There were 600 students in a school. Each offered either English or hindi or both. If `75%` offered English and `45%` Hindi how many offered both?

Options

A. 48
B. 60
C. 80
D. 120

Answer

To solve the problem, we will use the principle of inclusion-exclusion to find the number of students who offered both English and Hindi.

### Step-by-Step Solution:

1. **Identify the total number of students**:
- Total students = 600

2. **Calculate the number of students who offered English**:
- Percentage of students who offered English = 75%
- Number of students who offered English = \( \frac{75}{100} \times 600 \)
- Calculation:
\[ \frac{75}{100} \times 600 = 450 \]
- Therefore, 450 students offered English.

3. **Calculate the number of students who offered Hindi**:
- Percentage of students who offered Hindi = 45%
- Number of students who offered Hindi = \( \frac{45}{100} \times 600 \)
- Calculation:
\[ \frac{45}{100} \times 600 = 270 \]
- Therefore, 270 students offered Hindi.

4. **Use the principle of inclusion-exclusion**:
- Let \( N(E) \) be the number of students who offered English, \( N(H) \) be the number of students who offered Hindi, and \( N(E \cap H) \) be the number of students who offered both subjects.
- According to the principle of inclusion-exclusion:
\[ N(E \cup H) = N(E) + N(H) - N(E \cap H) \]
- Here, \( N(E \cup H) \) is the total number of students, which is 600.
- Substituting the known values:
\[ 600 = 450 + 270 - N(E \cap H) \]

5. **Solve for \( N(E \cap H) \)**:
- Rearranging the equation gives:
\[ N(E \cap H) = 450 + 270 - 600 \]
- Calculation:
\[ N(E \cap H) = 720 - 600 = 120 \]
- Therefore, the number of students who offered both English and Hindi is 120.

### Final Answer:
The number of students who offered both English and Hindi is **120**.