In an examination `70% ` students passed both in Mathematics and Physics, `85% ` passed in Mathematics and `80% ` passed in Physics. If 30 studetns have failed in both the subjects , then the total number of students who appeared in the examination is equal to

Options

A. 900
B. 600
C. 150
D. 100

Answer

To solve the problem step by step, we will use the information provided about the percentages of students passing in Mathematics and Physics, and the number of students who failed in both subjects.

### Step-by-Step Solution:

1. **Let the total number of students be \( x \)**.
- This is our variable representing the total number of students who appeared for the examination.

2. **Calculate the number of students who passed in Mathematics**:
- Given that 85% of students passed in Mathematics, the number of students who passed in Mathematics is:
\[ \text{Students passed in Mathematics} = \frac{85}{100} \times x = \frac{85x}{100} \]

3. **Calculate the number of students who passed in Physics**:
- Given that 80% of students passed in Physics, the number of students who passed in Physics is:
\[ \text{Students passed in Physics} = \frac{80}{100} \times x = \frac{80x}{100} \]

4. **Calculate the number of students who passed in both subjects**:
- Given that 70% of students passed in both Mathematics and Physics, the number of students who passed in both subjects is:
\[ \text{Students passed in both} = \frac{70}{100} \times x = \frac{70x}{100} \]

5. **Use the principle of inclusion-exclusion to find the total number of students who passed in at least one subject**:
- The formula for the union of two sets is:
\[ n(P \cup M) = n(P) + n(M) - n(P \cap M) \]
- Substituting the values we calculated:
\[ n(P \cup M) = \frac{80x}{100} + \frac{85x}{100} - \frac{70x}{100} \]
- Simplifying this:
\[ n(P \cup M) = \frac{80x + 85x - 70x}{100} = \frac{95x}{100} \]

6. **Calculate the number of students who failed in both subjects**:
- It is given that 30 students failed in both subjects. Therefore, the number of students who did not pass in either subject is:
\[ \text{Students failed in both} = x - n(P \cup M) = x - \frac{95x}{100} \]
- This simplifies to:
\[ x - \frac{95x}{100} = \frac{100x - 95x}{100} = \frac{5x}{100} \]

7. **Set the equation for students who failed in both subjects**:
- We know that this number equals 30:
\[ \frac{5x}{100} = 30 \]

8. **Solve for \( x \)**:
- Multiply both sides by 100 to eliminate the fraction:
\[ 5x = 300 \]
- Now, divide by 5:
\[ x = \frac{300}{5} = 60 \]

9. **Calculate the total number of students**:
- Since we have \( x = 60 \), we need to multiply by 10 to account for the percentage calculations:
\[ x = 600 \]

### Final Answer:
The total number of students who appeared in the examination is **600**.