In an examination `80%` of the students passed in physics `70%` of the students passed in chemistry `15%` of the students failed in both the subjects . If 325 students passed in both subjects. Then find the total students.

Options

A. `600`
B. `700`
C. `500`
D. `800`

Answer

To solve the problem step by step, we will follow the logical flow of the information provided.

### Step 1: Understand the given data
We know:
- 80% of the students passed in Physics.
- 70% of the students passed in Chemistry.
- 15% of the students failed in both subjects.
- 325 students passed in both subjects.

### Step 2: Calculate the percentage of students who passed at least one subject
Since 15% of the students failed in both subjects, it means that 85% of the students passed at least one subject (either Physics or Chemistry or both).

### Step 3: Set up the equation for students passing both subjects
Let \( X \) be the percentage of students who passed in both subjects. According to the principle of inclusion-exclusion, we can express the total percentage of students who passed at least one subject as:
\[ \text{Percentage of students passing at least one subject} = \text{Percentage passing Physics} + \text{Percentage passing Chemistry} - \text{Percentage passing both subjects} \]
This gives us:
\[ 85 = 80 + 70 - X \]
Solving for \( X \):
\[ 85 = 150 - X \]
\[ X = 150 - 85 = 65 \]
Thus, 65% of the students passed in both subjects.

### Step 4: Relate the percentage of students passing both subjects to the number of students
We know that 325 students passed in both subjects, which corresponds to 65% of the total number of students. Let \( T \) be the total number of students. We can write the equation:
\[ 0.65T = 325 \]

### Step 5: Solve for the total number of students
To find \( T \), we rearrange the equation:
\[ T = \frac{325}{0.65} \]
Calculating this gives:
\[ T = 500 \]

### Conclusion
The total number of students is **500**.

---