Statement:\nHalf of the villagers have their own houses. One-fifth of the villagers cultivate paddy. One-third of the villagers are literate. Four-fifths of the villagers are below twenty-five years of age.\n\nWhich of the following is certainly true?\nI. All villagers who have their own houses are literate.\nII. Some villagers under twenty-five are literate.\nIII. A quarter of the villagers who have their own houses cultivate paddy.\nIV. Half of the villagers who cultivate paddy are literate.

Introduction / Context:
We must identify a statement that is certainly true given four proportions. Without overlap data, only deductions forced by the Pigeonhole Principle or obvious subset relations can be made.

Given Data / Assumptions:

• Own houses: 1/2 of villagers.
• Cultivate paddy: 1/5 of villagers.
• Literate: 1/3 of villagers.
• Below 25 years: 4/5 of villagers.

Concept / Approach:
Check each claim for inevitability considering maximum/minimum overlaps.

Step-by-Step Solution:

Verification / Alternative check:
Use extreme-case overlap to test certainty; only II survives all placements.

Why Other Options Are Wrong:
They assert specific overlaps not guaranteed by totals.

Common Pitfalls:
Assuming proportional overlap without justification.

Final Answer:
Only II.