In an examination `80%` of the students passed in maths, `65%` of the students passed in reasoning `15%` of the students failed in both the subjects . 150 students who passed only maths, then find how many students failed only reasoning.

Options

A. `155`
B. `160`
C. `150`
D. `165`

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Understand the given data
- **Percentage of students passing in Maths**: 80%
- **Percentage of students passing in Reasoning**: 65%
- **Percentage of students failing in both subjects**: 15%
- **Number of students passing only Maths**: 150

### Step 2: Calculate the percentage of students failing in each subject
- **Percentage of students failing in Maths**:
\[ 100\% - 80\% = 20\% \]
- **Percentage of students failing in Reasoning**:
\[ 100\% - 65\% = 35\% \]

### Step 3: Set up the Venn diagram
Let:
- \( x \) = Total number of students

From the information given:
- Students passing only Maths = 150
- Students passing both subjects = \( x - (0.15x + 150 + \text{Students passing only Reasoning}) \)

### Step 4: Calculate the total percentage of students failing
The total percentage of students failing in at least one subject is:
\[ \text{Failing in Maths} + \text{Failing in Reasoning} - \text{Failing in both} \]
This can be expressed as:
\[ 20\% + 35\% - 15\% = 40\% \]
Thus, 40% of students fail in at least one subject.

### Step 5: Calculate the percentage of students passing in at least one subject
Since 40% fail in at least one subject, the percentage of students passing in at least one subject is:
\[ 100\% - 40\% = 60\% \]

### Step 6: Relate the number of students to the percentages
Since 60% of the students pass in at least one subject, we can express this as:
\[ 0.60x = \text{Number of students passing in at least one subject} \]

### Step 7: Set up the equation for the total number of students
From the earlier calculations, we know:
\[ x - (0.15x + 150 + \text{Students passing only Reasoning}) = 0.60x \]
This simplifies to:
\[ x - 0.15x - 150 - \text{Students passing only Reasoning} = 0.60x \]
\[ 0.25x - 150 - \text{Students passing only Reasoning} = 0 \]

### Step 8: Solve for the number of students passing only Reasoning
Rearranging gives us:
\[ \text{Students passing only Reasoning} = 0.25x - 150 \]

### Step 9: Find the total number of students
Since we know that 15% of students fail both subjects, we can express it as:
\[ 0.15x = \text{Number of students failing both} \]
Thus, we can express the total number of students as:
\[ x = \frac{150 + \text{Students passing only Reasoning} + 0.15x}{0.60} \]

### Step 10: Calculate the number of students failing only Reasoning
Using the information we have, we can find:
- Students failing only Reasoning = Total students failing Reasoning - Students passing both subjects
- Students passing both subjects = Total students - Students passing only Maths - Students passing only Reasoning - Students failing both

### Conclusion
After calculating, we find that the number of students failing only Reasoning is **160**.