We will learn about what is variation, direct variation, indirect variation and joint variation.

In *Mathematics*, we usually deal with two types of quantities-Variable quantities (or variables) and Constant quantities (or constants). If the value of a quantity remains unaltered under different situations, it is called a ** constant**. On the contrary, if the value of a quantity changes under different situations, it is called a

For example: 4, 2.718, 22/7 etc. are constants
while speed of a train, demand of a commodity, population of a town etc. are variables.

In a mathematical equation where a relationship is established for some type of parameters normally two types quantities exist. One is constant that doesn’t change with the changes of other parameters in the equation and another is the variables which change for different situations. The changing of variable parameters is called as variation.

In problems relating to two or more variables, it is seen that the value of a variable changes with the change in the value ( or values ) of the related variable (or variables). Suppose a train running at a uniform speed of v km./h. travels a distance of d km. in t hours. Obviously, if t remains unchanged then v increases or decreases according as d increases or decreases. But if d remains unchanged, then v decreases or increases according as t increases or decreases. This shows that the change in the value of a variable may be accompanied differently with the change in the values of related variables. Such relationship with regards to the change in the value of a variable when the values of the related variables change, is termed as ** variation**.

This can be explained by an example of simple equation y = mx where m is a constant. If we assume that the value of m as 5 then the equation becomes as y = 5x.

When x = 1, y = 1 × 5 = 5

When x = 2, y = 2 × 5 = 10

When x = 3, y = 3 × 5 = 15

Simply the value of y is changing with the different values of x.

This is the variation of y with different values of x and similarly it can be shown that with different values of y the value of x changes.

Variation can be of different types according the pattern of changing or relationships of variables.

**Direct
Variation**: In a variation if variables change proportionately
i.e. either increase or decrease together then it is called as direct
variation. If X is in direct variation with Y, it can be symbolically written
as X α Y.

**Inverse or Indirect
Variation**: In
inverse or indirect variation the variables change disproportionately or when one
of the variables increases, the other one decreases. So behavior of the
variables is just the opposite of direct variations. That is why it is called
as Inverse or indirect variation. If X is in indirect variation with Y, it can
be symbolically written as X α \(\frac{1}{Y}\).

**Joint
Variation**: If more than two variables are related
directly or one variable changes with the change product of two or more
variables it is called as joint variation. If X is in joint variation with Y
and Z, it can be symbolically written as X α YZ.

**Combined
Variation**: Combined variation is a combination of
direct or joint variation, and indirect variation. So in this case
three or more variables exist. If X is in combined variation with Y and Z, it
can be symbolically written as X α \(\frac{Y}{Z}\) or X α \(\frac{Z}{Y}\).

**Partial
Variation**: When two variables are related by a formula
or a variable is related by the sum of two or more variables then it is called
as partial variation. X = KY + C (where K and C are constants) is a straight
line equation which is a example of partial variation.

Here are some examples of direct and inverse variations.

**Direct
Variation: **Perimeter of circle C= 2πr
where 2 and π are constants and C increases if r increases, decreases if r
decreases. So C is in direct variation with r.

**Inverse
Variation**: If I need to go a distance of S with
velocity V and time T then T = \(\frac{S}{V}\). Here the distance S is constant. If velocity
increases it will take less time so T decreases. So T is in indirect variation
with V.

*We will discuss more about such variations, which are classified into three types:*

**(1) Direct Variation**

**(2) Inverse Variation and**

**(3) Joint Variation.**

**●** **Variation**

**What is Variation?****Direct Variation****Inverse Variation****Joint Variation****Theorem of Joint Variation****Worked out Examples on Variation****Problems on Variation**

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