 # Inverse Variation

In our day-to-day life, we observe that the variation in values of some quantity depends upon the variation in values of some other quantity. Inverse variation means that a variable is inversely varying with respect to another variable. Hence, a variable is inversely proportional to another variable. For example: if the distance travelled by train at constant speed increases then the time taken by it increases too and vice versa. If the number of people is added to a job, the time taken to accomplish the job decreases.

## Inverse Variation Definition

Unlike the direct variation, where one quantity varies directly as per changes in another quantity, in case of inverse variation, the first quantity varies inversely as per another quantity. Hence, it is also called the inverse proportion. ## Inverse Variation Equation

Sometimes, we observe that the variation in values of one quantity is just opposite to the variation in the values of another quantity. If the value of one quantity increases, the value of other quantity decreases in the same proportion and vice versa. This is termed as inverse variation and the two quantities are said to be inversely proportional to each other. Two quantities existing in inverse variation can be expressed as, $$x~∝~\frac{1}{y}$$

$$\Rightarrow~ xy~=~k$$

Where x and y are the value of two quantities and k is a constant known as the constant of proportionality. If x1, y1 are initial values and x2, y2 are final values of quantities existing in inverse variation. They can be expressed as,

$$\frac{x_1}{x_2}~=~\frac{y_2}{y_1}$$

## Inverse Variation Formula

As per the inverse variation formula, if any variable x is inversely proportional to another variable y, then the variables x and y are represented by the formula:

xy = k or y=k/x

where k is any constant value.

## Inverse Variation Word Problems

Illustration 1: 9 pipes are required to fill a tank in 4 hours. How long will it take if 12 pipes of the same type are used?

Solution: Let, the desired time to fill the tank be $$x$$ minutes. We know that as the number of pipes increases, the time taken to fill the tank will decrease. Hence, this is a case of inverse variation. In other words, the number of pipes is inversely proportional to the time taken. Thus,

$$\frac{x_1}{x_2}~=~\frac{y_2}{y_1}$$

$$\Rightarrow~\frac{9}{x}~=~\frac{12}{4}$$

$$x~=~3~hours$$

Illustration 2: If 24 workers can build a house in 40 days, how many workers will be required to build the same house in 20 days?

Solution: Let the number of workers employed to build the wall in 20 days be $$x$$. We know that time taken to build the house is inversely proportional to the number of workers required. Thus,

$$\frac{x_1}{x_2}~=~\frac{y_2}{y_1}$$

$$\Rightarrow~\frac{24}{x}~=~\frac{20}{40}$$

$$\Rightarrow~x~=~48~days$$

Illustration 3: There are 100 students in a hostel. Food provision for them is for 20 days. After 8 days 20 new students entered into the hostel. How long will this provision last now?

Solution: Suppose the provision lasts for x more days after the intrusion of 20 new students which would have lasted for 12 days if the number of students was 100. We know that the time till which food lasts is inversely proportional to the number of students in the hostel. Thus,

$$\frac{x_1}{x_2}~=~\frac{y_2}{y_1}$$

$$\Rightarrow~\frac{100}{120}~=~\frac{x}{12}$$

$$\Rightarrow~x~=~10~days$$

Total number of days for which provision lasts = $$10~+~8$$ = $$18~days$$

For a detailed discussion on the concept of inverse variation and Inverse Variation Formula and developing a relationship between two quantities, download BYJU’S – The learning App.

Quiz on Inverse Variation