The document explains joint and combined variation, where joint variation describes a relationship where one quantity varies directly with the product of two other quantities, exemplified by the area of a rectangle. Combined variation refers to a dependence on multiple variables, with some varying directly and others inversely, illustrated through given equations and examples. Specific calculations for joint and combined variations are provided to show how to derive the constant and final answers.

1. Joint and
Combined
Variation

2. Joint Variation
▪ It occurs when one quantity varies
directly as the product of two or more
other quantities

3. Joint Variation
y varies jointly as x and z if there
exists a nonzero number k such
that y = kxz, where x≠0 and z≠0

4. Joint Variation
Example:
The area of a rectangle varies jointly as
its length and width. Find the equation
of joint variation if A = 60 sq.ft, l = 15 ft,
and w = 4 ft

5. Joint Variation
Solution:
The area (A)
Varies jointly = (K)
as
its length (h) and (.)
Width (W)

6. Joint Variation
Solution:
A = klw
60 = k (15)(4)
60 = 60
k = 1

7. Combined Variation
▪ It 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 (when the rest of
the variables are held constant).

8. Combined Variation
If a situation is modelled by an equation
of the form:
y =
kx
z
Where k is a nonzero constant, we say
that y varies directly as x and inversely
as z.The number (k) is called the
constant variation

9. Combined Variation
Example:
If y varies directly as x and inversely as
z, and y = 22 when x = 4 and z = 6, find y
when x = 10 and z = 25

10. Combined Variation
Solution:
Step 1: y =
kx
z
Step 2: 22 =
k(4)
6
132 = 4k
33 = k

11. Combined Variation
Solution:
Step 3: y =
33x
z
Step 4: y =
33(10)
25
FinalAnswer: y =
66
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