Difficulty: Medium
Correct Answer: 25
Explanation:
Introduction / Context:
This is another set theory question using percentages, now in the context of examination results. It focuses on the relationship between passing in individual subjects and in both subjects using complements and the inclusion exclusion principle.
Given Data / Assumptions:
Concept / Approach:
If 5 percent fail both, then 95 percent pass in at least one subject. Passing in History or Hindi or both is the union of the passing sets. Using the formula for the union of two sets, we can find the intersection, that is, those who pass both subjects.
Step-by-Step Solution:
Let H be the set of students passing History.Let D be the set of students passing Hindi.P(H) = 65 percent, P(D) = 55 percent.Failing in both subjects = 5 percent.So passing in at least one subject = 100 - 5 = 95 percent.By inclusion exclusion, P(H ∪ D) = P(H) + P(D) - P(H ∩ D).Thus 95 = 65 + 55 - P(H ∩ D).So P(H ∩ D) = 65 + 55 - 95 = 25 percent.
Verification / Alternative check:
Assume 100 students. Then 25 students pass both, 40 pass only History, 30 pass only Hindi and 5 fail in both. Totals: History passes = 25 + 40 = 65, Hindi passes = 25 + 30 = 55, and fail both = 5. All conditions are satisfied, confirming that 25 percent pass both subjects.
Why Other Options Are Wrong:
Values 15, 20 and 30 percent do not satisfy the inclusion exclusion equation once the given individual percentages and total failing percentage are accounted for. They break the consistency between the sets or the complement.
Common Pitfalls:
Some candidates mistakenly use 5 percent as pass in both or misinterpret it as intersection instead of the complement. Others forget to subtract the intersection term when using P(H ∪ D) = P(H) + P(D) - P(H ∩ D). Always align each percentage with the correct region of the Venn diagram.
Final Answer:
25 percent of the students pass in both History and Hindi.
Discussion & Comments