**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**