1.
What I Need to Know
In this module, you will learn more about situations that involve
variations when you take the following lessons:
Lesson 1 • Direct Variation
Lesson 2 • Inverse Variation
In these lessons you will learn to:
• Illustrates situations that involve the following variations:
(a) direct; (b) inverse; (c) joint; (d) combined. (M9AL – IIa-1)
• translates into variation statement a relationship between two quantities
given:
(a) a table of values, (b) a mathematical equation;
(c) a graph, and vice versa. (M9AL – IIb-1)
Solve problems involving variations. (M9AL-II-c-1)
After answering the module, you must be able to
1. identify direct and inverse variations;
2. illustrate situations involving direct and inverse variations;
3. translates into variation statement a relationship involving direct and
inverse variations between two quantities given a table of values and a
mathematical equation; and
4. solve problems involving direct and inverse variations.
Lesson 1 DIRECT VARIATION
What’s In
Complete the table of values given that y = 2x.
x 1 2 3 4 5
y
Questions:
1. How did you get the different values of y?

2.
2. What do you notice with the values of y as x increases?
What’s New
A local government organization launches a recycling campaign of waste
materials to schools in order to raise students’ awareness of environmental
protection and the effects of climate change. Every kilogram of waste material
earns points that can be exchanged for school supplies and grocery items. Paper,
which is the number one waste collected, earns 5 points for every kilo.
The table shows the points earned by a Grade 9 class for every number of
kilograms of waste paper collected.
Number of kilograms (n) 1 2 3 4 5 6
Points (p) 5 10 15 20 25 30
Questions:
1. What happens to the number of points when the number of kilograms of
paper is doubled? tripled?
2. How many kilograms of paper will the Grade 9 class have to gather in order
to raise 500 points? Write a mathematical statement that will relate the two
quantities involved?
3. What happens to the number of points when the number of kilograms of
paper is doubled? tripled?
4. How many kilograms of paper will the Grade 9 class have to gather in order to
raise 500 points? Write a mathematical statement that will relate
the two quantities involved.
How was the activity? What new thing did you learn from the
activity? The next activities will help you understand the concepts
behind it.
What is it

3.
Let us consider the activity in What’s New section, the variation statement
involved between the number of kilograms (n) and the points (p) is p = 5n. In this
case the constant of variations k is 5.
Using a
convenient scale, the
graph of the relation p
= 5n is a line.
The graph above describes a direct variation of the form y =kx. For more
detailed solution of problems involving direct variation, let us see how this is done.
Illustrative Examples
1. If y varies directly as x and y = 24 when x = 6, find the constant of
variation and the equation of the variation. Solution:
a. Express the statement “y varies directly as x” as y =kx.
b. Solve for k by substituting the given values in the equation.
y =kx
24 = k(6)
Whenever a situation produces pairs of numbers in which their ratio is
constant, it means that one quantity varies directly as the other quantity, or the
two quantities are in direct variation.
The statements:
“y varies as x”
“y is directly proportional to x” and
“y is proportional to x”
may be translated mathematically as y =kx or = k, where k is the constant of
variation.
For two quantities, x and y, an increase in x causes an increase in y as
well. Similarly, a decrease in x causes a decrease in y.
points (p)
30
25
20
15
10
5
0 1 2 3 4 5 6 7
number of kilograms(n)

4.
24 = 6k
k = 246
k = 4
Therefore, the constant of variation is 4.
c. Form the equation of the variation by substituting 4 in the
statement, y =kx. Therefore, y = 4x.
2. The table shows that the distance d varies directly as the time t. Find the
constant of variation and the equation which describes the relation.
a.
Solution:
a. Since the income of the worker varies directly as the number of hours
he work, then I = kh.
b. Using one of the point (2,150), from the graph, substitute the values
of I and h in I = kh and solve for k.
= kh
150 = k (2)
2k = 150
k = 75
c. Form the equation of the variation by substituting 75 in the statement
I = kh. So, I = 75h
What’s More
Activity 1. Write an equation for the following statements.
1. The cost (c) of chicken varies directly as its weight (w).
2. The circumference (C) of a circle varies directly as the length of its diameter (
d).
3. Water pressure (P) varies directly as the height of the water. (H) 4. The fare (F)
of a passenger varies directly as the distance (d) of his destination.
5. The area (A) of a square varies directly as the square of its side (s).
Activity 2. Identify if the tables and graphs below illustrate a direct variation
between the variables. If they do, find the constant of variation and an equation
that defines the relation.
Time(hr) 1 2 3 4 5
Distance(km) 10 20 30 40 50

5.
1.
2.
3.
4.
Activity 3. Write an equation where y varies directly as x.
1. y =28 when x = 7
2. y = 24 when x =3
3. y =63 when x = 81
What I Have Learned
On a sheet of paper, summarize what you have learned from this lesson.
x 1 2 3 4
y -3 -6 -9 -12
x 7 14 21 28
y 3 6 9 12
x -15 10 -20 25
y -3 2 -4 5
x 2 3 4 5
y 1 2 3 4

6.
What I Can Do
Do the following.
A. Formulate a real –life problem which involves direct variation between two
quantities. Write an equation which relates the two quantities.
B. Make a table of values which describes the relation of the two quantities in
the problem.
C. And draw a graph which illustrates the relation.
Assessment
Choose the letter of the correct answer.
1 . 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 =
n
2. Which is an example of a direct variation?
2
A. xy = 10 B. y = 2 C. y = 5x
X
2
D. = x y
3. A car travels a distance of d km in t hours. The formula that relates d to t
is d = kt . What kind of variation is it?
A. direct B. inverse C. joint D. combined
4. y varies directly as x and y = 32 when x = 4. Find the constant of variation.
A. 8 B. 28 C. 36 D. 128
5 . 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
6. Bea’s income varies directly as the number of days that she works. If she
earns Php 8,000 in 20 days, how much will she earn if she worked 3
times as long?
A. Php 26,000 B. Php 24,000 C. Php 20,000 D. Php 16,000
7. Find the constant of variation k if y varies directly as x and x = 42 when y
=14.
1 1

7.
A. B. 2 C. 3 D. 4
3 2
8. What happens to T when h is doubled in the equation T =4h?
A. T is halved. B. T is tripled. C. T is doubled. D. T becomes zero.
9. Write an equation where y varies directly as x when y = 10 when x = 24.
5 1
A. y = x B. y = 2 x C. y = 2x D. y = 4x
12 2
10.Nestor uses 10 liters of gasoline to travel 100 kilometers, how many liters of
gasoline will he use on a trip of 550 kilometers?
A.55L B.50L C.40L D. 20L
11.The service fee f of a massage therapist varies directly as the number of
hours h of service rendered. A massage therapist charges Php 350 for 1 hour
service. The table shows the number of hours h and the service charge f .
What mathematical statement illustrates the relation between
the two quantities?
h 2 3 4 5
f 350 700 1050 1400 1750
A. f = 350h B. f = 250h C. f = 150h D. f = 50h
12.What is the constant of variation in problem number 11?
A. 1750 B. 1400 C. 700 D. 350
13. Write an equation which describes the graph.
100
80
60
40
20
0 1 2 3 4 5 6 7
number of hours (x)
A. y =20x B. y = 10x C. y = 5x D. y = x
pay (y)

8.
14.The weight w of an object is directly proportional to its mass m. Write an
equation which describes the relation.
A. m =kw B. w = km C. w = k + m D. k = mw
15.If x varies directly as y and x = 75 when y = 15, what is the value of y
when x = 25?
A.2 B.3 C.4 D.5
What’s In
Consider the table of values below.
x 1 2 3 4
y 3 6 9 12
1. What have you observed with the values of y as x increases?
2. What equation describes the relation between x and y?
What’s New
Let us consider the situation below.
Hans lives 20 kilometers away from the City where he works. Driving his
car takes him to reach his work depends on his average speed. Some possible
speed and the length of time it takes him are as follows:
Time in hours 1 5 5 5
6 7 8
Speed in kph
50 60 70 80
Questions:
1. How do the speed and time of travel affect each other?
2. Write a mathematical statement to represent the relation.
Lesson 2 INVERSE VARIATION

9.
3. Is there a constant number involved? Explain the process that you have used
in finding out.
The situation in the problem shows “ an increase in speed produces a decrease in
time in travelling.” The situation produces pairs of numbers, whose product is
constant. Here, the time t varies inversely as the speed s such that st = 50 (a
constant)
In this situation,” the speed s is inversely proportional to the time t ,” and is written
as s = k, where k is the proportionality constant or the constant of t variation. Hence,
the equation represented in the table and graph is s = 50
t
, where k = 50.
What is it
The situation in the previous activity is an example of inverse variation.
Inverse variation occurs whenever a situation produces pairs of
numbers whose product is constant.
For two quantities x and y, an increase in x causes a decrease in y or
𝑘
The statement, “y varies inversely to x,” translates to y =
𝑥
where k is the constant of variation.
Illustrative Examples
1. If y varies inversely as x and y = 5 and x = 15. Find the equation and solve
for k.
Solution:
𝑘
The relation y varies inversely as x translates to y= . Substitute the
𝑥 values to find k.
The equation of the variation is y = 75
x
vice versa. We can say that y varies inversely as x or in symbols,

10.
2. The table shows that y varies inversely as x. Find the constant of variation
and the equation which describes the relation.
x 2 4 6 8
y 36 18 12 9
Solution:
a. Since y varies inversely as x, so y =k
x
b. b. Using one of the pairs of values, (2, 36), from the table, substitute the
values of y and x in y = k x and
solve for k.
y = k
x
36 =k
2
36 (2) = K
K=72
Therefore, the constant of variation is 72 and the equation of the variation is y = 72
x
3. Find the constant of variation and write the equation which represents
the graph below.
0 1 2 3 4 5 6 7
width (w)
Solution:
a. Since the length l is inversely proportional as the width w, so we have
l = k w
length(l)
6 •
5
4 •
3 •
2 •
1
•

11.
b. Using one point, (2, 6), from the graph, substitute the values of w and l
in l = k and solve for k.
w
Therefore, the constant of variation is 12 and the equation of the variation is
What’s More
Activity 1. Express each of the following statements into mathematical
equation.
1. The number of hours h required to complete a certain job varies inversely
as the number of machines m used to do the work.
2. The number of rice cake slices r varies inversely as the number of
persons n sharing the whole rice cake.
3. The temperature t at which water boils varies inversely as the number of
feet h above the sea level.
Activity 2. Answer the following.
1. If y varies inversely as x, and y = 48 when x = 10, find y when x = 32.
2. Find the equation that expresses the variation below.
a 2 3 4 6
b 48 32 24 16
What I Have Learned
On a sheet of paper, summarize what you have learned from this
lesson. Provide real-life situation that involves inverse variation. Illustrate the
relationship between two quantities using the table of values, a graph and an
equation.

12.
What I Can Do
Read the phrases found in the first column of the table below, identify if it is an
inverse variation. Put check(√) if it is and (x ) if it is not in the second column.
Phrase √ or x
1.The number of hours to finish a job to the number of men
working
2. The distance an airplane flies to the time travelling
3. The number of persons sharing a pie to the number of slices of the pie
4. The area of a square to the measure of its side
5.The altitude of a cylinder with constant volume to the area of its
base
Assessment
Choose the letter of the correct answer.
1. The speed r of a moving object is inversely proportional to the time t
travelled is written as
2. The volume V of a gas at constant temperature varies inversely as the
Pressure (P). The formula that relates V to P is V = . What kind of
variation is it?
A. direct B. inverse C. joint D. combined
3. y varies inversely as x and y = 32 when x = 4. Find the constant of variation.
A. 8 B. 28 C. 36 D. 128
4. Which of the following describes an inverse variation? A.
x 2 3 4 5
y 5 10
5
2
3 2
B.
x 1 2 3 4
y 5 10 15 20

13.
C.
x 40 30 20 10
y 8 6 4 2
D.
x 4 8 10 12
y 2 4 5 6
5 . If y varies inversely as x and y = 12 when x =4, find y when x = 12. A.3 B.4 C.36
D.48

14.
A
Answer Key
Lesson 1
What’s In
1. Replace x with the given values.
2. The values of y increases as x increases.
What’s More
Activity 1

15.
REFERENCES
Jose-Dilao, Soledad and Bernabe , Julieta G, Intermediate Algebra ,Revised Edition
Mathematics 10 Learner’s Module Next
Generation Math http://www.mathisfun.com
{ HYPERLINK "http://www.mathbitsnotebook.com" } {
HYPERLINK "http://www.courses.lumenlearning.com" }
Development Team of the Module
Writer’s Name: Armil O. Turtor
Reviewer’s Name: Ismael K. Yusoph
Management Team: SDS: Ma. Liza R. Tabilon
ASDS: Judith V. Romaguera
OIC -ASDS: Ma. Judelyn J. Ramos
OIC -ASDS: Armando P. Gumapon
CID Chief: Lilia E. Abello,Ed.D.
LR : Evelyn C. Labad
PSDS: Antonina D. Gallo, Ed.D.
Principal: Daisy Flor J. Romaguera