The average marks of 140 students are 45 . If the average of passed students was 42 and failed students was 20 , then find the number of students who have passed .

Options

A. 124.5
B. 159.09
C. 1600/11
D. cannot be determined

Answer

To solve the problem step by step, we will follow the given information and use the formulas for averages and totals.

### Step 1: Understand the given data
- Total number of students = 140
- Average marks of all students = 45
- Average marks of passed students = 42
- Average marks of failed students = 20

### Step 2: Calculate the total marks of all students
The total marks can be calculated using the formula:
\[ \text{Total Marks} = \text{Average Marks} \times \text{Total Number of Students} \]
So, the total marks of all students is:
\[ \text{Total Marks} = 45 \times 140 = 6300 \]

### Step 3: Let the number of passed students be \( x \)
If \( x \) is the number of passed students, then the number of failed students will be:
\[ 140 - x \]

### Step 4: Calculate the total marks of passed students
The total marks of passed students can be calculated as:
\[ \text{Total Marks of Passed Students} = \text{Average Marks of Passed Students} \times \text{Number of Passed Students} \]
Thus, we have:
\[ \text{Total Marks of Passed Students} = 42x \]

### Step 5: Calculate the total marks of failed students
Similarly, the total marks of failed students can be calculated as:
\[ \text{Total Marks of Failed Students} = \text{Average Marks of Failed Students} \times \text{Number of Failed Students} \]
So, we have:
\[ \text{Total Marks of Failed Students} = 20(140 - x) \]

### Step 6: Set up the equation for total marks
The total marks of all students is the sum of the total marks of passed and failed students:
\[ 6300 = 42x + 20(140 - x) \]

### Step 7: Simplify the equation
Expanding the equation gives:
\[ 6300 = 42x + 2800 - 20x \]
Combining like terms results in:
\[ 6300 = 22x + 2800 \]

### Step 8: Solve for \( x \)
Subtract 2800 from both sides:
\[ 6300 - 2800 = 22x \]
\[ 3500 = 22x \]
Now, divide both sides by 22:
\[ x = \frac{3500}{22} \approx 159.09 \]

### Step 9: Conclusion
Since the number of students must be a whole number, we round \( x \) to the nearest whole number. Thus, the number of students who passed is approximately 159.