In a test, 14% of the students failed in Mathematics and 22% failed in Hindi. If 18% of the students failed in both subjects, what percentage of students passed in both Mathematics and Hindi?

Introduction / Context:
This problem uses the concept of set intersection and union with percentages, specifically the idea of overlapping groups of students failing in different subjects. It tests understanding of inclusion exclusion principles with percentages and helps develop reasoning about students passing or failing in multiple subjects simultaneously.

Given Data / Assumptions:

• 14% of students failed in Mathematics.
• 22% of students failed in Hindi.
• 18% of students failed in both Mathematics and Hindi.
• We assume percentages are taken out of the same total student population.
• We must find the percentage of students who passed in both subjects.

Concept / Approach:
Let the total student population represent 100%. Let F_M be the set of students failing in Mathematics and F_H be the set failing in Hindi. The percentage failing in at least one subject is given by the union F_M ∪ F_H. Using the inclusion exclusion principle, the percentage in the union is P(F_M) + P(F_H) - P(F_M ∩ F_H). Students who pass both subjects are those not in the union, so we subtract this union percentage from 100%.

Step-by-Step Solution:
Step 1: Let total students represent 100%. Step 2: Percentage failing in Mathematics, P(F_M) = 14%. Step 3: Percentage failing in Hindi, P(F_H) = 22%. Step 4: Percentage failing in both subjects, P(F_M ∩ F_H) = 18%. Step 5: Percentage failing in at least one subject, P(F_M ∪ F_H) = P(F_M) + P(F_H) - P(F_M ∩ F_H). Step 6: Substitute values: P(F_M ∪ F_H) = 14 + 22 - 18 = 18%. Step 7: Therefore, 18% of students fail in at least one of the two subjects. Step 8: Students who pass both subjects = 100% - 18% = 82%. Step 9: Hence, 82% of the students pass in both Mathematics and Hindi.

Verification / Alternative check:
We can check by assuming a total of 100 students. Then 14 fail Mathematics, 22 fail Hindi, and 18 fail both. The number failing at least one is 14 + 22 - 18 = 18 students. Therefore, 100 - 18 = 82 students pass both subjects. Converting back to percentage, this again gives 82%, matching our earlier result.

Why Other Options Are Wrong:
40%, 46%, 54%, and 64% would correspond to much lower pass rates, implying a significantly higher failure rate than the one computed. For example, 40% passing both would mean 60% either failing one or both subjects, which contradicts the computed 18% failing at least one. Only 82% is consistent with the given failure percentages and the inclusion exclusion principle.

Common Pitfalls:
A common error is to add failure percentages directly (14 + 22 = 36) and then subtract this from 100 to say 64% passed both subjects, which ignores the overlap of 18%. Another mistake is to subtract the intersection twice or to misinterpret the intersection value. Correct application of P(A ∪ B) = P(A) + P(B) - P(A ∩ B) is crucial. Always carefully distinguish between failing at least one subject and failing both subjects.

Final Answer:
The percentage of students who passed in both subjects is 82%.