### Theory:

Two quantities $$x$$ and $$y$$ are found to be inverse proportion when an increase of $$x$$ causes $$y$$ (and vice versa) to decrease proportionally. The product of their corresponding values remains constant. That is, if $$xy = k$$, then it is stated $$x$$ and $$y$$ to vary inversely proportional.
When the quantities $$x$$ and $$y$$ are in indirect proportion, we can write ${x}_{1}$${y}_{1}$ $$=$$ ${x}_{2}$${y}_{2}$ or $\frac{{x}_{1}}{{x}_{2}}$ $$=$$ $\frac{{y}_{2}}{{y}_{1}}$. Where ${y}_{1}$, ${y}_{2}$ are the values of $$y$$ corresponding to the values ${x}_{1}$, ${x}_{2}$ of $$x$$.
Example:
A farmer has enough food to feed $$20$$ hens in his field for $$6$$ days. How long would the food last if there were $$10$$ more hens in his field? Let the number of days be $$x$$.

Total number of hens $$= 20$$ $$+$$ $$10 =$$ $$30$$.

Increase in the number of hens decreases the food number of days the food lasts.

Thus, the number of hens and the number of days are in inverse proportion.
When the quantities $$x$$ and $$y$$ are in indirect proportion, we can write ${x}_{1}$${y}_{1}$ $$=$$ ${x}_{2}$${y}_{2}$ or $\frac{{x}_{1}}{{x}_{2}}$ $$=$$ $\frac{{y}_{2}}{{y}_{1}}$. Where ${y}_{1}$, ${y}_{2}$ are the values of $$y$$ corresponding to the values ${x}_{1}$, ${x}_{2}$ of $$x$$.
Substitute the known values.

$$\frac {20}{30}$$ $$=$$ $$\frac{x}{6}$$

$$\frac {2}{3}$$ $$=$$ $$\frac{x}{6}$$

$$3$$ $$×$$ $$x$$ $$=$$ $$2$$ $$×$$ $$6$$

$$3x =$$ $$12$$

$$x =$$ $$\frac{12}{3}$$

$$x =$$$$4$$

Hence, the food will last for four days.