In a certain examination, 34% of the students failed in Mathematics and 42% of the students failed in English. If 20% of the students failed in both subjects, what percentage of students passed in both Mathematics and English?

Introduction / Context:
This percentage question tests basic set theory and the inclusion exclusion principle for two overlapping groups. Many aptitude exams use this model of failure or pass percentages in two subjects such as Mathematics and English. You are asked to find the percentage of students who passed in both subjects when you know how many failed in each subject and how many failed in both subjects together.

Given Data / Assumptions:

We assume the total number of students is 100%, that is, the whole population of candidates.
34% of the students failed in Mathematics.
42% of the students failed in English.
20% of the students failed in both Mathematics and English.
Every student either passes or fails in a subject; there is no other category.

Concept / Approach:
The key concept is the inclusion exclusion rule for two sets. For any two events A and B, we have:
percentage in A or B = percentage in A + percentage in B – percentage in both A and B.
Here A is the set of students who failed in Mathematics and B is the set of students who failed in English. Once we know the percentage of students who failed in at least one of the two subjects, we can subtract that from 100% to obtain the percentage who passed in both subjects.

Step-by-Step Solution:
Step 1: Let total students be 100% for easy calculation.Step 2: Percentage of students who failed Mathematics = 34%.Step 3: Percentage of students who failed English = 42%.Step 4: Percentage of students who failed in both Mathematics and English = 20%.Step 5: Use inclusion exclusion to find the percentage who failed in at least one subject.Failed in at least one subject = 34% + 42% – 20% = 56%.Step 6: Percentage of students who passed in both subjects = 100% – 56% = 44%.

Verification / Alternative check:
If you imagine 100 students, then 56 students fail in at least one subject and 44 students do not fail in any subject, which means they pass in both Mathematics and English. This matches the computed 44%, so the answer is consistent and reasonable.

Why Other Options Are Wrong:
34% is only the Mathematics failure percentage and has nothing to do with students passing in both subjects.
54% is close but it would imply that only 46% failed in at least one subject, which does not fit the given data.
64% would mean only 36% failed in at least one subject, which contradicts the sum of individual failures.
56% is actually the percentage who failed in at least one subject, not those who passed in both.

Common Pitfalls:
Many students simply add 34% and 42% and subtract this from 100% without adjusting for the double counted 20%, which is incorrect. Another common mistake is to confuse the percentage who failed in at least one subject with the percentage who passed in both subjects. Always apply inclusion exclusion correctly and then subtract from 100% to get the group that passed both papers.

Final Answer:
The percentage of students who passed in both Mathematics and English is 44%.