In an examination 80% students passed in English and 85% students passed in mathematics. If 73% students passed in both these subjects, then what percent of students failed in both the subjects ?

Question

Accepted Answer

To determine the percentage of students who failed in both subjects, we can use the principle of inclusion-exclusion in set theory. Let's break down the problem:

1. **Given Data:**
   - 80% of students passed in English.
   - 85% of students passed in Mathematics.
   - 73% of students passed in both subjects.

2. **Objective:**
   - Find the percentage of students who failed in both subjects.

3. **Using the Principle of Inclusion-Exclusion:**
   - The formula to find the percentage of students who passed in at least one subject is:
     $$
     \text{Percentage passed in at least one subject} = (\text{Percentage passed in English}) + (\text{Percentage passed in Mathematics}) - (\text{Percentage passed in both})
     $$
   - Substituting the given values:
     $$
     \text{Percentage passed in at least one subject} = 80\% + 85\% - 73\% = 92\%
     $$

4. **Calculating the Percentage of Students Who Failed in Both Subjects:**
   - If 92% of students passed in at least one subject, then the percentage of students who failed in both subjects is:
     $$
     100\% - 92\% = 8\%
     $$

5. **Conclusion:**
   - Therefore, 8% of students failed in both subjects, which corresponds to Option 2.

6. **Explanation of Incorrect Options:**
   - **Option 1 (20%)**: This would imply a much higher failure rate than calculated. The correct calculation shows only 8% failed in both.
   - **Option 3 (15%)**: This percentage is also higher than the calculated 8% failure rate.
   - **Option 4 (27%)**: This is significantly higher than the calculated 8% and does not align with the given data.

By following the logical steps and calculations, we confirm that Option 2 (8%) is indeed the correct answer.