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Constant of Variation

"Variation" defines a concept that deals with variability in mathematics. Variation is defined by a  ny change in some quantity due to change in another. We often come across with different types of variation problems in mathematics. In one go, these problems are seemed to be really hard. But once we have the proper knowledge in this context, we find them quite simple.

The problems of variation include simple equations and relations involving usually one or two variables (though there may be more variables too).

Before dealing with variation problems, it is necessary to learn about constant of variation. Constant of variation is a constant quantity that is plugged in some variation relation in order to convert it into an equation.

The variation relation is to be multiplied by some constant number, (generally represented by "k") for getting an equation. This constant is known as a "constant of variation". Let us go ahead and learn more about constant of variation, its types and problems based on this.

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Definition

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The constant of variation is defined as a number which connects two (or even more) variables which are proportional to each other, either directly or inversely. We may also explain the concept of constant of variation as a ratio with which the two variables change in direct variation or direct proportion. The constant of variation is also referred to the constant of proportion.

A variation means the change in a variable with the change in another variable. Generally, the variation may be direct (same kind of change in one quantity with the similar change in another) or inverse (opposite change in one quantity with some change in another).
The constants of variation are classified into three major categories which are as follows:
i) Direct constant of variation
ii) Inverse constant of variation
iii) Joint constant of variation
These are discussed in detail in the sections below.

Joint Variation

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Another type of variation is "joint variation". It is similar to direct variation, but with at least two variables (i.e. two ore more variables). More elaborately, if it is said that a variable "c" is jointly proportional to two variables "a" and "b", then
c $\alpha$ ab
c = k ab


For example:
In equation z = -6 xy
Here, z is jointly proportional to two variables x and y. We can say that z is directly proportional to x and y both. The constant -6 is known as constant of joint variation.

Direct Constant of Variation

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The direct variation is the type of variation in which the one variable varies in the same way as another does; i.e. one variable increases with the increase in another and decreases with the decrease in another. We can say that in direct variation; when one quantity goes up, the another also goes up; similarly when one goes down, another too goes down.

The most common example of direct variation is distance and speed. More is the speed, the more will be the distance covered. Also, less speed leads to less distance covered. Direct variation is also known as direct proportion. We may express direct variation or proportion using relations or equations. The constant of variation applied in direct variation is termed as the direct constant of variation.

In other words, if we state that "a and b vary directly", then it means that with the increase in a, b increases by same factor and with the decrease in a, b decreases by same factor. We may say that a and b possess the same ratio. This factor or ratio is known as direct constant of variation. This variation can be expressed as:
a $\alpha$ b
a = k b
Where k is known as direct constant of variation.
The direct constant of variation may be defined as the ratio of two variables, i.e.
k = $\frac{a}{b}$
For example:
For equation y = 5x, the number 5 is referred as the direct constant of variation.

If we explore direct variation graphically, then we find that its general equation is similar to linear equation, y = k x which represents the slope of a straight line.

Inverse Constant of Variation

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In the indirect variation, one variable varies in oppositely or inversely as another does; i.e. one variable decreases with the increase in another and it increases with the decrease in another. Thus; if by some factor, one quantity is going up, then another goes down with that factor. Similarly, if one quantity is going down with some factor, then another will go up with that factor. This factor is known as inverse constant of variation or inverse coefficient of proportionality.

For example: Speed and time are in inverse variation; since when speed is greater, the time taken in covering some distance is less. Similarly, when speed is lesser, the time taken is more.

Let "a" and "b" are two variables that vary inversely. This means that when a increases, b decreases and when a decreases, b increase. Inverse variation is illustrated by the following relation:
a $\alpha \frac{1}{b}$
a = $\frac{k}{b}$
Where k is called the inverse constant of variation.

Inverse constant of variation may also be defined as the product of two variables, such that:
k = a.b
For example:
For equation xy = 1.4, the number 1.4 is the inverse constant of variation.

Word Problems

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The examples of word problems based on variation are as follows:

Problem 1: The fund collected for a charity is directly proportional to the number of people attended a cultural program. If the fund collected last year when 100 people attended the function was $\$$3000. Calculate the total amount if it is predicted that this year 120 people will attend the function?

Solution: Let us assume that the amount of money be denoted by p and number of people be expressed by q.
Then by direct variation, we have the equation -
p = k q ; where k is the constant of variation.
3000 = k x 100
k = 30

Now, q = 120
p = k q
p = 30 x 120
p = 3600
The amount of money will be $\$$3600.

Problem 2: A journey is covered in 4 hours if the average speed of a car is 90 km/hr. Estimate the time taken by the same car at an average speed of 120 km/hr?

Solution: The speed and time taken are inversely related to each other. Since if speed is more, the time taken would be less and vice versa. Let the time taken is represented by t and speed be denoted by s.
Then, by inverse variation equation, we have
s x t = k ; where k is the constant of variation.
90 x 4 = k
k = 360

Now, with s = 120 km/hr
s x t = k
120 x t = 360
t = 3 hours

Problem 3:  Hooke's Law states that the force is directly proportional to the stretch in a spring. A spring is stretched 5 inches in a force of 50 pounds is applied. Calculate the amount of stretch in the same spring if 150 pounds of force is applied?

Solution: Let us consider the variable denoting force be f and that denoting stretch be x. Then, the following equation is constructed by direct variation.
f k = x ; where k is the constant of variation.
50 = 5 k
k = 10

If f = 150 pounds
f k = x
150 k = 10

x = $\frac{150}{10}$

x = 15 inches
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