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. For example: if the distance traveled by train at constant speed increases then the time taken by it increases too and vice versa. If more number of people are added to a job, the time taken to accomplish the job decreases. 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 \(x_1\), \(y_1\) are initial values and \(x_2\), \(y_2\) are final values of quantities existing in inverse variation. They can be expressed as,

\(\frac{x_1}{x_2}~=~\frac{y_2}{y_1}\)

**Examples based on inverse variation**

**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 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 were 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 relation between two quantities, download Byju’s-the learning app.