# Direct and Inverse Variations

## What is Variation?

If the value of a quantity can change according to different situations, it is termed as a variable. If a change in the value of variable results in a change in the value of a related variable, their relationship is termed as a variation. If the value of variable increases with an increase in the value of a related variable, their relationship is termed as a direct variation. If the value of a variable decreases with an increase in the value of a related variable or vice versa, their relationship is termed as an inverse variation.

### Direct Variation

The variables in a direct variation are said to be directly proportional. If the value of one variable increases, the value of the other variable also increases and vice versa. Assuming k to be constant, a direct variation can be expressed as:

y=kx

### Inverse Variation

The variables in an inverse variation are said to be inversely proportional. If the value of one variable increases, the value of the other variable decreases and vice versa. Assuming k to be constant, an inverse variation can be expressed as:

xy = k or y = k/x

### Solved Examples

Example 1:

If the sales tax on a Rs 60 purchase is Rs. 4, what would it be on a Rs 300 purchase?

Solution:

Here, the sales tax increases if the cost of purchase increases. Therefore, this is a direct variation.

Sales tax on Rs 60 purchase = Rs 4

Sales tax on Rs 1 purchase =rs 4/60 = rs 1/15

Sales tax on Rs 300 purchase =  (1/15) x 300 = Rs 20

Therefore, sales tax on Rs 300 purchase is Rs 20.

Example 2:

If 15 men can complete a piece of work in 50 days. How many days will be required to complete the work if 30 men work together?

Solution:

Here, the number of days required to complete the work decreases if the number of men increases. Hence, this is an inverse variation.

15 men can complete a piece of work in 50 days.

1 man can complete the work in 50 x 15 = 750 days

30 men can complete the work in 750/30 = 25 days

Therefore, 30 men can complete the work in 25 days.

Example 3:

If y varies directly as x, and y = 25 when x = 75, find x when y = 80.

Solution:

First, write an equation of direct variation:

y=kx

Where k is a constant.

Substituting the values of x and y,

y = kx

25 = 75k

k = 25/75

k = 1/3

Therefore, an equation of the direct variation is y = ⅓ x

To find x when y = 80, substitute in y = ⅓ x

y = ⅓ x

80 = ⅓ x

x = 80 x 3

x = 240

Therefore, when y =80, x = 240

Example 4:

If y is inversely proportional to x, and y = 8 when x = 20, find x when y = 16.

Solution:

First, write an equation of inverse variation:

xy = k

Where k is a constant.

Substituting the values of x and y,

xy = k

20 x 8 = k

k = 160

Therefore, an equation of the direct variation is xy = 160

To find x when y = 16, substitute in xy = 160.

xy = 160

x(16) = 160

x = 160/16

x = 10

Therefore, when y = 16, x = 10

Example 5:

A man earns Rs. 3000 in 5 days.

i) How much will he earn in 30 days?

ii) How much time will be required by him to earn Rs. 15000?

Solution:

i)

 No.of days 5 30 Earnings (Rs) 3000 ?

Let the amount earned in 30 days be Rs x. The amount he earns increases as the number of days increases, hence this is direct variation. Therefore,

5/3000   =  30/x

x   = (30 x 3000)/5

x  = 18000

The man earns Rs. 18000 in 30 days.

ii)

 No.of days 5 ? Earnings (Rs) 3000 15000

Let the number of days be x. The number of days increases as the earnings increase, hence, this is a direct variation. Therefore,

5/3000   =   x/15000

x   =  (5 x 15000)/3000

x  = 25

The man earns Rs 15000 in 25 days.

Example 6:

A man driving at a speed of 60 km/h covers a distance in 40 minutes.

i) How long will he take to cover the same distance as a speed of 50 km/hr.

ii) What will be his speed if he covers the same distance in 60 minutes?

Solution:

i)

 Speed (Km/hr) 60 50 Time (min) 40 ?

Let the time taken be x minutes. As speed decreases, the time taken to travel a given distance increases, hence, this is an inverse variation. Therefore,

60 x 40 = 50 x x

x = (60 x 40)/50

x = 48

The man takes 48 min to travel if he drives at a speed of 50 km/hr.

ii)

 Speed (Km/hr) 60 ? Time (min) 40 60

Let speed be x km/hr. Since speed decreases as time increases, hence, this is an inverse variation. Therefore,

60 x 40 = x x 60

x = (60 x 40)/60

x = 40

The man drives at a speed of 40 km/hr.

### Did You Know

• A variation where one quantity varies directly as the product of two or more quantities is called a joint variation. For example: if x varies directly as y and square of z, then, x = kyz2, where, k is a constant. It can be said that z varies jointly as y and z2.

• Sums on direct and inverse variation can be solved using the unitary method or proportion.

• Some important physics laws like Newton’s law of gravitation combine joint and inverse variations. Newton’s law of gravitation is given by,

F = k (m1m2)/r2

It states that the force between two bodies varies jointly as their masses m1and m2 and inversely to the square of the distance between their centres.

• k is a non-zero constant in both direct and inverse variations.

• Simple interest vs time, density vs mass, force vs acceleration are some examples of direct variation.

• Density vs volume, the number of people vs time taken to complete work are some examples of inverse variations.

1. How can one recognize a direct or inverse variation?

In a direct variation, if the value of one variable increases, the value of another variable also increases. Consider the following table:

 x 2 4 6 y 10 20 30

It can be seen that on increasing the value of x, the value of y is also increasing and vice versa. Hence, this can be identified as a direct variation.

In an inverse variation, if the value of one variable increases, the value of other variable decreases and vice versa. Consider the following table:

 x 20 15 10 y 2 6 8

It can be seen that on decreasing the value of x, the value of y increases. Also, on increasing the value of y, the value of x decreases. Hence, this can be identified as an inverse variation.

2. How is the graph of a direct variation and an inverse variation?

For a direct variation, y = mx, therefore, the graph of a direct variation is a straight line passing through the origin. For an inverse variation, y = k/x where, x ≠ 0, k ≠ 0 Therefore, the graph of an inverse variation is neither a straight line nor does it pass through the origin as x can never be 0. It is in the shape of a curve or a hyperbola.