# Direct and Inverse Variation

In this post we will understand the difference between direct and inverse variation along with examples.

This chapter requires basic understanding of proportions, so make sure to have your basics cleared before reading this chapter.

## Difference between direct and inverse variation

(01) Introduction

Direct Variation
Direct variation is also called direct proportion in mathematics.

Inverse variation
Inverse variation is also called inverse proportion.

(02) Concept explanation

Direct Variation
As the name suggests, in direct variation the relationship between two entities is direct.

It means that multiplication of one entity by factor results in scaling up of the other entity by same factor.

The table gives the value of x and y.
Note that multiplication of x by 3 results in multiplication of y by 3.

Hence, there is a direct relationship between the entities.

Inverse Variation
In inverse variation, there is inverse relationship between the two entities.

It means that multiplication of one entity by a factor results in division of other entity by the same factor and vice-versa.

Note that multiplication of x entity results in division of y entity by factor 3.

Hence, there is inverse relationship between the entity.

(03) Mathematical expressions

Math expression of direct variation
If x and y are directly proportional entity, then the relation can be expressed as: \mathtt{y\ \varpropto \ x}

It says that y is directly proportional to x.

Replacing symbol \mathtt{\varpropto } with constant k.

In direct proportion the value of k = y/x.
This value remains constant for all data points.

Math expression for Inverse Variation

If x and y are inversely proportional, the relation can be expressed as:
\mathtt{y\ \varpropto \ \frac{1}{x}}

The expression tells that y is inversely proportional to entity x.

Replacing symbol \mathtt{\varpropto } with constant k.

For inverse proportional, the value of constant k = y . x

The value of k will be same for all data points of inverse variation.

(04) Graphical Representation

Direct Variation
The graph of direct variation is a straight line.

Let x and y are two directly proportional entities with k =5
The equation will be, y = 5x

plotting the equation in graph, we get;

Hence, the direct proportion equation has a straight line.

Inverse Variation
The graph of inverse variation is a curved line.

Let the entities x and y are inversely proportional to each other with k = 5.

The equation will be; y = 5 / x

Plotting the graph we get;

## Examples of Direct and Inverse Variation

(01) Check the below table and find if the relationship between x and y is direct or inverse variation.

Solution
To check if the entities are directly or inversely proportional, find the value of constant k.

Constant for direct variation is; k = y / x

Constant for inverse variation is; K = y . x

Observe the below table with the constant data.

Note that the value of k for direct variation is same for all data points.

Hence, entity x and y are directly proportional to each other.

(02) Check the below table and find if the entity x and y are directly or inversely proportional.

Solution
To check is the ratios are directly or inversely proportional, try to find values of respective constants k.

k formula for direct proportional
k = y / x

K formula for inverse proportional
K = y . x

Look at the below table with k values.

From the table you can note that the inverse variation has constant value of k = -8.

Hence, entity x and y are inversely proportional.

(03) Its given that entity x and y are directly proportional. Some of the values of x and y are given; ( 6 , 14) and ( 30 , y ).
Find the value of y when x = 30

Solution
Formula for direct proportion is given as:
y = k . x

Putting values of x and y.
14 = k . 6

k = 14/6 – – eq. (1)

Now let’s find value of y for x = 30

Putting the value of x and k in the below formula;
y = k . x

\mathtt{y\ =\ \frac{14}{6} \ \times \ 30}\\\ \\ \mathtt{y\ =\ \frac{14\ \times \ 30}{6}}\\\ \\ \mathtt{y\ =\ 14\ \times \ 5}\\\ \\ \mathtt{y\ =\ 60}

Hence, for x = 30, the value of y = 60

(04) Its given that x and y are in inverse variation. Some data points for x and y are ( 10, 100) and ( x, 10). Find the value of x when y = 10.

Solution
The formula for inverse variation is :
k = x . y

Putting the values of x and y in equation.
k = 10 . 100
k = 1000 – – – eq. (1)

Now lets find value of x for y = 10
Again using the formula k = x . y

1000 = x . 10

x = 1000/10

x = 100

Hence for y = 10, the value of x is 100