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 r**esults 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