The average marks obtained by 125 students in an exam is 29. If the average marks of passed students is 36 and that of failed students is 11. Find the numbers of failed students?

Answer

To solve the problem step by step, we will use the concept of averages and the allegation method.

### Step 1: Understand the given information
- Total number of students = 125
- Average marks of all students = 29
- Average marks of passed students = 36
- Average marks of failed students = 11

### Step 2: Calculate the total marks obtained by all students
To find the total marks obtained by all students, we can use the formula:
\[ \text{Total Marks} = \text{Average Marks} \times \text{Number of Students} \]
So, the total marks obtained by all students is:
\[ \text{Total Marks} = 29 \times 125 = 3625 \]

### Step 3: Set up the allegation method
We will use the allegation method to find the ratio of passed and failed students. The formula for allegation is:
\[ \text{Difference between average of passed and overall} : \text{Difference between overall and average of failed} \]
Calculating these differences:
- Difference between average of passed and overall:
\[ 36 - 29 = 7 \]
- Difference between overall and average of failed:
\[ 29 - 11 = 18 \]

### Step 4: Set up the ratio
Now we can set up the ratio of passed students to failed students:
\[ \text{Ratio} = 18 : 7 \]

### Step 5: Calculate the total parts in the ratio
The total parts in the ratio is:
\[ 18 + 7 = 25 \]

### Step 6: Find the number of passed and failed students
Let the number of passed students be represented by 18 parts and the number of failed students by 7 parts. We can find the number of students represented by each part:
\[ \text{Value of each part} = \frac{125}{25} = 5 \]
Now, we can find the number of passed and failed students:
- Number of passed students:
\[ 18 \times 5 = 90 \]
- Number of failed students:
\[ 7 \times 5 = 35 \]

### Step 7: Conclusion
The number of failed students is:
\[ \text{Number of failed students} = 35 \]