UGCNET Questions

Out of 100 students, 20% failed in both subjects, so 80 students passed in at least one of Mathematics or English. Let the number who passed in both subjects be x. Using inclusion–exclusion, the number who passed in at least one is 60 + 70 − x. This must equal 80, giving 130 − x = 80 and hence x = 50. Therefore, 50 students passed in both Mathematics and English.

Option A:
Option A, 40, would correspond to 60 + 70 − 40 = 90 students passing in at least one subject, which contradicts the given figure that only 80 passed at least one.

Option B:
Option B, 45, gives 60 + 70 − 45 = 85 students passing in at least one, again inconsistent with the total implied by 20 students failing both.

Option C:
Option C, 55, implies 60 + 70 − 55 = 75, which would mean 25 students failed in both subjects, not the stated 20.

Option D:
Option D, 50, exactly balances the inclusion–exclusion equation and yields 80 students passing at least one subject, matching the information derived from the 20 who failed both.

Let the total number of people be represented as 100%. If 20% like neither tea nor coffee, then 80% like at least one of the two beverages. By the inclusion–exclusion principle, the percentage who like at least one is the sum of tea-likers and coffee-likers minus those who like both. Thus, 80 = 70 + 40 − (both). Solving this gives (both) = 70 + 40 − 80 = 30%.

Option A:
Option A, 20%, equals the proportion who like neither, not those who like both, and so confuses the outside region with the intersection.

Option B:
Option B calculates the intersection using inclusion–exclusion and ensures that the total does not exceed 100%. It matches the only value that makes the percentages internally consistent.

Option C:
Option C, 40%, would give 70 + 40 − 40 = 70% liking at least one, which contradicts the information that 20% like neither and thus only 80% like at least one.

Option D:
Option D, 50%, creates an even larger overlap and leads to a sum exceeding the allowed total, making the data inconsistent.