In a class 60% of the students pass in English, 45% pass in Hindi and 25% pass in both subjects; what percentage of students fail in both English and Hindi?

Introduction / Context:
This problem involves overlapping sets and percentages, a classic topic in quantitative aptitude. It deals with students passing in two different subjects, English and Hindi, and asks for the percentage that fails in both. The question tests understanding of the principle of inclusion and exclusion in the context of percentages instead of absolute numbers, which is very common in competitive exams.

Given Data / Assumptions:

• 60% of the students pass in English.
• 45% of the students pass in Hindi.
• 25% of the students pass in both English and Hindi.
• We assume the total number of students is 100 for convenience, because percentages are involved.
• We need to determine what percentage fail in both subjects.

Concept / Approach:
The key concept is the principle of inclusion and exclusion for two sets. For sets A and B, representing passing in English and passing in Hindi, the formula for the percentage of students who pass in at least one subject is: Percentage in A union B = Percentage in A + Percentage in B minus Percentage in intersection of A and B. Once we know the percentage who pass in at least one subject, we subtract that from 100% to find the percentage who fail in both subjects.

Step-by-Step Solution:
Let the total number of students be 100 for simplicity.Percentage passing in English (E) = 60%.Percentage passing in Hindi (H) = 45%.Percentage passing in both subjects (E and H) = 25%.Using inclusion and exclusion: percentage passing in at least one subject = 60 + 45 - 25 = 80%.Therefore percentage failing in both subjects = 100% - 80% = 20%.

Verification / Alternative check:
Consider an actual class of 100 students. If 60 pass in English, 45 in Hindi and 25 pass in both, then number who pass only in English is 60 - 25 = 35, and number who pass only in Hindi is 45 - 25 = 20. Students passing in at least one subject are 35 + 20 + 25 = 80. Hence students failing in both are 100 - 80 = 20. The calculation matches the percentage approach exactly, confirming that 20% is the correct value for students who fail in both English and Hindi.

Why Other Options Are Wrong:
25% would mean that only 75% pass in at least one subject, which contradicts the computed union of 80% from the formula.

30% would imply that only 70% pass in at least one subject, again inconsistent with the inclusion and exclusion calculation.

15% assumes a smaller failure group than supported by the data and does not satisfy the requirement that the sum of exclusive and overlapping pass groups must equal the known totals.

Common Pitfalls:
A common mistake is to simply add 60% and 45% and think that 105% is impossible, without subtracting the overlap of 25%. Some learners also subtract the 25% incorrectly or misinterpret it as either only English or only Hindi. To avoid these errors, always draw a simple Venn diagram with two intersecting circles representing English and Hindi, place the 25% in the intersection first and then complete the rest. This visual method reinforces the inclusion and exclusion formula and prevents double counting.

Final Answer:
The percentage of students who fail in both English and Hindi is 20%.