Inverse Variation

Inverse Variation Meaning

In your day to day life, you would observe that the variation in values of different quantities depends on the variation in values of another quantity. While the direct variation describes a linear relationship between two variables, the inverse variation describes another sort of relationship.

When there are two different quantities with an inverse variation, when one quantity increases, the other one decreases.

For example, when you are travelling to a particular location, as your speed increases, the time it takes to arrive at that specific location decreases. If you decrease your speed, the time it takes to arrive at that location tends to increase. Hence, the quantities are inversely proportional. In this article, we would discuss about the inverse variation definition, inverse variation formula, and some inverse variation examples word problems.

Inverse Variation Definition

When two variables change in the inverse variation, it is called an inverse proportion. In the inverse variation, one variable is constant times the inverse of the other. If one variable increases, the other will decrease, and if one decreases other will increase. This means that the variables tend to change in the same ratio but inversely.

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Inverse Variation Equation

The generalized inverse variation equation is given by:

Y = K \[\frac{1}{x}\], which can also be written as XY = K. 

Here, K = constant.

Hence, the product of two variables is a constant in the case of inverse variation. 

A variable quantity A is known to vary inversely to another variable quantity B when the quantity A varies as the reciprocal of the quantity  B, that is when A varies as \[\frac{1}{B}\].

Therefore, if A is inversely proportional to B, you can write it as A ∝ \[\frac{1}{B}\]

A = m\[\frac{1}{B}\] or AB = m.

where m is the constant and is not equal to zero.

Therefore, if one variable varies inversely to the other, the product of the values of the variables is constant.

Here, x ∝ \[\frac{1}{y}\].

Therefore, xy = k.

Here, x and y are the values of two quantities and k is a constant which is called as the constant of proportionality. If x1 and y1 are initial values and x2 and y2 are the final values of quantities existing in inverse variation, you can express them as:

\[\frac{x1}{x2}\] = \[\frac{y2}{y1}\]

Inverse Variation Formula

According to the inverse variation formula, if a variable x is inversely proportional to the other variable y, the variables x and y are represented by the formula:

xy = k or

y = \[\frac{k}{x}\]

where k is a constant value.

Inverse Variation Word Problems

Given below are some of the inverse variation examples for your better understanding.

Example 1:

In a factory, 10 men can do a job in 30 days. How many days will it take for 20 men to do the same job?

Solution:

In this case, when the manpower increases, they would need less than 30 days for completing the same job. Hence, this is an inverse variation.

Consider x to be the number of men workers and y to be the number of days to complete the given work.

Hence, 

x1=10  

x2=20

y1=30

Using the product rule of inverse variation,

(10) (30) = (20) (y2)         

Hence, 300=20y2

Solving for y2, you get

y2 = \[\frac{300}{20}\]

      = 15

Hence, it would take 15 days for 20 men to finish the same job.

Example 2:

Consider that a car runs at the speed of 40 kmph and takes 3 hrs to complete a distance. What time it would take to run at the speed of 60 km?

Solution:

Consider T to be the time taken to cover the distance, S to be the distance, and V to be the speed of the car. 

The indirect variation equation would be

S = VT 

Here, 

S is constant

V and T are variables

According to the case given, the distance that car covers is

S = VT 

= 40 × 3 

= 120 km.

Hence, when the speed of the car is 60 kmph, it will take

S = VT 

120 = 60 × T

T = 2 hrs.

Therefore, the car will take 2 hours at a speed of 60 kmph to complete the distance.