Variation
Two quantities A and B may be such that as
one quantities changes in value, the other quantity also changes in value
depending on the change in the value of the first quantity. .
Direct variation
One quantity A is said to vary directly as
another quantity B, if the two quantities depend upon each other in such a
manner that if B is increased in a certain ratio, A is increased in the same
ratio and if B is decreased, A is decreased in the same ratio. .
This is denoted as A a B (A varies directly
as B). .
If A a B then A = kB, where k is a constant.
It is called a constant of proportionality. .
Inverse variation
A quantity A is said to vary inversely as
another quantity B, if the two quantities depend upon each other in such a
manner that if B is increased in a certain ratio, A is decreased in the same
ratio and if B is decreased, then A is increased in the same ratio. It is the
same as saying that A varies directly with 1/B. .
It is denoted as A a 1/B i.e., A = k/B where
is k is a constant of proportionality. .
Joint variation
If there are three quantities A, B and C such
that A varies with B when C is constant and varies with C when B is constant,
then A is said to vary jointly with B and C when both B and C are varying. .
Then A a BC or A = KBC where k is the
constant of proportionality. .
Worked out examples
EXAMPLE 1
The volume of a solid varies jointly with its
height as well as the base area. s1 and s2 are two such
solids. The base area of solid s1 is 100 sq.m and the height is 5 m,
the volume is 1500 cu.m. Find the base area of solid s2 whose height
is 10 m and volume is 4500 cu.m. .
Solution
Let the
volume, height and base area of the solid be denoted by V, h and a
respectively. .
&⇒ V m h a
&⇒ 
.
Taking V1 = 1500 cu m, h1
= 5m,
a1 = 100 sq m, h2 = 10m
and
V2 = 4500 cu m. .
a2 = 
= 
= 150 sq m. .
EXAMPLE 2
A varies directly as b2. When b =
3, a = 6. Find a when b = 6. .
Solution
a µ b2. .
&⇒ 
Taking a1 = 6, b1 = 3,
and b2 = 6;
a2 = a1 
= 24
