9
Mathematics
Quarter 2 - Module 1:
VARIATIONS:
Direct and Inverse Variations
About the Module
This module was designed and written with you in mind. It is here to help you master
about VARIATIONS: Direct and Inverse Variations. 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.
This module is divided into two lessons:
Lesson 1 – Direct Variation
Lesson 2 – Inverse Variation
After working through this module, you are expected to:
• illustrate situations that involve direct and inverse variation;
• translate into variation statement a relationship between two quantities given by:
(a) a table of values; (b) a mathematical equation; and (c) a graph, and vice versa
• solve problems involving variations.
ii
What I Know (Pre-Test)
DIRECTIONS: Read and analyze each item carefully. Choose your answer in the
given choices. Write your solutions in some of the items using a separate sheet of
paper. NOTE: If there is no possible answer, solve it on your own and show your
solution.
_____1.) The cost (c) varies directly as the number (n) of candies is written as:
𝑘
𝑘
A.) 𝑛 = 𝑘𝑐
B.) 𝑐 = 𝑘𝑛
C.) 𝑛 =
D.) 𝑐 =
𝑛
𝑛
_____2.) Which of the following situations represents a direct variation?
A.) The number of pechay plants in a row to the space between them
B.) The cost per person of renting a private resort to the number of persons
sharing the rent
C.) The service fee of a physical therapist to the number of hours of service
rendered
D.) The number of days a bag of bread lasts to the number of persons who
consume it
_____3.) The statement “m is proportional to s” may be translated mathematically as
𝑘
A.) 𝑚 = 𝑘𝑠
B.) 𝑘𝑦 = 𝑠
C.) 𝑠 = 𝑘𝑚
D.) 𝑚 =
𝑠
_____4.) M varies directly as p. What effect will it have on M if p is doubled?
A.) M is quadrupled
B.) tripled
C.) halved
D. doubled
_____5.) M varies inversely as p. What effect will it have on M if p is doubled?
A.) M is quadrupled
B.) tripled
C.) halved
D. doubled
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
_____6.) What kind of variation equation is 𝑠𝑝𝑒𝑒𝑑 =
if the distance is held
𝑡𝑖𝑚𝑒
constant?
A.) Direct Variation
C. Inverse Variation
B.) Combined Variation
D. Joint Variation
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
_____7.) What kind of variation equation is 𝑠𝑝𝑒𝑒𝑑 =
if speed is held
𝑡𝑖𝑚𝑒
constant?
A.) Direct Variation
C. Inverse Variation
B.) Combined Variation
D. Joint Variation
_____8.) Which of the following table of values describes a direct variation?
A.)
C.)
x
1
2
3
4
x
1
2
3
4
y
2
4
6
8
y
12
6
4
3
B.)
D.)
x
y
10
12
20
60
30
40
40
30
1
x
y
2
5
3
10
3
4
5
2
5
2
_____9.) What is the graph of a direct variation?
A.) Line
B.) Parabola
C. Hyperbola
D. Ellipse
_____10.) If y is directly proportional to x and y = 6 when x = 3, find y when x = 12?
A.) 3
B.) 4
C.) 24
D. 48
_____11.) If y varies inversely as x, and 𝑦 = −3 when x = 8, find y when x = 12.
A.) 9
B.) – 9
C.) 2
D. – 2
_____12.) If 4 men can do a job in 10 days, how many men can do the same job in 5
days?
A.) 1
B.) 5
C.) 7
D.) 8
_____13.) Jenny’s salary varies directly as the number of hours that she works. If her
salary for 8 hours is ₱400.00, how much will she earn if she works four times
longer?
A.) ₱100.00
B.) ₱400.00
C.) ₱1, 600.00
D.) ₱4000.00
For item numbers 14 – 15, refer to the situation inside the box.
The number of days a sack of rice lasts varies inversely as the number of
people who consume it. The sack of rice lasts ten days for 6 people.
_____14.) What is the constant of the variation?
A.) 60
B.) 30
C. 20
D. 10
_____15.) How long will it last for 2 people?
A.) 60
B.) 30
D. 10
2
C. 20
Lesson
1
Direct Variation
What I Need To Know
At the end of this lesson, you are expected to:
1. identify related quantities in real-life situations
2. recognize relationships between two quantities that involve direct variation
3. translate statements that involve variations to
a. a table of values
b. a mathematical equation;
c. a graph, and vice versa
What’s In
Do you believe in the saying that “The only thing that is constant in this world
is change”?
Nothing in this world is permanent; one changes because of one variable that
changes.
Mathematics deals with two types of quantities: variables and constants. If
the value of the quantity remained unchanged under different situations, then it is
called a constant. While the value of the quantity changes under different situations,
it is called a variable.
Around us, some quantities are constant, and the others are variables. For
example, the number of hours in a day is constant, that is 24 hours, but the number
of hours of daylight in a day is not because it varies from time to time.
This relationship with regards to the change in the value of the variable when
the values of the related variables change is called Variation.
4 Types of Variation:
a.) Direct variation
b.) Inverse Variation
c.) Joint Variation
d.) Combined variation
In this module, you will explore the types of relationships between variables.
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What’s New
❖
Just Explore!
Related Quantities
Which of the following quantities are related to each other?
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Clipart 1: Online Pictures via Microsoft 365
the number of a ballpen and the
amount of money paid in buying
a ballpen
Socio-economic status and number
of friends
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Clipart 3: Online Pictures via Microsoft 365
speed and distance covered by
a bicycle in an hour
The number of people volunteered
in planting a crop and the number
of crops planted.
All the situations above, except for “Socio-economic status and the number of
friends” are related quantities because their friends will not depend on his/her socioeconomic status.
What Is It
Consider the relationship between speed and distance covered by a bicycle in
an hour.
Example 1: A bicycle travels at a speed of 10 kilometers per hour. How far
has it traveled after:
a.) half an hour?
b.) 2 hours?
c.) 3 hours?
d.) 4 hours?
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Solution:
Organize the answers to the questions using the table below.
1
2
5
Time (hr)
Distance (km)
1
10
2
20
3
30
4
40
Notice that as time increases, the distance also increases. As time
decreases, the distance decreases. In fact, as the time is doubled, the distance
is doubled. And as time is halved, the distance is halved.
÷2
Time (hr)
Distance
(km)
𝑥2
10 20 30
=
=
1
2
3
1
2
5
1
10
÷2
2
20
3
30
4
40
The ratio of
distance over
time is 10
𝑥2
The relationship between the time and distance in the example above can be
presented by the equation 𝑑 = 10𝑡. The equation is called direct variation. We can
say that distance varies directly as time. Since the distance traveled depends on the
length of the bicycle traveling at a constant speed, the distance traveled is called the
dependent variable. The length of time the bicycle travels is called the independent
variable.
DIRECT VARIATION
There is a direct variation whenever a situation produces pairs of
numbers in which their ratio is constant.
Denote x and y as two quantities, so the statements:
“y varies directly as x”
“y is directly proportional to x” and
“y is proportional to x”
can be translated mathematically as y = kx, where the number k is called
the constant of proportionality or the constant of variation.
For two quantities, x and y, an increase in x causes an increase in y.
Similarly, a decrease in x causes a decrease in y as well.
In Example 1, 10 is the constant of variation, and 𝑑 = 10𝑡 is the equation of
the variation. In this equation, as the value of t increases, the value of d also
increases. Also, as the value of t decreases, d decreases.
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Graph 𝑑 = 10𝑡.
Notice that the graph
starts at the origin and
rises from left to right.
The
graph
of
𝑦 = 𝑘𝑥, 𝑘 > 0, always
goes through the origin
and moves upward to
the right, and the
constant of variation is
the slope of the given
line.
Example 2: The table below shows that the service fee (f) of a physical therapist is
directly proportional to the number of hours (n) of service rendered. Find:
A. the constant of variation, and
B. the equation which describes the relation.
Time (hr.)
Service Fee (km)
1
800
2
1600
3
2400
4
3600
5
4000
Solution:
Since the service fee (f) of a physical therapist is directly proportional to the
number of hours (n) of service rendered, then
𝑓 = 𝑘𝑛
Using one of the pairs of values, (3, 2400), from the table, substitute the
values of f and n in 𝑓 = 𝑘𝑛.
𝑓 = 𝑘𝑛
2400 = 𝑘(3)
2400 = 3𝑘
𝟐𝟒𝟎𝟎
𝟑𝒌
=
𝟑
𝟑
𝑘 = 800
A. Therefore, the constant of variation is 800. It means that the physical
therapist is paid ₱ 800.00 per hour.
B. To form the mathematical equation of the variation, substitute 800 in the
statement 𝑓 = 𝑘𝑛. So, the equation of variation that describes the
relation is 𝒇 = 𝟖𝟎𝟎𝒏
You can see that the constant of variation can be solved if one pair of values
of x and y is known.
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a.) What is the cost of one
kilogram of Bangus?
b.) Mrs. Rama buys 5
kilograms of Bangus,
how much does she pay?
c.) Mang Eric has ₱540.00.
How many kilograms of
Bangus can he buy?
Amount
Example 3: The graph at the right shows a direct linear variation between the
number of kilograms of Bangus to its cost.
1080
990
900
810
720
630
540
450
360
270
180
90
0
Solution:
Look at the graph on the right to
get the answers to the questions
above.
a.) 1 kilogram of Bangus is
₱180.00
0
1
2
3
4
Kilograms
b.) Mrs. Rama will pay
₱900.00 for the 5
kilograms of Bangus
c.) Mang Eric will buy 3 kilograms of Bangus with his money.
5
6
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In the given examples, you can see that Direct Variation is a situation that if
one quantity increases, the other quantity increases. And if one quantity decreases,
the other quantity also decreases.
Try It!
Determine whether the relationship between the quantities involved in the
situations below is a direct variation.
1.) The number of workers in building a house and the amount of work
finished on the day
2.) The quality of the cellphone and the price of the phone
3.) The number of family members and the length of time to consume 1 sack
of rice
Answer and explanation of the items in “Try It!”.
1.) Direct Variation, because as the number of people building the house
increases, the amount of work that will be done on that day will increase.
And if there will be fewer people, then the amount of work will also
decrease.
2.) Direct variation, because if you want to have a cellphone with good
quality, you should expect a high price. So, if the quality of the phone gets
higher, the price also gets increased.
3.) Not a direct variation, because as the number of the family increases, the
length of time that one sack of consumption will decrease.
In the direct variation, if one quantity/variable increases, the other
quantity/variable will also increase. And if one quantity/variable decreases, the
other quantity/variable will also decrease.
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Translate Me!
Translate the direct variations below into its mathematical statement
using the variable indicated in the statement.
1.) The cost (C) of carrots is proportional to its weight (w) in kilograms.
2.) The area (A) of a square varies directly as the square of its side (s).
3.) The fare (F) of a passenger is directly proportional to the distance (d) of his
destination.
4.) The distance (D) traveled by a motorcycle varies directly as its speed (s).
5.) An employer’s salary (S) is directly proportional to the number of days (d)
he has worked.
Answer to “Translate Me!”.
1.) 𝐶 = 𝑘𝑤
2.) 𝐴 = 𝑘𝑠2
3.) 𝐹 = 𝑘𝑑
4.) 𝐷 = 𝑘𝑠
5.) 𝑆 = 𝑘𝑑
Were you able to get all the answers correctly?
I think you are ready to answer situations or word problems involving Direct
Variations. You can use these steps in solving variation problems.
STEPS IN SOLVING A DIRECT VARIATION PROBLEM
1.) Write the equation of variation.
2.) Construct a table of values.
3.) Substitute known values to solve for k.
4.) Solve for the desired value.
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Example 1: If y varies directly as x and y = 28 when x = 7, what is the value of y if
x = 20?
Solution:
Step 1. Write the equation of variation
y = kx
Step 2. Construct a table of values.
x
7
20
y
28
?
Step 3. Solve for k.
Use the pairs of values (7, 28) from the given, substitute the
values of x and y in y=kx, then solve for k.
𝑦 = 𝑘𝑥
28 = 𝑘(7)
28 = 7𝑘
28
7𝑘
=
7
7
4=𝑘
Substitute the values
Multiplying
Solve for k.
Step 4. Solve for the desired value.
Replace the obtained value of k in the equation in STEP 1.
Solve for y, when k = 4 and x = 20
𝑦 = 𝑘𝑥
𝑦 = (4)(20)
𝑦 = 80
Hence, y = 80 when x = 20.
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Example 2: The weight of an object on Mars varies as its weight on Earth. An
object that weighs 115kg on Earth weighs 44kg on Mars. How much
would a person weigh 75kg on Earth weights on Mars?
Solution:
Step 1. Write the equation of variation
First, find the constant of variation.
Let e = weight of an object on Earth
m = weight of an object on Mars
m = ke
Step 2. Construct a table of values.
e
115
75
m
44
?
Step 3. Solve for k.
Use the pairs of values (115, 44) from the given, substitute the
values of x and y in m = ke, then solve for k.
m = ke
44 = 𝑘(115)
44
𝑘=
115
Substitute the values
Solve for k.
Step 4. Solve for the desired value.
Replace the obtained value of k in the equation in STEP 1.
44
Solve for y, when 𝑘 =
and e = 75.
115
𝑚 = 𝑘𝑒
44
𝑦 = ( ) (75)
115
𝑦 = 28.70
Hence, the person's weight on Mars is 28.70kg if he is 75kg here
on Earth.
What’s More
Activity 1.1: Translate Me!
INSTRUCTION: Write an equation for the following statements:
1.) The perimeter (P) of a square is directly proportional to the length of its sides
(s).
2.) The circumference (c) of a circle is proportional as the diameter (d).
3.) y is proportional to the square root of u.
4.) The cost of electricity (C) varies directly as the number of kilowatt-hour
consumption (h).
5.) The weight (W) of an object is directly proportional to its mass (m).
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Activity 1.2: NOW, IT’S YOUR TURN!
INSTRUCTIONS: Determine if the tables and graphs below express a direct variation
between the variables. If they do, find the constant of variation and the equation that
defines it. If not, write not a direct variation.
50
1.)
4.)
x
y
-3
6
-4
8
-5
10
-6
12
40
30
20
2.)
10
x
y
1
12
2
6
3
4
4
3
x
y
2
10
4
20
8
40
12
60
0
0
1
2
3
4
5
6
7
3.)
5.)
50
40
30
20
10
0
0
1
2
3
4
5
6
Activity 1.3: NOW, IT’S YOUR TURN!
INSTRUCTIONS: Solve the following problems. Show your solution.
1.) The number of kilograms of water in a person’s body varies directly as the mass
of the person. If a person with a mass of 99 kilograms contains 60 kilograms of
water in his body, how many kilograms of water are in a person with a mass of
130 kilograms?
2.) The amount of water that has leaked from the faucet varies directly with time. In
2 hours, 1 gallon of a water leak. How many gallons of water leaks in 100 hours?
3.) A tutor teacher charges ₱200.00 per hour of tutorial service to a Grade 9 Junior
High School student. If he spends 2 hours in his tutorial service per day, how
much would he receive in 15 days?
10
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What I Need To Remember
•
•
•
The situation is Direct Variation if for two quantities, x and y, an
increase in x causes an increase in y. And a decrease in x causes a
decrease in y also.
To get the constant of variation, get the ratio of the quantities or
variables
Follow the following steps before solving any given word problem:
a.) Write the equation of variation.
b.) Construct a table of values.
c.) Substitute known values to solve for k.
d.) Solve for the desired value.
Lesson
2
Inverse Variation
What I Need To Know
At the end of this lesson, you are expected to:
1. identify related quantities in real-life situations
2. recognize relationships between two quantities that involve direct variation
and inverse variation
3. translate statements that involve variations to
d. a table of values
e. a mathematical equation;
f. graph, and vice versa
What’s In
❖ Flashback!
Determine whether the two quantities in the situation are directly related.
Write Direct variation or Not Direct.
Situation
Direct or
Not Direct?
1. The age of a used car to its resale value
2. The number of days to finish painting a house to the
number of men working
3. The time a teacher spends checking the answer sheets to
the number of students
4. The area of the wall to the amount of paint used to cover it
5. The number of people sharing a pizza to the number of
slices of the pizza
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Answer and explanation of the situation inside the table.
1.) Not Direct, because as the age of the car increases, the price the owner
can get in selling it decreases.
2.) Not Direct, because as the number of people doing the job increases, the
time needed to finish the job decreases.
3.) Direct, because as the number of answer sheets to be checked increases,
the teacher's time in checking increases.
4.) Direct, because as the area of the wall increases, the amount of paint
will also increase.
5.) Not Direct, because as the number of people sharing the pizza increases,
the size of the slice each will get decreases.
What’s New
❖ Who’s Got More Share?
Two groups of people bought 1 box of
pizza each, one group has 4
members, and the other group
has 8. if they will divide the
8 slices of pizza equally,
how many slices of pizza
will each of the member get
in the group of:
A.) 4 members?
B.) 8 members?
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Whose group got more share of pizza each?
What is It
Take a look at the situation being presented in Who’s Got More Share, if the
pizza is divided into 8 slices, let’s look at the table below on the number of slices that
a person will get based on the number of people that will be dividing the whole pizza,
given that they will divide the slices equally among themselves.
Number of
person (n)
Number of
slices (s)
(8)(1) = (2)(4) = (4)(2) = (8)(1)
1
2
4
8
8
4
2
1
The product of the number of
people and the number of
slices is 8.
This situation shows that an increase in the number of people sharing the
whole pizza results in a decrease in the number of slices each group member will
12
receive. At the same time, the decrease in the number of people sharing the whole
pizza produces an increase in the number of slices that each member will get.
𝑛𝑠 = 8 ( 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
In the given situation, “the
number of pizza slices varies
inversely to the number of
persons sharing a whole pizza”
𝑘
𝑛
50
40
and is written as 𝑠 = , where k is
the constant of variation. Thus,
the equation represented in the
30
8
𝑛
20
table, and the graph is 𝑠 = ;
where k = 8.
10
Note that the product of
the two quantities is constant.
0
The graph at the right goes down
0
1
2
3
4
5
6
7
8
from left to right. It shows that
the share per slice decreases as the number of people sharing a whole pizza
increases.
INVERSE VARIATION
Inverse Variation occurs when a situation produces pairs of numbers
whose product is constant.
The statement:
“y varies inversely as x”
𝒌
can be translated mathematically as y = , where the number k is called the
𝒙
constant of variation.
Try It!
Determine whether the relationship between the quantities involved in the
situations below is an inverse variation.
1.) The height of a person and its weight
2.) The number of person and the number of days in consuming food
3.) The speed of the person in typing a certain manuscript to the time spent
in typing
Answer and explanation of the items in “Try It!”.
1.) Not inverse variation, because the height of a person varies directly
as its weight. If the height of the person increases, its weight also
increases. If the height decreases, its weight also decreases.
2.) Inverse Variation, because as the number of people increases, the
number of days to consume a food decrease. And if the number of
people decreases, the number of days in consuming food increases.
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3.) Inverse variation, because as the person's speed increases, the time
spent in typing decreases, and if the speed decreases, the time spend
increases.
In the inverse variation, if one quantity/variable increases, the other
quantity/variable will decrease. And if one quantity/variable decreases, the other
quantity/variable will increase.
Translate Me!
Translate the direct variations below into its mathematical statement
using the variable indicated in the statement.
1.) E is inversely proportional to z.
2.) The gravitational for (F) between two bodies varies inversely as the
square of the distance (d) between them.
3.) D varies inversely as the square root of s.
Answer to “Translate Me!”.
𝑘
1.) 𝐸 =
𝑧
2.) 𝐹 =
𝑘
𝑑2
3.) 𝐷 =
𝑘
√𝑠
Were you able to get all the answers correctly?
In the next part of this module, you will be learning how to answer word
problems in inverse variation. To solve it, use the same steps in solving Direct
Variation problems.
STEPS IN SOLVING AN INVERSE VARIATION PROBLEM
1.) Write the equation of variation.
2.) Construct a table of values.
3.) Substitute known values to solve for k.
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Example 1: If N varies inversely as M, and N = 6 when M = 3, find N when M = 10.
Solution:
Step 1. Write the equation of variation
𝑘
𝑁=
𝑀
Step 2. Construct a table of values.
M
3
10
N
6
?
Step 3. Solve for k.
𝑘
Substitute the values of N = 3 and M = 3 in 𝑁 = , then solve for k.
𝑀
𝑘
𝑁=
𝑀
𝑘
6=
Substitute the values
3
𝑘 = (6)(3)
Solve for k
𝑘 = 18
.
Step 4. Solve for the desired value.
Replace the obtained value of k in the equation in STEP 1.
14
Solve for N, when k = 18 and M = 10
𝑁=
𝑁=
𝑁=
Hence, 𝑵 =
𝑘
𝑀
18
10
9
5
𝟗
when M = 10.
𝟓
Example 2: The number of hours required to do a job varies inversely as the number
of people working together. If it takes 6 hours for 10 people to paint a
house, how long would it take 12 people to paint the same house?
Solution:
Step 1. Write the equation of variation
First, find the constant of variation.
Let t = number of hours or time
p = number of people
𝑘
𝑡=
𝑝
Step 2. Construct a table of values.
p
10
12
t
6
?
Step 3. Solve for k.
Use the pairs of values (10, 6) from the given, substitute the
𝑘
values of p and t in 𝑡 = , then solve for k.
𝑝
𝑘
𝑝
𝑘
10 =
6
𝑘 = (10)(6)
𝑘 = 60
𝑡=
Substitute the values
Solve for k.
Step 4. Solve for the desired value.
Replace the obtained value of k in the equation in STEP 1.
Solve for t, when 𝑘 = 60 and p = 12.
𝑘
𝑡=
𝑝
60
𝑡=
12
𝑦=5
Therefore, 10 people can finish the painting in 5 hours.
15
What’s More
Activity 1.4: Translate Me!
INSTRUCTIONS: Write an equation to the following statements.
1.) The base (b) of a triangle is inversely proportional as its height (h).
2.) F varies inversely as the cube root of S.
3.) The number (n) of plants in a row varies inversely as the space (s) between
them.
4.) The length (l) of a rectangular field is inversely proportional as its
width (w).
5.) The mass (m) of an object varies inversely as the acceleration due to
gravity (g).
Activity 1.5: NOW, IT’S YOUR TURN!
INSTRUCTIONS: Solve the following word problems below by following the steps
presented. Answer on a separate sheet of paper.
1.) A doll manufacturing company found that the demand for the doll varies
inversely as the price of the doll. When the price is ₱100.00 per doll, the
demand is 400 pieces of dolls. Compute the demand of dolls if each doll
would cost ₱150.00 a piece.
2.) If x varies inversely as y, and x = 13 when y = 9, what is the value of x when
y = 3?
What I Need To Remember
•
•
Inverse variation occurs if one quantity in the situation increases while
the other quantity decreases. And if one quantity decreases the other
quantity increases.
To get the constant of an increase variation, just multiply the values of
the given variables or quantities.
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What I Can Do
INSTRUCTIONS: What new learnings do you have about the real-life applications of
Direct and Inverse Variation? How would you relate this to your daily life? How would
you use this in making decisions? Write your answer on a piece of paper.
Your reflection or output will be graded using the rubric.
Skills
Depth of
Reflection
5
4
The response
demonstrates an
in-depth
reflection on and
personalization
of the theories,
concepts, and
strategies
presented in the
module.
The response
demonstrates a
general
reflection on,
personalization
of the theories,
concepts, and
strategies
presented in the
module.
Required
Components
The response
surpasses the
required
components of
the selected
topic.
The response
includes the
required
components of
the selected
topic.
Quality of
Information
The response
clearly relates
to the main
topic. It
includes all
the necessary
supporting
examples.
The response
is stated
clearly,
concisely, and
properly.
Reasons are
expressed
logically.
The response
clearly relates
to the main
topic. It
includes 1-2
necessary
supporting
examples.
The response
is stated
clearly,
concisely, and
properly.
Reasons are
expressed
illogically.
Conventions
17
3
The response
demonstrates
a minimal
reflection on,
personalization
of the theories,
concepts, and
strategies
presented in
the module.
The response
includes only
few
components of
the selected
topic.
The response
clearly relates
to the main
topic. No
examples are
given.
The response
is unclear and
not organized.
Thoughts are
not expressed
logically.
2
The response
demonstrates
a lack of
reflection on
the topic, with
no details
provided.
The response
does not
include the
required
components of
the selected
topic.
The response
has little to do
or no relations
to the main
topic.
The response
is unclear and
disorganized.
Thoughts
make little to
no sense.
Assessment (Post Test)
DIRECTIONS: Read and analyze each item carefully. Choose your answer in the
given choices. Write your solutions in some of the items using a separate sheet of
paper. NOTE: If there is no possible answer, solve it on your own and show your
solution.
_____1.) c varies inversely as n is written as:
A.) 𝑛 = 𝑘𝑐
B.) 𝑐 = 𝑘𝑛
C.) 𝑐 =
𝑘
𝑛
D.) 𝑐=𝑘𝑛
_____2.) What is the constant of variation shown in the table?
x
y
A.) 80
4
20
5
25
7
35
B.)5
5.5
27.5
C.)
1
10
D.)
1
2
_____3.) The statement “m is proportional to the square root of s” may be translated
mathematically as
𝑘
𝑘
𝐴. ) 𝑦 = 𝑘√𝑠
B.) 𝑦 = 𝑘𝑠2
C.) 𝑦 =
D.) 𝑦 = 2
√𝑠
𝑠
_____4.) m varies inversely as n such that when m = 12, n = 9. Find m when n = 27.
A.) 18
B.) 9
C.) 4
D. 3
_____5.) h is directly proportional to the square of v such that v = 3, h = 8. What is
the constant of variation?
8
8
A.)
B.)
C.) 24
D. 72
9
3
_____6.) The cost (c) per person of hiring a taxi varies inversely as the number of
persons (N) sharing the cost. What equation relates to the situation?
𝑘
𝑘
A.) 𝑐 = 𝑘𝑁
B.) 𝑁 = 𝑘𝑐
C.) 𝑐 =
D.) 𝑁 =
𝑁
𝑐
_____7.) What kind of variation is the situation inside the box?
Alice can type 30 words in 1 minute. At this rate, how many words can
she type in 10 minutes?
A.) Direct Variation
B.) Combined Variation
C. Inverse Variation
D. Joint Variation
_____8.) Which of the following table of values describes an inverse variation?
A.)
C.)
x
1
2
3
4
x
1
2
3
4
y
2
4
6
8
y
12
6
4
3
B.)
D.)
x
y
3
4
6
8
12
16
24
32
18
x
y
2
10
3
5
4
5
2
5
2
_____9.) What is the graph of a direct variation?
A.)
C.)
B.)
D.)
_____10.) P is inversely proportional to x, and P = 9, when x = 4. Find P when x = 18.
81
A.) 2
B.) 4
C.) 36
D.
2
_____11.) If m varies directly as n, and 𝑚 = −6 when n = 2, find m when 𝑛 = −4.
A.) – 12
B.) – 3
C.) 3
D. 12
_____12.) The length of time a bag of cat food lasts varies inversely as the number of
cats. If a bag of cat food will feed 5 cats in 10 days, how long will it feed 10
cats?
A.) 5
B.) 10
C.) 20
D. 30
For item numbers 13 – 15, refer to the situation inside the box.
Three college students decided to rent an apartment near the school where
they are studying. They found the nearest and cheapest apartment that costs
₱6000. 00 a month, which they found too much for their monthly budget. How many
students will they need to share with so that each will pay only for ₱1200. 00 a
month?
Let n = be the number of students
s = share of rent per student
_____13.) What equation represents the problem above?
𝑘
𝑠
A.) 𝑛 = 𝑘𝑠
B.) 𝑛 =
C.) 𝑛 =
𝑠
𝑘
D.) 𝑠 =
𝑘
𝑛
_____14.) How much is the constant amount of the variation of the given problem
inside the box?
A.) ₱6 000. 00
B.) ₱3000. 00
C.) ₱2000. 00
D.) ₱1000. 00
_____15.) How many students are needed to share with so that each will pay only
for ₱1200. 00 a month?
A.) 3
B.) 4
C.) 5
D.) 6
19
ℎ
𝑘
𝑠
𝑘
√𝑆
𝑘
Text Text Text text
Text Text
20
𝑘
𝑦
𝑘
𝑦
𝑘 = (13)(9)
𝑘 = 117
𝑥=
13
9
𝑥=
Solve for x, when y = 3
𝑘
𝑥=
𝑦
117
𝑥=
3
𝑥 = 39
Solve for k:
x
y
2.) 39
3
Activity 1.5: Now, It’s Your
Turn!
1.) Approximately 267 dolls
𝑤
𝑘
5.) 𝑚 =
𝑔
4.) 𝑙 =
3.) 𝑛 =
2.) 𝐹 = 3
2
1
3.) ₱6,000.00
Solve for k:
𝑤 = 𝑘𝑡
2 = 𝑘(1)
1
𝑘=
2
Solve for w, when t = 100
𝑤 = 𝑘𝑡
1
𝑤 = ( ) (100)
2
𝑤 = 50 𝑔𝑎𝑙𝑙𝑜𝑛𝑠
t
w
𝑤 = 𝑘𝑡
100
2.) 50 gallons
Sol’n: Let 𝑤 = 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟
𝑡 = 𝑡𝑖𝑚𝑒
Activity 1.3: Now, It’s Your
Turn!
1.) 78.79 kg.
Activity 1.1: Translate Me!
1.) 𝑃 = 𝑘𝑠
2.) 𝐶 = 𝑘𝑑
3.) 𝑦 = 𝑘√𝑢
4.) 𝐶 = 𝑘ℎ
5.) 𝑤 = 𝑘𝑚
Activity 1.2: Now, It’s Your Turn!
1.) Not Direct Variation
2.) Not Direct Variation, because as the values of
x increase the values oy y decrease.
3.) Direct Variation
4.) Direct Variation, because the graph increases
from left to right.
5.) Not Direct Variation
What I can Do
Activity 1.4: Translate Me!
𝑘
1.) 𝑏 =
Answer Key (With Explanation)
Remember: This portion of the module contains all the answers. Your HONESTY is
required.
References
Clipart
Clipart 1. Accessed November 2, 2020. Online Pictures via Microsoft 365.
Clipart 2. Accessed November 2, 2020. Online Pictures via Microsoft 365.
Clipart 3. Accessed November 2, 2020. Online Pictures via Microsoft 365.
Clipart 4. Accessed November 2, 2020. Online Pictures via Microsoft 365.
Clipart 5. Accessed November 2, 2020. Online Pictures via Microsoft 365.
Clipart 6. Accessed November 20, 2020. Online Pictures via Microsoft 365.
Clipart 7. Accessed November 20, 2020. Online Pictures via Microsoft 365.
Books
Merde L. Bryant, et. Al, Mathematics 9 Leaner’s Material 9 (Meralco Avenue, Pasig
City: Vibal Group, Inc.), 190 – 212.
Nivera, Gladys C. and Lapinid, Minie Rose C. 2013. Grade 9 Mathematics: Pattern
and Practicalities (Chino Roces Avenues, Makati City: Salesiana Books by Don
Bosco Press.), 113 – 138, 160 – 162.
Orines, Fernando B. and Diaz, Zenaida E. 2004. Next Century Mathematics
Intermediate Algebra (J Panis Strees: Phoneix Publishing House, Inc.), 235 – 246,
253 – 255.
Online Source
iRubric: Reflective Essay Rubric. Accessed August 9, 2021.
https://www.rcampus.com/rubricshowc.cfm?sp=yes&code=L34935
Congratulations!
You are now ready for the next module. Always remember the following:
1.
o
o
o
2.
3.
4.
Make sure every answer sheet has your
Name
Grade and Section
Title of the Activity or Activity No.
Follow the date of submission as agreed with your teacher.
Keep the modules with you.
Return them at the end of the school year.
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