Passed Both Subjects Percentage Calculation
This solution finds the percentage of students who achieved passing marks in both English and Mathematics, using the given failure statistics from an examination.
Given Data:
- Percentage of students who failed in English, $P(E) = 20\%$
- Percentage of students who failed in Mathematics, $P(M) = 60\%$
- Percentage of students who failed in both subjects, $P(E \cap M) = 15\%$
Step 1: Calculate the percentage of students who failed in at least one subject.
We use the Principle of Inclusion-Exclusion for sets:
Percentage failed in at least one subject $= P(E \cup M) = P(E) + P(M) - P(E \cap M)$
Substituting the given values:
$P(E \cup M) = 20\% + 60\% - 15\% = 80\% - 15\% = 65\%$
Step 2: Calculate the percentage of students who passed in both subjects.
The percentage of students who passed in both subjects is the complement of the percentage who failed in at least one subject.
Percentage passed in both subjects $= 100\% - (\text{Percentage failed in at least one subject})$
Percentage passed in both subjects $= 100\% - P(E \cup M)$
$= 100\% - 65\% = 35\%$
Thus, 35% of the students passed in both subjects.
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