600 students took the test on Physics and Chemistry.`35%` students failed in Physics and `45%` students failed in Chemistry and `40%` of those who passed in Chemistry also passed in Physics, then how many students failed in both?

Options

A. 162
B. 138
C. 60
D. none of these

Answer

To solve the problem step by step, we will analyze the information provided and use it to find the number of students who failed in both Physics and Chemistry.

### Step 1: Calculate the number of students who failed in Physics and Chemistry
- Total students = 600
- Percentage of students who failed in Physics = 35%
- Percentage of students who failed in Chemistry = 45%

**Calculating the number of students who failed in each subject:**
- Students failed in Physics = 35% of 600 = (35/100) * 600 = 210
- Students failed in Chemistry = 45% of 600 = (45/100) * 600 = 270

### Step 2: Calculate the number of students who passed in each subject
- Percentage of students who passed in Physics = 100% - 35% = 65%
- Percentage of students who passed in Chemistry = 100% - 45% = 55%

**Calculating the number of students who passed in each subject:**
- Students passed in Physics = 65% of 600 = (65/100) * 600 = 390
- Students passed in Chemistry = 55% of 600 = (55/100) * 600 = 330

### Step 3: Calculate the number of students who passed in both subjects
- Given that 40% of those who passed in Chemistry also passed in Physics.
- Students who passed in both = 40% of students who passed in Chemistry = (40/100) * 330 = 132

### Step 4: Use the inclusion-exclusion principle to find the number of students who failed in both subjects
- Let A be the set of students who failed in Physics, and B be the set of students who failed in Chemistry.
- We know:
- |A| = 210 (failed in Physics)
- |B| = 270 (failed in Chemistry)
- |A ∩ B| (failed in both) = ?

Using the formula for inclusion-exclusion:
\[ |A ∪ B| = |A| + |B| - |A ∩ B| \]

Where |A ∪ B| represents the total number of students who failed in at least one subject. Since we need to find |A ∩ B|, we can rearrange the formula:
\[ |A ∩ B| = |A| + |B| - |A ∪ B| \]

### Step 5: Calculate |A ∪ B|
- The total number of students is 600.
- Students who passed in both subjects = 132
- Therefore, students who failed in at least one subject = Total students - Students who passed in both = 600 - 132 = 468

### Step 6: Substitute the values into the inclusion-exclusion formula
\[ |A ∩ B| = 210 + 270 - 468 \]
\[ |A ∩ B| = 480 - 468 = 12 \]

### Conclusion
The number of students who failed in both Physics and Chemistry is **12**.

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