Variation

What is Variation?

In the field of Mathematics, we usually deal with numbers, figures, elements, groups, items, etc. Elements of a group or individuals that remain the same, over time are called constants. As the name suggests they remain consistent that is constant.

Examples- 4, 2.5, 22/7, 3.142, 1.732 are all constants, they do not change at all.

Whereas, in the field of Mathematics, when elements of a group or individually do not remain the same, over time are called variables. As the name suggests their values vary from time to time.

Examples- Speed of a moving bus, Time taken by the train to reach a specific location.

In a mathematical equation when bonding is built between two given quantities that exist. One quantity tends to show a change with the change in the limitations in the equation and these quantities are usually termed as variables, while the other quantity that shows no change with the change in the limitations of the equation is called constant quantities. 

A change in the limitations(parameters) of a given variable of an equation is called a Variation. In other words, such a relation that shows changes in the values of a quantity when the values of the related variables are changed is called Variation.

An Example Of Variation-

Given is a simple equation a = bc where b is a constant. We assume that the value of ‘b’ is taken as 5(constant, remains the same) then the equation becomes a = 5c.

Consider the value of c = 1

Therefore, a = 5(1) = 5

Consider the value of c = 2, 

Therefore, a = 5(2) = 10

Consider the value of c = 3, 

Therefore, a = 5(3) = 15

From the above-given example, we understand that the value of ‘a’ changes concerning the different given values of ‘c’.

This is a perfect example of a variation of ‘a’ with different values of ‘c’ and thus, we can say that with change in the values of ‘a’ the values of ‘c’ change too.

In problems related to two or more variables, it can be observed that the value of an element changes with the change in the value of the related element. This tells us that the change in a parameter of an element may be associated not similarly with the change in the value of an associated parameter.

Different Types Of Variation

Direct Variation-

If in a given equation, the parameters of ‘x’ and ‘y’ increase or decrease at the same time, that is at one or the same time then, it is called direct variation.

We can also say that ‘x’ and ‘y’ are directly proportional to each other.

It is denoted by writing it as x ∝ y

Indirect Variation-

If in a given equation, the parameters of ‘x’ increase and ‘y’ decreases or the parameters of ‘y’ increase and ‘x’ decrease at the same time, that is at one or the same time then, it is called an indirect variation or inverse variation. We can also say that ‘x’ and ‘y’ are indirectly proportional to each other.

It is denoted by writing it as x ∝ 1/y

Joint Variation

When in a given equation more than two quantities are linked directly to each other or one quantity changes with the change in the product of two or more quantities it is called joint variation.

Combined Variation

When a combination of direct variation or joint variation, and indirect variation is present in an equation that is called a combined variation. 

In this case, there exist three or more 3 variables.

Partial Variation

If two quantities are linked to each other with the help of a formula or when a quantity is related to another by the sum of two or more elements, then it is called partial variation.

Example of questions related to Variation

Q1. Re-write the given statements with the help of variation symbols only.

1. The area (a) of a cube is directly proportional to the edges of a cube (s).
2. The use of diesel (d) by a car and the distance the same car travels (m) are in direct variation.

A2. Solution:

Ans i) a ∝ s

Ans ii) d ∝ m

Conclusion-

Variation can teach us a lot of new things. This topic can also be studied via visual observation and practical knowledge by observing our surroundings in day-to-day life.

A train shows us variation, moving cars, the phases of the moon, cloud patterns, etc all are daily observations and examples of variation in the practical/real world.