A survey was conducted among 500 civil service aspirants in a coaching institute regarding… — State PSC Exam

A survey was conducted among $500$ civil service aspirants in a coaching institute regarding their daily newspaper reading habits among three publications: *The Hindu*, *The Indian Express*, and *Business Standard*.

The survey revealed the following data:
- The total number of aspirants reading *The Hindu*, *The Indian Express*, and *Business Standard* are $240$ , $210$ , and $180$ respectively.
- Exactly $20$ aspirants read all three newspapers.
- The ratio of the number of aspirants who read ONLY *The Hindu* and *The Indian Express* to those who read ONLY *The Indian Express* and *Business Standard* to those who read ONLY *The Hindu* and *Business Standard* is $3 : 2 : 4$ .
- The number of aspirants who read ONLY *Business Standard* is $70$ .
- The number of aspirants who read ONLY *The Hindu* is equal to the number of aspirants who read ONLY *The Indian Express*.

Based on the data provided, how many aspirants in total do NOT read any of the three newspapers?

$45$ Answer
B
$65$
C
$35$
D
$55$

Answer

The total number of aspirants who do not read any of the three newspapers is 45.

By resolving the 7 disjoint regions of the Venn diagram using the given ratio

3k : 2k : 4k

, we find

k = 15

from the total for Business Standard (

180 = 70 + 4k + 2k + 20

). This yields exclusive intersection values of

45

30

, and

60

. Substituting these into the total for Indian Express (

210

) yields

\text{Only}(E) = 115

, which also equals

\text{Only}(H)

. Summing all seven disjoint regions gives

115 + 115 + 70 + 45 + 30 + 60 + 20 = 455

. Subtracting from the total population of

500

gives

45

aspirants who do not read any of the three newspapers.

Step-by-Step Solution

Define disjoint regions of the Venn diagram using given variables.

Let

H

E

, and

B

denote the sets of readers. Let

\text{Only}(H \cap E) = 3k

\text{Only}(E \cap B) = 2k

, and

\text{Only}(H \cap B) = 4k

. The center intersection

\text{All Three} = 20

Setting up explicit variables for mutually exclusive regions allows algebraic representation of total set sizes.

Calculate the ratio constant

k

using the total for set

B

(Business Standard).

|B| = \text{Only}(B) + \text{Only}(H \cap B) + \text{Only}(E \cap B) + \text{All Three} \implies 180 = 70 + 4k + 2k + 20 \implies 90 = 6k \implies k = 15

All sub-regions comprising set

B

are known except

k

, making it possible to solve for

k

directly.

Determine the exact counts of two-set exclusive intersections.

\text{Only}(H \cap E) = 3(15) = 45

\text{Only}(E \cap B) = 2(15) = 30

, and

\text{Only}(H \cap B) = 4(15) = 60

Multiplying the ratio multipliers by

k = 15

gives exact counts for each overlap.

Calculate

\text{Only}(E)

and

\text{Only}(H)

|E| = \text{Only}(E) + 45 + 30 + 20 = 210 \implies \text{Only}(E) = 115

. Since

\text{Only}(H) = \text{Only}(E)

\text{Only}(H) = 115

Subtracting known intersection counts of set

E

from

|E| = 210

gives

\text{Only}(E)

, which equals

\text{Only}(H)

Calculate the total number of aspirants who read at least one newspaper and find the complement.

|H \cup E \cup B| = 115 + 115 + 70 + 45 + 30 + 60 + 20 = 455

. Aspirants reading none

= 500 - 455 = 45

Summing all 7 disjoint regions gives the union. Subtracting the union from the universal set gives the number of aspirants reading no newspaper.

Key Concept

3-Set Principle of Inclusion-Exclusion and Disjoint Region Analysis

Question

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