In an examination 80% candidates passed in English and 85% candidates passed in Mathematics. If 73% candidates passed in both these subjects, then what per cent of candidates failed in both the subjects ?

Options

A. 8
B. 15
C. 27
D. 35

Answer

To solve the problem, we will use the principle of inclusion-exclusion to find the percentage of candidates who failed in both subjects.

**Step 1: Define the percentages**
- Let \( P(E) \) be the percentage of candidates who passed in English = 80%
- Let \( P(M) \) be the percentage of candidates who passed in Mathematics = 85%
- Let \( P(E \cap M) \) be the percentage of candidates who passed in both subjects = 73%

**Step 2: Calculate the percentage of candidates who passed in at least one subject**
To find the percentage of candidates who passed in at least one subject, we can use the inclusion-exclusion principle:

\[ P(E \cup M) = P(E) + P(M) - P(E \cap M) \]

Substituting the values we have:

\[ P(E \cup M) = 80\% + 85\% - 73\% \]

**Step 3: Perform the calculation**
Now, we will perform the calculation:

\[ P(E \cup M) = 80 + 85 - 73 = 92\% \]

This means that 92% of the candidates passed in at least one of the subjects (either English or Mathematics or both).

**Step 4: Calculate the percentage of candidates who failed in both subjects**
To find the percentage of candidates who failed in both subjects, we subtract the percentage of candidates who passed in at least one subject from 100%:

\[ \text{Percentage of candidates who failed in both} = 100\% - P(E \cup M) \]

Substituting the value we calculated:

\[ \text{Percentage of candidates who failed in both} = 100\% - 92\% = 8\% \]

Thus, the percentage of candidates who failed in both subjects is **8%**.

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