As one amount increases, another amount decreases at the same rate or as one amount decreases, another amount increases at the same rate is known as inverse proportion.
Two quantities, \(x\) and \(y\), are inversely proportional if \(x \propto \frac{1}{y}\) \(\Rightarrow xy = k\), where \(k\) is positive. We can also write it as \(\frac{x_1}{x_2} = \frac{y_2}{y_1}\) where \(x_1, x_2\) are values of \(x\) corresponding to \(y_1,y_2\) are values of \(y\).
If \(10\) people can do a work in \(10\) days, find how many days will it take to do the same work if there are \(20\) people.
Let the number of days be \(x\).
Logically, if \(10\) people can do work in \(10\) days, then \(20\) people can do the same work in less than \(10\) days.
Thus, an increase in one quantity causes a decrease in other quantity. This is inverse proportion.
Let us use the formula \(\frac{x_1}{x_2} = \frac{y_2}{y_1}\).
Substituting the known values, we have:
\(\frac{10}{20} = \frac{x}{10}\)
\(\frac{1}{2} = \frac{x}{10}\)
\(\frac{10}{2} = x\)
\(5 = x\)
Therefore, \(20\) people can do the same work in \(5\) days.