CSAT Solved Papers/ 2021/Q46

2021 CSAT — Q46

Quant Statement validity 2.5 marks Hard

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-fifth of the villagers are under $25$ years of age. Which one of the following statements is certainly correct?

A All the villagers who have their own houses are literate.
B Some villagers under 25 years of age are literate. Answer
C Only half of the villagers who cultivate paddy are literate.
D No villager under 25 years of age has his own house.

Worked rationale

The fractions are $\tfrac12$ houses, $\tfrac15$ paddy, $\tfrac13$ literate, $\tfrac45$ under $25$ . Test each claim for being forced by these sizes alone.

Check (b): under- $25$ is $\tfrac45 = 80\%$ and literate is $\tfrac13 \approx 33.3\%$ . Two subsets of the same village whose sizes sum to $80\% + 33.3\% = 113.3\% > 100\%$ must overlap by at least

80\% + 33.3\% - 100\% = 13.3\% > 0,

so some under- $25$ villagers are literate. This is forced — certainly correct.

The others are not forced:

(a) houses ( $50\%$ ) and literate ( $33\%$ ) need not nest; not all house-owners must be literate.
(c) no information links paddy-cultivators to literacy.
(d) under- $25$ is $80\%$ and house-owners $50\%$ ; they must overlap (same pigeonhole), so it is false that no under- $25$ owns a house.

Answer: (b) Some villagers under 25 years of age are literate.

Visual solution

The same solve, worked by hand — read it, then trace it.

Hand-drawn worked solution for UPSC 2021 CSAT Q46 — Statement validity — Tap the drawing to open it full size for the fine detail.

Why the other options miss

A

claimed more than the numbers allow: asserts a containment ( $50\%\subseteq 33\%$ ) that the sizes cannot force (and in fact forbid).
C

solved the wrong question: invents a precise overlap between two unrelated subsets.
D

missed a case: ignores that $80\%+50\%$ forces an overlap, making the claim demonstrably false.

Specialist insight

This is a minimum-overlap (pigeonhole) test dressed as a verbal claim: two subsets of one universe whose sizes exceed $100\%$ must intersect. Only $\tfrac45$ and $\tfrac13$ over-fill the village ( $113.3\%$ ), guaranteeing a literate under- $25$ villager — that is the one certain statement. The traps either assert containment the sizes can’t justify (a, c) or deny an overlap the sizes actually force (d). Add the two fractions: if they top $1$ , “some overlap” is certain; otherwise nothing is forced.

The trap, in one line

$\tfrac45 + \tfrac13 > 1$ forces a literate under- $25$ villager $\Rightarrow$ (b); the rest assert unforced containments.

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