Variations

Definition 1.1 A variable y is
in direct variation with another
variable x (or y is directly
proportional to x) if y and x
are related by an equation in
the form y=kx, where k is a
nonzero constant.

Definition 1.2 The constant
k in the equation y = kx is
called the constant of
variation or the constant of
proportionality.

EXAMPLE 1
A quantity y varies directly as another
quantity x. If x = 20 then y = 8.
1.Write an equation that represents
the relationship.
2.Find the value of y when x = 125.

SOLUTION
Given that y varies directly as x, this means
that y = kx where k is a nonzero constant.
1.First, find k given that y = 8 if x = 20.
Substitute these values in y = kx to find
the value of k.
y = kx
8 = k • 20

Divide both sides of the equation by 20.
k = or
Thus, y =
20
8
5
2
χ
5
2

2. To find the value of y when x = 125,
substitute 125 for x in the equation
y =
25
y =
Hence, y = 50 when x = 125.1
50521
5
2
=///•
/
χ
5
2

Definition 1.3 A variable is in
direct nth power variation with
another variable x (or y is directly
proportional to the nth power of
x) if y and x are related by an
equation in the form y = kx ,ⁿ
where k is a nonzero constant
and n is a positive real number.

EXAMPLE 1
Assume that S varies directly as
the second power of r. If the
constant of variation is equal to
4π. Write an equation that
represents the relationship
between S and r.

Solution
The given relationship is a
direct nth power variation in
which n = 2 and the constant
of variation k is equal to .
Hence, the required equation
is S = r
π4
π4 2

EXAMPLE 2
Suppose L varies directly as
the cube of t. If t = 5 then
L = 162.5. If t = 2 what is
the value of L?

SOLUTION
Since L varies directly as the
cube of t, the equation is L = kt
where k is a nonzero
constant. First, find k by
substituting the values t = 5 and
L = 162.5 in the equation L =
3
3

L = kt
162.5 = k•(5)
162.5 = 125k
k = or 1.3
3
3
125
5.162

Hence, L = 1.3t . To find L when
t = 2, substitute 2 for t in the
equation.
L = 1.3t = 1.3(2) = 1.3(8) = 10.4
Thus, when t = 2, L = 10.4.
3
33

Definition 1.4 A variable y is in
inverse nth power variation with
another variable x (or y is
inversely proportional to the nth
power of x) if the variables are
related by an equation in the form
y= , where k is a nonzero
constant and n is a positive real
number.
n
x
k

EXAMPLE
Suppose y varies inversely
as x and y = when x = 4.
Find the value of x when
y = .
5
1
35
1

SOLUTION
Since y varies inversely as x, then y =
where k is a nonzero constant. Find k by
substituting x = 4 and y = .
x
k
5
1
x
k
y =
45
1 k
=

Multiply both sides of the equation by the
LCM of 4 and 5, which is 20. Then solve
for k.
4 = 5k
k =
Therefore the equation f the inverse
variation is y = or y = .
5
4
x
5
4
x5
4

To find the value of x when y = ,
substitute for y in the equation y = .
y =
=
Multiplying both sides of the resulting
equation by 35x, which is the LCM of 35
and 5x , you get
x = 28.
35
1
35
1
x5
4
x5
4
35
1
x5
4

In instances where a variable varies
directly or inversely, or both, with
two or more other variables in the
same formula, the relationship
among the variables is called
combined variation. When a variable
varies directly with two or more
variables, the relationship among the
variables is called a joint variation.

That is, a variable y varies
jointly as variables x and z
if y = kxz, where k is a
nonzero constant. A joint
variation is also a
combined variation.

EXAMPLE
Suppose z varies directly as
the square of t and inversely
as u. When t = 3 and u = 12,
z = 3. Find the value of z if
t = 5 and u = 15.

SOLUTION
The equation representing the variation is
z = where k is a nonzero constant still to
be found. To find the value of k, substitute
3 for t, 12 for u, and 3 for z in the equation
z = .
u
kt2
u
kt2

z =
3 =
Multiply both sides of the equation by 12.
Then divide both sides of the equation by 9.
36 = k•9
= k Therefore, z =
k = 4
12
)3( 2
k
u
kt2
9
36
u
t2
4

To find the value of z if t = 5 and u = 15,
substitute 5 for t and 15 for u in the
equation z = .
Thus, when t = 5 and u = 15.
u
t2
4
3
20
15
100
15
254
15
)5(44 22
or
u
t
z =
•
===
3
20
=z

Variations

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Variations