What is Variation


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

Variable and Constant:

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 variable.

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



11 and 12 Grade Math 

From What is Variation to Home Page


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Multiplication of a Number by a 3-Digit Number |3-Digit Multiplication

    Mar 28, 24 06:33 PM

    Multiplying by 3-Digit Number
    In multiplication of a number by a 3-digit number are explained here step by step. Consider the following examples on multiplication of a number by a 3-digit number: 1. Find the product of 36 × 137

    Read More

  2. Multiply a Number by a 2-Digit Number | Multiplying 2-Digit by 2-Digit

    Mar 27, 24 05:21 PM

    Multiply 2-Digit Numbers by a 2-Digit Numbers
    How to multiply a number by a 2-digit number? We shall revise here to multiply 2-digit and 3-digit numbers by a 2-digit number (multiplier) as well as learn another procedure for the multiplication of…

    Read More

  3. Multiplication by 1-digit Number | Multiplying 1-Digit by 4-Digit

    Mar 26, 24 04:14 PM

    Multiplication by 1-digit Number
    How to Multiply by a 1-Digit Number We will learn how to multiply any number by a one-digit number. Multiply 2154 and 4. Solution: Step I: Arrange the numbers vertically. Step II: First multiply the d…

    Read More

  4. Multiplying 3-Digit Number by 1-Digit Number | Three-Digit Multiplicat

    Mar 25, 24 05:36 PM

    Multiplying 3-Digit Number by 1-Digit Number
    Here we will learn multiplying 3-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. 1. Multiply 201 by 3 Step I: Arrange the numb…

    Read More

  5. Multiplying 2-Digit Number by 1-Digit Number | Multiply Two-Digit Numb

    Mar 25, 24 04:18 PM

    Multiplying 2-Digit Number by 1-Digit Number
    Here we will learn multiplying 2-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. Examples of multiplying 2-digit number by

    Read More