# Inverse Variation || Concept, definition and examples of inverse proportion

## What is Inverse Variation (Inverse Proportion) ?

Two entities are inverse variation when scaling up one entity by a factor results in scaling down of other entity by same factor.

For example;
Consider the table of two entity x and y.

If you observe the data carefully you will find that the entity x and y are in inverse variation.

Note that multiplication of x by 2 results in division of y value by 2.

Similarly, the multiplication of entity x by 7 results in division of y by 7.

Conclusion
In Inverse Proportion, the scaling of one entity by a factor will result in scaling down of other entity by same factor.

### Why the property called inverse variation?

Because the entities are inversely related to each other.

Scaling up one entity results in down scaling of other entity by the same factor.

### How to represent inverse proportion?

The proportionality is represented by symbol \mathtt{\varpropto } .

Since the entities x and y are inversely represented, mathematically we can write as:

\mathtt{y\ \varpropto \ \frac{1}{x}}

Converting the expression into equation by replacing symbol \mathtt{\varpropto } with constant k.

Here the constant k is equal to multiplication of the two entities.

The above formula equations are important as they help to solve textbook problems. So make sure to remember both the equations.

#### Cases of Indirect proportion with graph

(a) When k > 0
When k is positive, the increase in value of x results in decrease in y value.

Below is the graph of y = 2 / x;

(b) When k < 0;
When k is negative, the increase in value of x results in increase in value of y

### Explain difference between inverse and direct variation

Direct variation
In direct proportion, if we multiply one entity by a factor, the other entity value will also be scaled up by that factor.

Hence, there is direct relationship between the two entity.
If you multiply one entity, the other entity will also get multiplied.

Inverse variation
In inverse proportion, if we multiply one entity by a factor, the other entity will scale down by the same factor.

Hence, there is inverse relationship between entities.
If we multiply one entity, the other entity gets divided.

## Examples of Inverse Variation

Example 01
Study the below table and check if the entities are inversely proportional

Solution
If the entities are inversely proportional, the value of k = x . y is constant.

Below is the table with value k.

Note that for all the data points the value of k is same.
Hence, the entities are inversely proportional.

Alternate Method
Another way to check inverse proportionality is to check if multiplication of one entity results in division of other entity or not.

The entity x is multiplied by 3 to get 9.
On the other hand the entity y is divided by 3 to get 7.
Hence, there is inverse relationship between the entity x and y.

Example 02
Check if the given data are in inverse variation or not.

Solution
If the entities are inversely proportional, then the value of k = x .y will be constant.

Given below is the table with k values.

Note that all the data points have same value of k = -6.
Hence, the entities x and y are inversely proportional.

Alternate Method
For indirect variation, check if multiplication of one entity results in division of another entity and vice versa.

The entity x is multiplied by 2 to get -12.
On the other hand, entity y is divided by 2 to get 0.5.

Let us look at another data point.

Here the entity x is divided by 2 to get -3.
And entity y is multiplied by 2 to get number 2.

Hence, there is inverse relationship between entity x and y.

Example 03
The entity x and y are in inverse variation.
For x = 7, the value of y is 66.
Find value of y, if x value is 77.

Solution
Its given that entity x and y are inversely proportional.

The mathematical expressions for inverse proportion is:
k = x . y

Insert value of x and y in the equation.
k = 7 . 66
k = 462

Hence the value of constant is 462.

Now find value of y for x = 77.

Again using the equation;
k = x . y

Put value of k and x in equation;
462 = 77 . y
y = 462 / 77
y = 6

Hence for x = 77, the value of y is 6.

Example 04
Entity x and y are inversely proportional. Find the value of x for given data points: (14, -9) and (x, -27)

Solution
The equation for inversely proportion is:
k = x . y

Let’s find value of k by putting values of x and y.

k = 14 . -9
k = -126

Now let’s find value of x for y = -27.

Again using the equation:
k = x . y

Putting values of k and y in the equation.
-126 = x . -27

x = -126 / -27

x = 4.67

Hence, for y = -27, the value of x is 4.67

You cannot copy content of this page