### Theory:

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 $$y$$.
Example:
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.

Solution:

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.