In an examination, 19% students fail in Mathematics and 10% students fail in English. If 7% of all students fail in both subjects, then the number of students passed in both subjects is

Options

A. 36% of all students
B. 64% of all students
C. 71% of all students
D. 78% of all students

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion to find the percentage of students who pass in both subjects.

### Step 1: Identify the percentages of failures
- Let \( P(M) \) be the percentage of students failing in Mathematics = 19%
- Let \( P(E) \) be the percentage of students failing in English = 10%
- Let \( P(M \cap E) \) be the percentage of students failing in both subjects = 7%

### Step 2: Calculate the percentage of students failing in at least one subject
Using the inclusion-exclusion principle:
\[ P(M \cup E) = P(M) + P(E) - P(M \cap E) \]
Substituting the values:
\[ P(M \cup E) = 19\% + 10\% - 7\% = 22\% \]
This means 22% of students fail in at least one subject (either Mathematics or English or both).

### Step 3: Calculate the percentage of students passing in both subjects
To find the percentage of students passing in both subjects, we subtract the percentage of students failing in at least one subject from 100%:
\[ P(\text{Pass in both}) = 100\% - P(M \cup E) \]
Substituting the value we found:
\[ P(\text{Pass in both}) = 100\% - 22\% = 78\% \]

### Conclusion
The percentage of students who pass in both subjects is **78%**.

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