Understanding the Problem
To determine the maximum number of students who did not pass either physics or chemistry, we begin with the given data:
- Total students = 120
- Students who passed in physics = 50% of 120 = 60 students
- Students who did not pass in chemistry = 20% of 120 = 24 students
Calculating Students Passing Chemistry
- Students passing in chemistry = Total students - Students not passing in chemistry
- Students passing in chemistry = 120 - 24 = 96 students
Finding Students Passing Only One Subject
- Students passing physics = 60
- Students passing chemistry = 96
- Since some students may pass both subjects, we need to optimize the overlap to maximize those who fail both.
Using the Inclusion-Exclusion Principle
We represent:
- Let A = Students passing physics = 60
- Let B = Students passing chemistry = 96
- Total students = 120
We want to find the maximum number of students who fail both subjects (let's denote this as N).
Using the Formula
The formula for students passing at least one subject is given by:
N = Total students - (A + B - X)
Where X is the number of students passing both subjects.
To maximize N, we minimize X.
Finding the Minimum Overlap
The minimum overlap (X) occurs when:
X = A + B - Total students
X = 60 + 96 - 120 = 36
Calculating Students Failing Both
Thus, substituting into the formula:
N = 120 - (60 + 96 - 36)
N = 120 - 120
N = 0
However, we want to maximize students failing both:
Conclusion
If we set the maximum value of students passing both subjects to 60 (the maximum for physics), we find:
N = Total students - A - B
N = 120 - 60 - 96 = -36 (not possible)
To find the maximum failing both, we analyze:
- Students failing physics = 120 - 60 = 60
- Students failing chemistry = 24
The maximum number of students failing both is achieved when overlaps are minimized, leading to:
60 (failing physics) + 24 (failing chemistry) - 120 (total) = 24 students.
Thus, the maximum possible number of students who can neither pass physics nor chemistry is 24.