In an examination, 35% of the total students failed in Hindi, 45% failed in…
2010
In an examination, 35% of the total students failed in Hindi, 45% failed in English and 20% in both. The percentage of those who passed in both subjects is
Answer: D. 40 — Concept — When two conditions can overlap, the inclusion–exclusion principle gives the share of a group that meets at least one of them as n(A ∪ B) = n(A) +…
- A.
10
- B.
20
- C.
30
- D.
40
Attempted by 4 students.
Show answer & explanation
Correct answer: D
Concept — When two conditions can overlap, the inclusion–exclusion principle gives the share of a group that meets at least one of them as n(A ∪ B) = n(A) + n(B) − n(A ∩ B); the common part is subtracted once because it has already been counted inside both individual figures. The share that meets neither condition is the complement of that union, 100% − n(A ∪ B).
Application — Take A as the students who failed in Hindi and B as the students who failed in English; a student passes in both subjects exactly when that student belongs to neither A nor B.
n(A) = 35% failed in Hindi and n(B) = 45% failed in English.
n(A ∩ B) = 20% failed in both subjects, and this 20% is already counted inside each of the two figures above.
Failed in at least one subject: n(A ∪ B) = 35% + 45% − 20% = 60%.
Passed in both subjects = 100% − n(A ∪ B) = 100% − 60% = 40%.
Cross-check — Break the cohort into blocks that do not overlap and confirm that they add up to the whole:
Failed in Hindi only: 35% − 20% = 15%.
Failed in English only: 45% − 20% = 25%.
Failed in both subjects: 20%.
Passed in both subjects: 100% − (15% + 25% + 20%) = 40%, and 15% + 25% + 20% + 40% = 100%, so the split is consistent.
Hence 40% of the students passed in both subjects.