In an examination, 35% of total students failed in Hindi, 45% failed in English, and 20% failed in both. What percentage of students passed in both subjects?

Explanation

Treat the students who failed in Hindi and the students who failed in English as two sets. NCERT's set theory formula for finite sets is that the number in the union of two sets equals the sum of the two sets minus their intersection. Here, the percentage who failed in at least one subject is 35% + 45% - 20% = 60%; the 20% is subtracted once because those students were counted in both failure groups. Students who passed in both subjects are exactly the complement of students who failed in at least one subject, so 100% - 60% = 40%.

Why the other options are wrong

• (A) 50% would result from subtracting only one failure percentage from 100%, but the question requires combining both failure sets and removing their overlap.
• (B) 45% is the percentage who failed in English, not the percentage who avoided failure in both subjects.
• (D) 35% is the percentage who failed in Hindi, so it cannot represent students who passed both Hindi and English.

Concept

This tests the union formula in set theory, especially avoiding double-counting in overlapping groups. It recurs in RAS reasoning because exam, survey and eligibility questions often reduce to percentages in two intersecting sets.

Source

• NCERT Mathematics Exemplar Problems Class XI, Chapter 1: Sets

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