Joint variation is when one quantity varies proportionally with two or more other quantities. Some key points: - Joint variation can be represented by an equation of the form y = kxz, where y varies jointly as x and z with k as the constant of variation. - Examples include the cost of pencils varying jointly with the number bought, and the area of a triangle varying jointly with its base and height. - To solve a joint variation problem, the equation is set up and the constant of variation is determined using given values to find the value of the variable for a new set of quantities. - Joint variation has many real world applications like the force on an object varying jointly with its

1. JOINT VARIATION

2. Definition Of Joint
Variation
Joint variation is the same as direct
variation with two or more quantities.
Direct variation occurs when two quantities
change in the same manner.
That is:
Increase in one quantity causes an increase
in the other quantity
Decrease in one quantity causes a
decrease in the other quantity

3. Joint Variation :
Joint variation is a variable which is
proportional to the product of two or
more other variables
Example :
 Y = K X Z Z = 11a X 15b

4. Introduction
 The meaning of the phrase "Joint Variation" can
be gleaned from the meaning of the two words
"Joint" and "Variation". In other words, it refers
to a case where one variable or quantity varies
jointly with several other variables or
quantities. Symbolically, when we say that the
variable Z varies jointly with the variables X and
Y, we mean that if either X, or Y or both X and Y
are varied, then Z will vary accordingly. This is
an example of a variable varying jointly with two
other variables. In general. a variable may vary
jointly with many variables (two or more).

5. For Example:
 The cost of a pencil and the number of
pencils you buy.
Buy more pay more.....Buy less pay less.
Direct variation between variables x and y
can be expressed as:
y = kx, where 'k' is the constant of
variation and k ≠ 0
y = kxz represents joint variation. Here, y
varies jointly as x and z.

6. More Examples on Joint Variation
 y = 7xz, here y varies jointly as x and z
y = 7x2z3, here y varies jointly as x2 and z3
Area of a triangle = is an example of joint
variation. Here the constant is 1. Area of a
triangle varies jointly with base 'b' and
height 'h'
Area of a rectangle = L x M represents
joint variation. Here the constant is 1. Area
of a rectangle varies jointly with length 'l'
and width 'w'.

7. Solved Example on Joint Variation
 Ques: Assume a varies jointly with b and c. If b = 2
and c = 3, find the value of a. Given that a = 12 when
b = 1 and c = 6.
 Solution:
 Step 1: First set up the equation. a varies jointly
with b and c a = kbcStep 2: Find the value of
the constant, k. Given that a = 12 when b = 1
and c = 6
a = kbc
12 = k x 1 x 6
⇒ k = 2Step 3: Rewrite the equation using the
value of the constant 'k'
a = 2bc

8.  Step 4: Using the new equation, find the
missing value.
If b = 2 and c = 3, then a = 2 x 2 x 3 = 12Step
5: So, when a varies jointly with b and c and If b
= 2 and c = 3, then the value of a is 12.
Real-world Connections for Joint
Variation
 Force = mass × acceleration. The force exerted
on an object varies jointly as the mass of the
object and the acceleration produced.

9. exercise
 If y varies jointly as x and z, and y = 33 when x
= 9 and z = 12, find y when x = 16 and z = 22.
 If f varies jointly as g and the cube of h, and f =
200 when g = 5 and h = 4, find f when g = 3
and h = 6.
 Wind resistance varies jointly as an object’s
surface area and velocity. If an object traveling
at 40 mile per hour with a surface area of 25
square feet experiences a wind resistance of
225 Newtons, how fast must a car with 40
square feet of surface area travel in order to

10.  For a given interest rate, simple interest varies
jointly as principal and time. If $2000 left in an
account for 4 years earns interest of $320, how
much interest would be earned in if you deposit
$5000 for 7 years?
 If a varies jointly as b and the square root of c,
and a = 21 when b = 5 and c = 36, find a when
b = 9 and c = 225.
 The volume of a pyramid varies jointly as its
height and the area of its base. A pyramid with
a height of 12 feet and a base with area of 23
square feet has a volume of 92 cubic feet. Find
the volume of a pyramid with a height of 17 feet
and a base with an area of 27 square feet.

11. answer
868
1.y=—
9
2.f=405
3.velocity=30 miles per hour
4. interest= $1400
5.a=126
6.volume=153 cubic feet

12. Importance of Joint Variation in Real
Life
 We have seen several real world examples of Joint
Variation. This concept is widely used in what-if
analysis. We will illustrate what-if analysis using an
example. If we look at the investment example, we
saw that the interest earned varies jointly with the
amount deposited, the rate of interest and the
period of investment. A what-if analysis would
involve varying each of the variables, in turn, and
finding out the effect on the rest of the variables.
 For example we could have several investment
plans where the rate of interest would be different,
the period of investment would be different and so
on. The what-if analysis would find the best option
for the investor, depending on his/her
requirements, by varying the variables in turn and

13. Thankyou for understanding
 By; EUNICE LIBAO
 G9- ILANG-ILANG