The average marks obtained by 180 students in an examination is 50. If the average marks of passed students is 80 and that of the failed students is 40, then what is the number of students who failed the examination ?

Options

A. 90
B. 135
C. 100
D. 45

Answer

To find the number of students who failed the examination, we can follow these steps:

### Step 1: Understand the given information
- Total number of students = 180
- Average marks of all students = 50
- Average marks of passed students = 80
- Average marks of failed students = 40

### Step 2: Calculate the total marks obtained by all students
To find the total marks obtained by all students, we can use the formula:
\[ \text{Total Marks} = \text{Average Marks} \times \text{Number of Students} \]
Substituting the values:
\[ \text{Total Marks} = 50 \times 180 = 9000 \]

### Step 3: Set up equations for passed and failed students
Let:
- \( p \) = number of passed students
- \( f \) = number of failed students

From the problem, we know:
\[ p + f = 180 \]

### Step 4: Express total marks in terms of passed and failed students
The total marks can also be expressed as:
\[ \text{Total Marks} = (\text{Average Marks of Passed} \times p) + (\text{Average Marks of Failed} \times f) \]
Substituting the known values:
\[ 9000 = (80 \times p) + (40 \times f) \]

### Step 5: Substitute \( f \) in terms of \( p \)
From the first equation \( p + f = 180 \), we can express \( f \) as:
\[ f = 180 - p \]
Now substitute \( f \) in the total marks equation:
\[ 9000 = (80 \times p) + (40 \times (180 - p)) \]

### Step 6: Simplify the equation
Expanding the equation:
\[ 9000 = 80p + 7200 - 40p \]
Combine like terms:
\[ 9000 = 40p + 7200 \]
Subtract 7200 from both sides:
\[ 1800 = 40p \]
Now, divide by 40:
\[ p = \frac{1800}{40} = 45 \]

### Step 7: Find the number of failed students
Now that we have \( p \), we can find \( f \):
\[ f = 180 - p = 180 - 45 = 135 \]

### Final Answer
The number of students who failed the examination is **135**.

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