Statement: Half of the villagers of a certain village have their own houses.…

2024

Statement: 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. Four-fifths of the villagers are below twenty-five. Then, which one of the following is certainly true?

Conclusions:

I. All the 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.

  1. A.

    If only Conclusion I follows.

  2. B.

    If only Conclusion II follows.

  3. C.

    If either I or II follows.

  4. D.

    If neither I nor II follows.

Show answer & explanation

Correct answer: B

Concept

In statement-and-conclusion questions built from proportions of the same population, when the given proportions of two different groups add up to more than the whole population (their sum exceeds 1), the two groups cannot stay entirely separate — some members must belong to both, so a conclusion of the form "Some of Group A are Group B" is certainly true. If the two proportions add up to at most the whole population, no overlap between those groups is guaranteed, so neither a "some" nor an "all" claim about them can be established as certain.

Application

  1. Conclusion I: house-owners (1/2) vs. literates (1/3) -> sum = 5/6 (does not exceed 1) -> no overlap is guaranteed, so an "all" claim is even further out of reach. Conclusion I is not certain.

  2. Conclusion II: villagers under 25 (4/5) vs. literates (1/3) -> sum = 4/5 + 1/3 = 17/15 (exceeds 1) -> an overlap of at least 2/15 of the population is guaranteed -> "Some villagers under 25 are literate" is certainly true. Conclusion II holds.

  3. Conclusion III: house-owners (1/2) vs. paddy cultivators (1/5) -> sum = 7/10 (does not exceed 1) -> no overlap is guaranteed; moreover III claims an exact quarter, a specific value that is nowhere derivable from the given proportions. Conclusion III is not certain.

  4. Conclusion IV: paddy cultivators (1/5) vs. literates (1/3) -> sum = 1/5 + 1/3 = 8/15 (does not exceed 1) -> no overlap is guaranteed; IV also claims an exact half, which is likewise not derivable. Conclusion IV is not certain.

Only Conclusion II is certain, so the answer is ‘If only Conclusion II follows.’

Cross-check

Re-verifying the arithmetic: 4/5 = 12/15 and 1/3 = 5/15, so their sum is 17/15, which is more than 15/15 (= 1). Since the two groups together occupy 17/15 of a population that totals only 1 (15/15), they must overlap by at least 2/15 no matter how the groups are arranged, confirming the overlap is unavoidable. None of the other pairs (5/6, 7/10, 8/15) reaches or exceeds 1, confirming that only Conclusion II is forced to be true.

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