Statement: Half 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. Which of the following is certainly true? I. All villagers who have their own houses are literate. II. Some villagers under twenty-five are literate. III. A quarter of the villagers who have their own houses cultivate paddy. IV. 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.