Variation

1. MATHEMATICS
GRADE 9
VARIATIONS
Quarter 2
Lesson 1

2. OBJECTIVES
01
02
03
M9AL – IIa – 1
Illustrates situations that involve variations.
M9AL – IIa – b – 1
Translates into variation statement a relationship
between two quantities given a table of values, a
mathematical equation, a graph, and vice versa.
M9AL – IIb – c – 1
Solves problems involving variation.

3. CHANGE IS INEVITABLE!
Don’t be afraid of change.
Sometimes changes in the values of two variables can be
related. When a change in a value of one quantity corresponds
to a predictable change in the value of the other, then we say
that the quantities are related.
“
“
VARIATIONS
GRADE 9 Mathematics

4. Nature of
Activity
Heartbeat
Related Quantities
Which of the following quantities are related? Explain how they are related.
Speed
Distance
covered by
a car
Socio-economic
status
Number of
Friends
Hardwork
Success

5. ACTIVITY
AGREE or DISAGREE?
That is the Question!

6. The more hours you play
online games, the more
money you pay.
AGREE or DISAGREE?
That is the Question!
If you increase a recipe for
more people, the more
ingredients you need.
The less water you drink, the
less trips to the bathroom
you have to make.
The less time you study,
the lower scores you will
get in the exam.

7. THE LENGTH
OF TIME A CAR
TRAVELS AT A
CONSTANT
SPEED AND THE
DISTANCE IT
COVERS
A car travels at 60
kilometers per hour.
How far has it
traveled after:
a. half an hour?
b. 2 hours?
c. 3 hours?
d. 4 hours?
Prepared
by:
Ms.
Kristine
S.
Layson
PCS
–
Junior
High
School
,
01
October
2017

8. Let us organize the answers in a table:
Notice that as the time increases, the distance also increases. As
the time decreases, the distance also decreases.
In fact, as the time or the number of hours is doubled, the distance
travelled is also doubled.
TIME (hours) ½ 1 2 3 4
DISTANCE (km) 30 60 120 180 240
The relationship between time and distance, can be illustrated by
the equation:
D = 60t

9. Dependent & Independent
VARIABLES
DIRECT
VARIATION
Y varies directly as x means that y = kx
where k is the constant of variation.
Prepared by: Ms. Kristine S. Layson PCS – Junior High School , 01 October 2017

10. REMEMBER!
Please take note of this one..

11. REMEMBER!
Please take note of this one..
When one value increases, the other value
also increases OR
When one value decreases, the other value
also decreases.
When one quantity always changes
by the same factor,
the two quantities are directly proportional.
01
02

12. The more time I drive
(at a constant rate), the more
miles I go.
01
02
03
04
05
06
Real World Examples
OF DIRECT VARIATION
The more hours I work,
the more money I make.
The more BTS Albums I
purchase, the more
money it costs.
The less cheese I buy at
the deli, the less money I
pay.
The less bead necklaces
I sell, the less profit I will
earn.
The less calls and texts I
make, the less amount I
need to pay.

13. Characteristics of Direct Variation
Graph…
The graph will always go through the
ORIGIN on the coordinate plane…this
means when x=0, y=0 on the graph.
The graph will always be in Quadrant I
and Quadrant III.
The graph will always be a straight line.
As the “x” value increases, the “y” value
always increases.
How do we know if we have a direct variation?
We can look at three different aspects..

14. LET US WORK
ON SOME
EXAMPLES..

15. TELL IF THE FOLLOWING GRAPH IS A DIRECT
VARIATION OR NOT.
NO NO
NO NO

16. NO YES
YES NO
TELL IF THE FOLLOWING GRAPH IS A DIRECT
VARIATION OR NOT.

17. x is the independent variable
Y is the dependent variable
DIRECT
VARIATION
Y varies directly as x means that:
y = kx
where k is the constant of variation.
Prepared by: Ms. Kristine S. Layson PCS – Junior High School , 01 October 2017

18. EXAMPLES OF DIRECT
VARIATION
y = 4x  k = 4
y = x  k = 1
y = 2x  k = 2
y = 2.5x  k = 2.5
y = ⅝ x  k = ⅝
y = 0.75  k = 0.75
x

19. As “x” increases in
value, “y” also
increases in value.
Direct Variation & Tables of Values
You can make a table of values for “x” and “y” and see how the values behave.
You could have a direct variation if…
As “x” decreases in
value, “y” also
decreases in value.

20. EXAMPLES OF DIRECT VARIATION:
X Y
6 12
7 14
8 16
Note: As “x” increases,
6 , 7 , 8
“y” also increases.
12, 14, 16
What is the constant of
variation of the table above?
12/6 = k → k = 2
Now you can write the equation
for this direct variation:
y = 2x

21. EXAMPLES OF DIRECT VARIATION:
Note: As “x” decreases,
30, 15, 9
“y” also decreases.
10, 5, 3
What is the constant of
variation of the table above?
10/30 = k → k = 1/3
Now you can write the equation
for this direct variation:
y = 1/3x
X Y
30 10
15 5
9 3

22. Using Direct Variation to find unknowns
Given y varies directly with x,
and y = 28 when x = 7,
Find x when y = 52.
X Y
7 28
? 52
1. Find the constant of variation
y = kx → 28 = k · 7
k = 4
2. Use y = kx. Find the unknown (x)
52 = 4x
x = 13
Therefore:
X = 13 when Y = 52

23. Using Direct Variation to find unknowns
Given that y varies directly
with x, and y = 3 when x = 9,
Find y when x = 40.5
X Y
9 3
40.5 ?
1. Find the constant of variation
y = kx → 3 = k ·9
k = 1/3
2. Use y = kx. Find the unknown (x)
y = 1/3 (40.5)
y = 13.5
Therefore:
Y = 13.5 when x = 40.5

24. If a varies directly as b, and
a = 7 when b = 5, find the
value of a when b = - 3.
01
02
03
04
05
EXERCISES on DIRECT VARIATION
Solve for the indicated variable in each problem:
If y varies directly as t, and
y = 80 when t = 2, find the
value of y if t = 10.
If p varies directly as H, and
p = 1 000 when h = 20, find p
when h = 45.
If d is directly proportional
to the square of t, and
d=20 when t = 2, find d
when t = 8.
If x is directly proportional
to the cube of y, and x = 64
when y = 2, find x when
y = 5.

25. A car uses 8 gallons of
gasoline to travel 240 miles.
How much gasoline will the
car use to travel 400 miles?
Find points in table
PROBLEM #1:
PROBLEM SOLVING
Using Direct Variation to solve word problems
Find the constant of
variation and equation
Use the equation
to find the unknown

26. Julia wages vary directly as the
number of hours she works.
If her wage for 5 hours is
Php 150, how much will be her
wage for 12 hours?
Find points in table
PROBLEM #2:
PROBLEM SOLVING
Using Direct Variation to solve word problems
Find the constant of
variation and equation
Use the equation
to find the unknown

27. Joy walks to school while Jeffrey runs.
Who will reach the school first?
If they both ran to school at the same
time, would they reach the school at
the same time?
How does the speed at which one travels
relate to the time it takes to travel a
certain distance?
Jeffrey and Joy are siblings who live in the same house
and go to the same school by the same route.

28. SPEED and
TIME are
inversely
related to each
other.

29. Please take note
of this one…
When one value INCREASES, the other value
also DECREASES OR
When one value DECREASES, the other value
also INCREASES.
When one quantity always changes
by the SAME FACTOR.

30. REMEMBER!!!!

31. Decide whether the given
quantities are directly or
inversely related.
EXAMPLE #1

32. Characteristics of INVERSE
VARIATION Graph…
The graph forms a CURVE LINE.
None of the graphs contain (0, 0). In
other words, (0, 0) can never be a
SOLUTION on a inverse equation.
How do we know if we have an inverse
variation?
We can look at three different aspects..
Prepared by: Ms. Kristine S. Layson S.Y. 2018 – 2019

33. Let’s try working on the following examples..
1 If y varies inversely as x and x = 18 when y = 6,
find y when x = 16.
2 Given that a varies inversely with b and
a = 30 when b= 3. Find a when b = 10.
3 If q varies inversely as r and q = 15 when r = 4,
find q when r = 12.

34. If 48 men can do a
piece of work in 24
days,
In how many days
will 36 men
complete the same work?
TYPE OF
VARIATION?
CONSTANT OF
VARIATION?
EQUATION OF
VARIATION?
FINAL ANSWER?
2 QUANTITIES
BEING COMPARED?

35. The number of days a bag of
bread lasts varies inversely as
the number of people who
consumes it.
if a bag of bread lasts
3 days for 6 people,
how long will it lasts for
2 people?
TYPE OF
VARIATION?
CONSTANT OF
VARIATION?
EQUATION OF
VARIATION?
FINAL ANSWER?
2 QUANTITIES
BEING COMPARED?

36. Thank you