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Two quantities $$x$$ and $$y$$ are found to be inversely proportional 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 $$y$$ 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 much longer would the food last if the field contained an additional $$10$$ hens?

Let the number of days be $$x$$.

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

The length of time that food is consumable reduces as hen numbers rise.

As a result, the relationship between the number of hens and the number of days are 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 $$x$$.
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.