GATE 2023 EC GA Q.4: solution and answer

GATE 2023 · ECGA Q.4 · 1 mark · MCQ

In a class of 100 students,

(i) there are 30 students who neither like romantic movies nor comedy movies,

(ii) the number of students who like romantic movies is twice the number of students who like comedy movies, and

(iii) the number of students who like both romantic movies and comedy movies is 20.

How many students in the class like romantic movies?

(A)
40
(B)
20
(C)
60
(D)
30

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Answer and solution

ANSWER

60 students like romantic movies, so the answer is (C).

Official answer: CSource: Official answer key, GATE 2023 EC (IIT Kanpur)

Let $R$ be the set of students who like romantic movies and $C$ the set of students who like comedy movies. Write $x = n (C)$ . From (ii), $n (R) = 2 x$ . From (iii), $n (R \cap C) = 20$ .
Complement rule: the students who like at least one of the two kinds of movie are all the students except the 30 who like neither (condition (i)), so $n (R \cup C) = 100 - 30 = 70$ .
Inclusion–exclusion for two sets: $n (R \cup C) = n (R) + n (C) - n (R \cap C)$ .
Substitute: $2 x + x - 20 = 70 \Rightarrow 3 x = 90 \Rightarrow x = 30$ So 30 students like comedy movies.
Hence the number who like romantic movies is $n (R) = 2 x = 60$ .
Check: romantic only $= 60 - 20 = 40$ , comedy only $= 30 - 20 = 10$ , both $= 20$ , neither $= 30$ ; total $= 40 + 10 + 20 + 30 = 100$ , and $60 = 2 \times 30$ as required by (ii). The answer is 60, option (C).

KEY IDEA

Two-set inclusion–exclusion: $n (R \cup C) = n (R) + n (C) - n (R \cap C)$ . Get $n (R \cup C)$ by subtracting the 'neither' group from the total. Then the ratio condition leaves one unknown and one equation.

WATCH OUT FOR

One slip is to forget to subtract the overlap: $2 x + x = 70$ gives $x = 23.33$ , which is not a whole number. Another is to read 'twice' as applying to the students who like only romantic and only comedy movies. That gives $r_{only} = 2 c_{only}$ , so $3 c_{only} = 50$ , which is also not a whole number. Some candidates also mark 40, which is the number who like ONLY romantic movies. The question asks for everyone who likes romantic movies, so the overlap must be counted too.

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GATE 2023 EC GA Q.4: solution and answer

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