This document provides guidance on using a strategic intervention material (SIM) to learn about combined variation. It begins with reminders on using the SIM independently and thoroughly. The least mastered skill is solving problems involving combined variation, where a variable depends on two or more other variables in a direct and inverse relationship. Examples are provided of writing equations to represent relationships like "W varies jointly as c and the square of a and inversely as b." Several practice activities involve writing equations and solving for unknowns. The document concludes with an assessment to check understanding of combined variation concepts.

1. Strategic Intervention Material in Mathematics-9
𝑨 = 𝝅𝒓 𝟐
Prepared by: Approved by:
MARICEL T. MAS JULITA M. VALDEZ
T-III/Lipay Natioanl High School Principal-II

2. Hello!
I am Olaf.
I am here to be your guide as you try to
unravel and be familiarize the lesson on:
Combined Variation
GOOD LUCK!You got this, man.
Guide
Card
2

3. A few reminders…
This learning package is intended to supplement your classroom
learning while working independently . The activities and exercises will widen your
understanding of the different concepts you should learn.
Purpose of this Strategic Intervention Material
(SIM)
How To Use This Strategic Intervention Material (SIM)
•Keep this material neat and clean.
• Thoroughly read every page.
• Follow carefully all instructions indicated in every activity.
• Answer all questions independently and honestly.
• Write all your answers on a sheet of paper.
• Be sure to compare your answers to the KEY TO
CORRECTIONS only after you have answered the given tasks.
• If you have questions or clarifications, ask your teacher.
3

4. Task Analysis
LEAST MASTERED SKILLS
solve problems involving combined variations
SUB TASKS
translate into variation statement a relationship
involving combined variation between two quantities
given by a mathematical equation
find the unknown value/variable of the variation given
the mathematical equation
4

5. Let’s start
Never say
“I can’t”
Always say
“I’ll try”
5

6. Overview
Inverse:
Bottom
Combined variation is another physical relationship among
variables. This is the kind of variation that involves both the
direct and inverse variations.
The statement “z varies directly as x and inversely as y”
means 𝑧 =
𝑘𝑥
𝑦
or 𝑘 =
𝑧𝑦
𝑥
, where k is the constant of variation.
Hi! Here are an
introduction
about our topic.
K is
always on
the top
6

7. Whatto
know?
These are some examples of relationship among
variables:
Examples: 1. Translating statements into mathematical equations
using k as the constant of variation .
a. “T varies directly as a and inversely as b”.
𝑇 =
𝑘𝑎
𝑏
b. Y varies directly as x and inversely as the square of z.
𝑌 =
𝑘𝑥
𝑧2
Hi! Do you still
remember this?
7

8. Activitycardno.1:
DVandIVCombined!
Using k as the constant of variation, write the equation of variation for
each of the following.
1. W varies jointly as c and the square of a and inversely as b.
2. P varies directly as the square of x and inversely as s.
3. The electrical resistance R of a wire varies directly as its length l
and inversely as the square of its diameter d.
4. The acceleration A of a moving object varies directly as the
distance d it travels and inversely as the square of the time t it travels.
5. The pressure P of a gas varies directly as its temperature t and
inversely as its volume V.
8

9. Whatto
process
At this phase, you will be provided with example that will lead
you in solving the problems you will encounter in this section.
Example
1. If z varies directly as x and inversely as y, and z = 9 when x = 6 and
y = 2, find z when x = 8 and y = 12.
𝑧 =
𝑘𝑥
𝑦
Using the equation, 𝒛 =
𝟑𝒙
𝒚
(9) =
𝑘(6)
(2)
𝑧 =
(3)(8)
(12)
18
6
=
6𝑘
6
𝒛 = 𝟐
k= 3
Now, by this time, you’re
going to apply your skill
in writing an equation to
solve the unknown
variable. You may use
this example to guide you
in solving the activities
that follow
9

10. Now, its your
turn. I know you
can do it!
Activity card no. 2
Solve the following
1. If r varies directly as s and inversely as the square of u, and r = 2 when
s = 18 and u = 2, find r when u = 3 and s = 27.
2. p varies directly as q and the square of r and inversely as s, and p = 40
when q = 5, r = 4 and s = 6, find p when q = 8, r = 6 and s = 9.
3. w varies directly as x and y and inversely as 𝑣2 and w = 1200 when x =
4, y = 9 and v = 6. Find w when x = 3, y = 12, and v = 9.
4. Suppose p varies directly as 𝑏2
and inversely as 𝑠3
. If 𝑝 =
3
4
when
b=6 and s= 2, find b when p=6 and s=4
5. If x varies as the square of y and inversely as z and x = 12 when y = 3
and z = 6, find x when y = 9 and z = 6.
10

11. Activity card no. 3
Example.
The volume V of a given mass of gas varies directly as the temperature T and
inversely as the pressure P. If V= 250𝑐𝑚3
when T= 500
and P = 24𝑘𝑔/𝑐𝑚2
, find the
volume when T= 600
and P=30𝑘𝑔/𝑐𝑚2
.
Steps: a. Translate the statement into equation.
b. Then, apply the substitution property to find the what is asked.
V=
𝑘𝑇
𝑃
V=
𝑘𝑇
𝑃
250 =
(𝑘)(50)
24
V=
(120)(60)
30
6000 = 50k V = 240𝑐𝑚3
50 50
k= 120
Having developed your
knowledge of the
concepts in the previous
activities, your goal now
is to apply these
concepts in various real-
life situations.
11

12. Now , it’s your
turn!
Solve the following:
1. The current I varies directly as the electromotive force E and
inversely as the resistance R. If in a system a current of 20 amperes
flows through a resistance of 20 ohms with an electromotive force
of 100 volts, find the current that 150 volts will send through the
system.
2. The force of attraction, F of a body varies directly as its mass m
and inversely as the square of the distance d from the body. When
m = 8 kilograms and d = 5 meters, F = 100 Newtons. Find F when
m = 2 kilograms and d = 15 meters.
12

13. This is it!
Your final
task! Good
luck! 
Assessment
cardDIRECTIONS: Choose the letter of the correct answer. Write your answer before the
number
1. A variation which describes a situation where a variable depends on two (or more) other
variables, and varies directly with some of them and varies inversely with others.
A. Direct B. Inverse C. Joint D. Combined
2. If w varies directly as the square of x and inversely as p and q then which of the following
equations describes the relation ?
A. W= K𝑥2
pq B. W =
𝑘𝑥2
𝑝𝑞
C. w=
𝑘𝑥
𝑝𝑞
D. w= Kxpq
3. w varies directly as the square of x and inversely as p and q. If w = 12 when x = 4, p = 2 and
q = 20, find k.
A. 3 B. 30 C. 300 D.
1
30
4. Refer to item # 3, find w when x = 3, p = 8 and q = 5.
A. 10 B. 9 C.
27
4
D. 5
13

14. Don’t give
up. I know
you can do it!

5. The maximum load that a cylindrical column with a circular cross section can hold varies
directly as the fourth power of the diameter and inversely as the square of the height. A 9 meter
column 2 meters in diameter will support 64 metric tons. How many metric tons can be
supported by a column 9 meters high and 3 meters in diameter?
A. 324 mt B. 234 𝑚𝑡 C. 342 𝑚𝑡 D. 432𝑚𝑡
6. The volume of gas varies directly as the temperature and inversely as the pressure. If the
volume is 230 cubic centimeters when the temperature is 300ºK and the pressure is 20 pounds
per square centimeter, what is the volume when the temperature is 270 ºK and the pressure is 30
pounds per square centimeter?
A. 138cm B. 138𝑐𝑚3
C. 138 D. 183cm
7. If h varies jointly as 𝑗2 and i and inversely as g, and h = 50 when j = 2, i = 5, and g = ½ , find
h when j=4, I = 10, g=
1
4
A. 25 B. 100 C. 800 D. 805
14

15. Try this
one!
Enrichment
cardPhotographers are always looking for ways to make their pictures sharper. One way is
to focus the lens at the hyperfocal distance. Focusing the lens at this distance will produce a
photograph with the maximum number of objects in focus. Photographers often use the
hyperfocal distance when taking pictures of landscapes. The two images at the right were shot
with the same camera at the same settings, except the one on the bottom was taken with the lens
focused at the hyperfocal distance, giving the extra degree of sharpness. Example 2 explores
how the hyperfocal length H can be calculated using the focal length L of the camera lens in
millimeters and the f-stop, or aperture, f. The aperture is a setting that tells you how wide the
lens opening is on a camera. It is the ratio of the focal length to the diameter of the lens opening
and so has no units.
The hyperfocal length H in meters varies directly with the square of the focal length L in
millimeters, and inversely with the selected aperture f.
a. Write a general variation equation to model this situation.
b. Find the value and unit of k when the hyperfocal length H is 10.42 m when using a 50 mm
lens (L) and the aperture f is set at 8.
c. Find the hyperfocal length needed if you want to shoot with a 300 mm lens at an aperture
setting of 2.8. Include the units in your calculations.
15

16. Referencecard
Learner’s Material Mathematics – Grade 9
First Edition, 2014 pp. 220-222.
Website links:
http://www.mesacc.edu/~scotz47781/mat120/notes/variation/combined/combined_
practice.html#
https://www.google.com.ph/search?ei=QkqrW4iKENn-
9QPUhKOYAw&q=metric+ton+symbol&oq=metric+ton+symbol&gs_l=psy-
ab.3..0j0i22i30k1.2429.4069.0.4763.7.7.0.0.0.0.637.1005.3-1j0j1.2.0....0...1c.1.64.psy-
ab..5.2.1004....0.l-x6ubAOgGA
http://www.mesacc.edu/~scotz47781/mat120/notes/variation/combined/combined.html
16

17. ANSWER CARD
Activity Card no. 1
1. 𝐴 =
𝑘𝑎2 𝑐
𝑏
2. 𝑃 =
𝑘𝑥2
𝑠
3. 𝑅 =
𝑘𝑙
𝑑2
4. 𝐴 =
𝑘𝑑
𝑡2
5. 𝑃 =
𝑘𝑡
𝑉
Activity Card no. 2
1. r=
4
3
2. p=96
3. w=533
1
3
4. b= 48
5. x= 108
Activity card no. 3
1. 𝐼 = 30 𝑎𝑚𝑝𝑒𝑟𝑒𝑠
2. 𝐹 =
25
9
newtons
Assessment Card
1. D 6. B
2. B 7. C
3. B
4. C
5. A
Here are the key
checkpoints! I
hope you made it!
Enrichment Card
a. 𝐻 =
𝑘𝐿2
𝑓
b. K= 0.033 m/𝑚2
c. H= 1060.7 m
17