Half of the villagers of a certain village have their own houses. One-fifth of the villagers cultivate paddy. One-third of the villagers are literate. Fourth-fifth of the villagers are under 25 years of age. Which one of the following statements is certainly correct?

The correct answer is: Some villagers under 25 years of age are literate.

Let us analyze the given information step-by-step using proportions or set overlaps.

Let the total number of villagers in the village be represented by $N$.

From the given data, we know the following fractions of the total population:
1. Fraction of villagers who are literate:

$P\left(\text{Literate}\right)=\frac{1}{3}$

2. Fraction of villagers who are under 25 years of age:

$P\left(\text{Under 25}\right)=\frac{4}{5}$

Now, let us determine if there must be an overlap between the set of villagers who are under 25 years of age and the set of villagers who are literate.

The minimum possible overlap (intersection) between two subsets of a population can be found using the principle of inclusion-exclusion:

$P(\text{Under 25}\cap \text{Literate})\ge P\left(\text{Under 25}\right)+P\left(\text{Literate}\right)-1$

Substituting the given fraction values into the inequality:

$P(\text{Under 25}\cap \text{Literate})\ge \frac{4}{5}+\frac{1}{3}-1$

To add the fractions, find a common denominator (15):

$\frac{4}{5}+\frac{1}{3}=\frac{12}{15}+\frac{5}{15}=\frac{17}{15}$

Subtracting 1 from this sum:

$P(\text{Under 25}\cap \text{Literate})\ge \frac{17}{15}-1=\frac{2}{15}$

Since $\frac{2}{15}>0$, at least $\frac{2}{15}$ of the total population must be both under 25 years of age and literate.

Therefore, it is mathematically certain that some villagers under 25 years of age are literate.