) 3 hours? (d) 4 hours?
Let us organize the answer to the questions in this table.
time (hr)
1
2
3
4
distance (km)
60
120
180
240
Notice that as the time increases the
distance also increases. And the time
decreases the distance also decreases.
The relationship between time and distance in this example is shown by the equation 𝒅 =
𝟔𝟎𝒕 since the time is multiplied by 60 to obtain the equivalent distance travelled. This
relationship shows a direct variation and the distance travelled (d) by the car varies directly
as the length of time (t).
In this problem, 60 is the constant of variation.
Now let us graph 𝑑 = 60𝑡.
Notice that the graph starts at the origin and rises
from left to right. Actually, the graph of 𝑦 = 𝑘𝑥, 𝑘 > 0,
always goes through the origin and moves upward
to the right and the constant of variation (𝑘) serves
as the slope of the line.
Example 1. If y varies directly as 𝑥 and 𝑦 = 15 when 𝑥 = 3, find 𝑦 when 𝑥 = 5.
Let us first solve for k: If y varies directly as
x, then 𝑦 = 𝑘𝑥 and substituting values of x
and y, we can solve for k.
Substitute the computed value of k and the
second value of y, then solve for x
𝑦 = 𝑘𝑥
15 = 𝑘(3)
𝑘=5
𝑦 = 𝑘𝑥
𝑦 = 5(5)
𝑥 = 25
Example 2. If 𝑥 varies directly as 𝑦 and 𝑥 = 9 when 𝑦 = 3, find x when 𝑦 = 12.
𝑥 = 𝑘𝑦
9 = 𝑘(3)
𝑘=3
Let us first solve for k: If x varies directly as
y, then 𝑥 = 𝑘𝑦 and substituting values of x
and y, we can solve for k.
𝑥 = 𝑘𝑦
𝑥 = 3(12)
𝑥 = 36
Substitute the computed value of k and the
second value of y, then solve for x
To watch a video tutorial on Direct Variation by Anywhere Math (2014), visit this link
https://youtu.be/cFAE8RiKtj8
Example 3. The weight of the human body (𝑏) is directly proportional to the total amount of
water (𝑤) in a human body. A person who weighs 50 kg contains 36 kg of water. How
many kg of water are in a person weighing 58 kg?
What is the working equation? Will it be 𝑤 = 𝑘𝑏 or 𝑏 = 𝑘𝑤?
Yes. Its 𝑏 = 𝑘𝑤, since the amount of water will depend on the
weight of the human body. So,
Then, what’s next? If 𝑘 =
25
and 𝑤 = 58 kg, then how will you
18
get the weight of water? Yes, substitute the computed value of k
and the second weight of the person which is 58, so we have,
Inverse Variation
This table shows the time taken for a car to travel a
distance of 120 km at different speeds.
Speed (km/h)
Time taken (hours)
10
12
20
6
30
4
40
3
1. If the speed of the car increases, will the time
taken increase or decrease?
2. If the speed of the car is doubled, how will the
50
2
60
1
𝑏 = 𝑘𝑤
50 = 𝑘(36)
25
𝑘=
18
𝑏 = 𝑘𝑤
25
58 =
𝑤
18
18
𝑤 = 58 ∙
25
19
𝑤 = 41
𝑘𝑔
25
time taken change? Hint: Compare the time taken when the speeds of the car are 20
km/h and 40 km/h.
3. If the speed of the car is tripled, what will happen to the time taken?
4. If the speed of the car is halved, how will the time taken changed?
5. Hint: Compare the time taken when the speed of the car are 60 km/h and 30 km/h.
6. If the speed of the car is reduced to
1
3
of its original speed, what will happen to the time
taken?
We notice that as the speed of the car increases, the time taken decreases proportionally,
that is, if speed is doubled then the time taken is halved, if the speed is tripled the time taken
is reduced to
1
3
of its original value.
Similarly, as the speed of the car decreases the time taken increases proportionally. And if
the speed is halved, the time taken is doubled, also if the speed is reduced to
1
3
of its
original, then the time taken is tripled.
This relationship is called inverse variation.
As we have seen from previous discussion, variables can vary with another variable in
different ways. One possible way is through inverse variation. Inverse variation is similar to
inverse proportion.
Let 𝑥 and 𝑦 denote two quantities, 𝑦 varies inversely as 𝑥, or 𝑦 is inversely proportional to 𝑥,
if there is a nonzero constant 𝑘 such that:
𝑘
𝑦 = 𝑥, the number k is called the constant of variation.
Example 1: Decide whether the two quantities in the situation are directly or inversely
related.
a. The side of the square related to its perimeter.
b. The number of people sharing a pizza related to the size of slice each one
gets.
c. The age of used cellular phone related to its selling price.
d. The time a painter paints a wall related to the area of the wall painted.
Solutions:
Quantity 1
Quantity 2
Kind of
Relationship
a. Side of
square
Perimeter
Direct
b. Number of
people
Size of slice
Inverse
c. Age of a
cellular
phone
Selling price
Inverse
d. Time to
paint
Area of the
wall
Direct
Explanation
As the side increase the perimeter
increases
As the number of person sharing
pizza increase the size of slice each
one gets decreases
As the age of used cellular phone
increases the selling price
decreases
The more time a painter spends in
painting a wall, the greater the area
of the wall painted
Example 2: A car is travelling a distance of 120 kilometer
a. How long will it take the car to reach its destination if it travels at a speed of 20kph?
40kph? 60 kph? 80 kph?
b. Graph
Solution:
a. Since 𝑡 =
Rate (kph)
Time (hr)
𝑑
𝑟
, we can have the following table of values:
20
6
40
3
60
2
80
1½
b.
Let us find the product of rate(r) and
time (t)
𝑟 ⋅ 𝑡: 20(6) = 120, 4(3) = 120
1
2
60(2) = 120, 80 (1 ) = 20
Note that the product of the two
quantities is constant. The graph at
the right goes down from left to right.
This shows that the travel time
decreases as the rate of the car
increases.
Example 3. If 𝑧 varies inversely as 𝑤, and 𝑧 = 10 when 𝑤 = 5. Find 𝑧 when 𝑤 = 8.
Equation for variation
:
Solving for k
:
Solving for z
:
𝑘
𝑤
𝑘 = 𝑧𝑤  𝑘 = 10(5) 𝑘 = 50
𝑧=
𝑘
50
25
𝑧 = 𝑤 𝑧 = 8  𝑧 = 4
Example 4: A crew of 12 can build a hut in 8 days. How long would it take a crew of 4 to
build the same hut?
Solution:
a. Understanding the problem:
Let : 𝑦 = number of days to build a hut
𝑥 = the number of person in a crew
𝑘 = constant of variation
This situation tells us that the more crew members working, the faster it will take to
build the hut. Thus, if we have lesser crew member the work will consume more time.
This implies that the relation is inversely proportional.
b. Write the equation: 𝑦 =
𝑘
𝑥
c. Solving for k:
𝑘
𝑥
𝑘
8=
12
96 = 𝑘
𝑦=
Substitute the value
Solve for 𝑘
This the constant of variation
𝑦=
d. Solving the equation:
96
𝑥
96
4
𝑦 = 24
𝑦=
This is the equation of variation.
Substitute 4 for 𝑥.
Solve for 𝑦
It will take 24 days for a crew of 4 to build a hut.
Let’s Dig In
Activity 1. Give the equation or formula for each of the following.
To watch a video tutorial on Direct and Inverse Variation by Khan Academy (2011),
visit this link https://youtu.be/92U67CUy9Gc
1.
2.
3.
4.
5.
The circumference (c) of a circle varies directly as the diameter (d).
The area (A) of a parallelogram whose base is 10cm varies directly as its altitude (a) and
its base.
The fare (F) of a passenger varies directly as the distance (d) of his destination.
The distance (D) travelled by a car varies directly as its speed (s).
The area (A) of a square varies directly as the square of its side s.
Activity 2. Determine the constant of variation.
1.
2.
3
4.
5.
𝑃 varies directly as the sum of 𝑢 and 𝑤. If 𝑢 = 4 and 𝑤 = 3, then 𝑃 = 14
𝑟 varies directly as the square of 𝑡. If 𝑡 = 16, then 𝑟 = 16.
𝑦 varies directly as the cube root of 𝑥. If 𝑥 = 27, then 𝑦 = 2.
𝑧 is proportional to the cube of 𝑑. If 𝑑 = 2, then 𝑧 = 5.
The surface area (𝑆𝐴) of a cube varies directly to the square of sides (𝑠). If 𝑠 = 4, then
𝑆𝐴 = 96.
Activity 3. Determine if the tables and graphs below express a direct variation between the
variables.
1.
6.
2.
3.
4.
7.
5.
Activity 4. Determine whether the relation between the quantities involve direct(D) or
inverse(I) variation.
1.
2.
3.
The number of workers and the time required to finish an amount of work.
The number of workers and the amount of work finished.
The price per kilogram of rice and the number of kilogram of rice to buy with ₱ 500.
4.
5.
The number of persons and the amount of food consumption.
The number of family members and the length of time to consume one sack of rice.
6.
7.
8.
9.
10.
The quality of the goods and the price of these goods.
The height of a person and his weight.
The circumference of a circle and its diameter.
The time elapsed for a car to cover a distance and the speed of the car.
The amount of interest on a principal amount and the length of time the principal is
invested.
Activity 5. If 𝑦 varies inversely as 𝑥 and the following condition exist. Give the equation of the
variation.
1
1.
𝑦 = 6 when 𝑥 = 15
6.
2.
𝑦 = 27 when 𝑥 = 18
7.
𝑦 = 4 when 𝑥 = 16
3.
𝑦 = 3 when 𝑥 = 5
8.
𝑦 = 3 when 𝑥 = 3
4.
𝑦 = 8 when 𝑥 = 10
9.
𝑦 = 0.4 when 𝑥 = 0.6
𝑦 = 14 when 𝑥 =
2
3
2
𝑦 = 40 when 𝑥 = 12
5.
10.
𝑦 = 0.8 when 𝑥 = 4
Activity 6. Find the constant of variation and write the equation representing the relationship
between the quantities in each of the following:
1.
2.
3.
5.
4.
Let’s Remember
•
Variation relates the value of one variable to those of the value of the other variable,
so that, if any variable 𝑦 varies directly as another variable 𝑥, then there exists a
constant 𝑘 such that 𝑦 = 𝑘𝑥, and in this case, y is said to be directly proportional to 𝑥
and 𝑘 is called the constant of variation.
•
In a direct variation, for 𝑘 > 0, as the value of 𝑥 increases, the value of 𝑦 also increases.
Similarly, as 𝑥 decreases, 𝑦 also decreases.
•
Let 𝑥 and 𝑦 denote two quantities. 𝑦 varies inversely as 𝑥, 𝑦 inversely proportional to
𝑥, if there is a non-zero constant k such that:
𝑦=
𝑘
𝑥
•
The number k is called the constant of variation.
•
In this relation, as the independent variable 𝑥 increases the dependent variable 𝑦
decreases. Similarly, if the value of 𝑥 decreases, then the value of 𝑦 increases.
Let’s Apply
Solve the following problems.
1. The force of gravity (𝑔) acting on an object varies directly with the mass of the object
(𝑚). The force on a mass of 10 kg is 98 N. What is the force acting on a mass of 25
kg?
2. The number of copies (𝑁) produced by a copier varies directly as the amount of time
the copier (𝑡) is operating. If the copier produces 150 copies in 4 minutes, how many
copies can it produce in 1 hour?
3. If an object dropped from a height of 20 meters takes 3 seconds to reach the ground,
from what height (ℎ) should it be dropped if it is to reach the ground in a time (𝑡) of 6
seconds?
4. The number of days (𝐷) a container of water lasts varies inversely as the number of
people (𝑝) who consume it. If a barrel of water lasts 2 days for 5 people, how long
will it last for 3 people?
5. The number of minutes (𝑚) that it takes for an ice cube to melt varies inversely as
the temperature (𝑡) of the water the ice cube is placed in. when the ice cube is
placed in 16℃ water, it takes 2.3 min to melt. How long would it take for this size of
ice cube to melt if it will be placed in 24℃?
Let’s Evaluate
Multiple Choice. Choose the letter of the correct answer.
1. It is a function which occurs between two quantities such that when one quantity
increases, the other increases in a definite way.
A. direct variation
C. direct square variation
B. inverse variation
D. joint variation
2. Which of the statement does not belong to the group?
A. the number of hours to finish a job to the number of man working.
B. the number of person sharing a pie to the size of the slices of the pie.
C. the time a teacher spends checking papers to the number of students.
D. the age of a used car to its resale value.
3. Which table shows direct square variation?
A.
𝑥
𝑦
1
2
2
8
3
18
4
32
C.
𝑥
𝑦
-6
4
8
3
12
2
24
1
B.
𝑥
𝑦
-3
1
-9
3
3
-1
6
2
D.
𝑥
𝑦
1
36
2
18
3
12
4
9
4. Consider the table of values below, what type of
variation represents the relationships of x and y?
A. direct variation
B. inverse variation
1
3
8
𝑦 8 4
3
C. direct square variation
D. joint variation
𝑥
2
4
2
5. Find the constant of variation 𝑘, if 𝑦 varies directly as 𝑥 and 𝑦, when 𝑦 = 24 and 𝑥 = 3.
A. 8
B. 72
C. 24/3
D. 1/8
6. If y varies directly as x. Solve for y.
A. 42
C. 49
B. 48
D. 56
𝑥
𝑦
3
5
21 35
7
?
7. Which of these graphs shows an inverse relationship?
A.
C.
B.
D.
8. Five men can build a hut in 6 days. In how many days can 10 men build the same hut?
A. 2
B. 3
C. 9
D. 12
9. What happens to 𝑦 when 𝑥 is tripled in the relation 𝑦 =
A. y is tripled
B. y is doubled
𝑘
𝑥
?
C. y is halved
D. y is divided by 3
10. If y varies directly as the square of x, how is y changed if x is increased by 20%?
A. 44% decrease in y
C. 0.44% decrease in y
B. 44% increase in y
D. 0.44% increase in y
https://forms.gle/Y4ARvaDgFBhM42787
787L6
References
Nivera, Gladys C., et. al., GRADE 9 MATHEMATICS (Patterns and Practicalities), Salesiana
Books, 2013
Orines, Fernando B., et. al., NEXT CENTURY MATHEMATICS (Intermediate Algebra),
Phoenix Publishing House, 2003
Oronce, Orlando A., et. al., E-MATH 9 (Work text in Mathematics), Rex Book Store, 2015
Tesorio, Ma. Luisa V., MATHEMATICS 2 ( An Alternative Learning System – The Modular
Approach) 2018 Edition, Tru-Copy Publishing House Inc., 2008
Yeo, Joseph, et. al, NEW SYLLABUS MATHEMATICS 9, Rex Book Store, 2017
MATHEMATICS GRADE 9, LEARNER’S MATERIAL, Department of Education, First
Edition, 2014
Development Team of the Module
Writer:
Editors:
JOSEPH C. LAGASCA
Content Evaluators: JOEY N. ABERGOS
AMELIA A. CANZANA
ALMA J. CAPUS
JENNIFER N. CONSTANTINO
MARIO D. DE LA CRUZ JR.
NAUMI G. LIGUTAN
DONALYN S. MIÑA
JULIUS PESPES
Language Evaluator: MARICAR RAQUIZA
Reviewers:
DR. LELINDA H. DE VERA
MIRASOL I. RONGAVILLA
JENNICA ALEXIS B. SABADO
DR. MELEDA POLITA
Illustrator: BERNARD MARC E. CODILLO
Layout Artist: BERNARD MARC E. CODILLO
Management Team: DR. MARGARITO B. MATERUM, SDS
DR. GEORGE P. TIZON, SGOD-Chief
DR. ELLERY G. QUINTIA, CID-Chief
MRS. MIRASOL I. RONGAVILLA, EPS-Mathematics
DR. DAISY L. MATAAC, EPS-LRMS / ALS
For inquiries, please write or call:
Schools Division of Taguig city and Pateros Upper Bicutan Taguig City
Telefax: 8384251
Email Address: sdo.tapat@deped.gov.ph