Exam Percentage Problem: Analyzing Pass and Fail Rates

This problem involves calculating the percentage of candidates who failed in both subjects given the pass percentages for individual subjects and the percentage who passed both. We need to find the group of candidates who are outside the 'passed' categories for both Physics and Reasoning.

Understanding the Given Information

• Percentage of candidates who passed in Physics: 70%
• Percentage of candidates who passed in Reasoning: 80%
• Percentage of candidates who passed in both Physics and Reasoning: 55%

Calculating Candidates Passing at Least One Subject

To find the percentage of candidates who failed in both, we first need to determine the percentage of candidates who passed in at least one subject (either Physics, Reasoning, or both). We can use the principle of inclusion-exclusion for this.

Let P be the event that a candidate passed in Physics.

Let R be the event that a candidate passed in Reasoning.

We are given:

• $P(P) = 70\%$
• $P(R) = 80\%$
• $P(P \cap R) = 55\%$ (Passed both)

The percentage of candidates who passed in at least one subject is given by the formula:

$P(P \cup R) = P(P) + P(R) - P(P \cap R)$

Substituting the given values:

$P(P \cup R) = 70\% + 80\% - 55\%$

$P(P \cup R) = 150\% - 55\%$

$P(P \cup R) = 95\%$

So, 95% of the candidates passed in at least one of the subjects.

Calculating Candidates Failing in Both Subjects

The total percentage of candidates is always 100%. If 95% of candidates passed in at least one subject, the remaining candidates must have failed in both subjects.

Percentage failed in both = Total Percentage - Percentage passed in at least one subject

Percentage failed in both = $100\% - P(P \cup R)$

Percentage failed in both = $100\% - 95\%$

Percentage failed in both = $5\%$

Conclusion

Therefore, 5% of the candidates failed in both Physics and Reasoning.