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. 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

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)

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

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

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

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)

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

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,

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

•

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.

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

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

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

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