The following data was observed from a study of car complaints received from 180 respondents at Colonel Verma's car care workshop, viz., engine problem, transmission problem or mileage problem. Of those surveyed, there was no one who faced exactly two of these problems. There were 90 respondents who faced engine problems, 120 who faced transmission problems and 150 who faced mileage problems.
How many of them faced all the three problems?

Options

A. 45
B. 60
C. 90
D. 20

Answer

To solve the problem, we need to find out how many respondents faced all three car problems: engine problems, transmission problems, and mileage problems. We will denote the number of people facing all three problems as \( w \).

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( x \) be the number of respondents who faced only engine problems.
- Let \( y \) be the number of respondents who faced only transmission problems.
- Let \( z \) be the number of respondents who faced only mileage problems.
- Let \( w \) be the number of respondents who faced all three problems.

2. **Set Up Equations Based on Given Data**:
- From the problem, we know:
- Total respondents = 180
- Respondents with engine problems = \( x + w = 90 \)
- Respondents with transmission problems = \( y + w = 120 \)
- Respondents with mileage problems = \( z + w = 150 \)

3. **Express \( x, y, z \) in terms of \( w \)**:
- From \( x + w = 90 \), we get \( x = 90 - w \).
- From \( y + w = 120 \), we get \( y = 120 - w \).
- From \( z + w = 150 \), we get \( z = 150 - w \).

4. **Total Respondents Equation**:
- The total number of respondents can be expressed as:
\[ x + y + z + w = 180 \]
- Substituting the expressions for \( x, y, z \):
\[ (90 - w) + (120 - w) + (150 - w) + w = 180 \]

5. **Simplify the Equation**:
- Combine like terms:
\[ 90 + 120 + 150 - w - w - w + w = 180 \]
\[ 360 - 2w = 180 \]

6. **Solve for \( w \)**:
- Rearranging gives:
\[ 360 - 180 = 2w \]
\[ 180 = 2w \]
\[ w = \frac{180}{2} = 90 \]

7. **Conclusion**:
- The number of respondents who faced all three problems is \( w = 90 \).

### Final Answer:
The number of respondents who faced all three problems is **90**.