UPSC Prelims 2010 Q80: Science & Technology – Logical Reasoning & Puzzles
- (a) All the villagers who have their own houses are literate
- (b) Some villagers under twenty - five are literate
- (c) A quarter of the villagers who have their own houses cultivate paddy
- (d) Half of the villagers who cultivate paddy are literate
Answer: (b) Some villagers under twenty - five are literate
Explanation
A village statistics logic puzzle
Given: half the villagers own houses, one-fifth cultivate paddy, one-third are literate, four-fifths are under 25 — we must find what's certainly true.
| Option | Verdict | Reason |
|---|---|---|
| (a) All house-owners are literate | Incorrect | With only 1/3 literate overall, it's not guaranteed that the entire 1/2 who own houses are all literate — these are separate, only partially overlapping groups |
| (b) Some villagers under 25 are literate | Correct | Since 4/5 are under 25 and 1/3 are literate, these two large groups must overlap at least partially — if no one under 25 were literate, all literate villagers (1/3) would have to come from the remaining 1/5 who are 25+, which is impossible since 1/3 > 1/5; so some under-25 villagers must be literate |
| (c) A quarter of house-owners cultivate paddy | Incorrect | This is a specific numeric claim not guaranteed by the given proportions alone |
| (d) Half the paddy cultivators are literate | Incorrect | This is also a specific numeric claim not guaranteed by the given data |
Why (b) must be true — the pigeonhole logic:
| Group | Proportion |
|---|---|
| Literate | 1/3 of villagers |
| 25 and above (NOT under 25) | Only 1/5 of villagers |
Since 1/3 (literate) is larger than 1/5 (the entire 25-and-above group), not all literate villagers can be 25 or older — so some literate villagers must be under 25.
Answer: (b) Some villagers under twenty-five are literate.
Why each option is right or wrong
- (a) Incorrect. There's no guarantee that house-ownership (1/2) and literacy (1/3) fully overlap; this claim overreaches the given data.
- (b) CORRECT. Since literate villagers (1/3) outnumber the entire 25-and-above group (1/5), some literate villagers must be under 25.
- (c) Incorrect. The exact fraction of house-owners who cultivate paddy isn't determinable from the given proportions.
- (d) Incorrect. The exact fraction of paddy cultivators who are literate also isn't determinable from the given data.
Key point
Since literate villagers (1/3) outnumber the entire 25-and-older group (1/5), it's logically certain that SOME villagers under 25 are literate — a pigeonhole-principle deduction, unlike the other unverifiable numeric claims.
Remember
Village puzzle: since literate (1/3) > 25-and-above (1/5), some under-25 villagers MUST be literate — the only certainly-true statement among the options.
Common mistakes
- Assuming proportions given separately (like house-ownership and literacy) must overlap in a specific guaranteed way.
- Missing the pigeonhole-style reasoning: a larger group (1/3) can't fit entirely inside a smaller group (1/5).
- Picking a specific, unverifiable numeric claim (like 'a quarter' or 'half') over the one logically certain statement.
Exam tip
For 'which is certainly true' puzzles with overlapping proportions, check if one group is too LARGE to fit entirely inside another — that forces overlap.
Sources
- NCERT Class XI Mathematics — Sets
- R. S. Aggarwal — A Modern Approach to Verbal and Non-Verbal Reasoning (Logical Venn Diagrams)