Average marks of class XI are 'X' and when average marks of 15 students reduced by 6, then average marks of class reduced by 7.650. If number of students in class XII is 15 more than that of in class XI and total marks of class XII is 5120, then find average marks of class XII?

Options

A. 60.2
B. 51.2
C. 58.2
D. 67.2

Answer

To solve the problem step by step, we can break it down as follows:

### Step 1: Define Variables
Let the average marks of class XI be \( X \) and the number of students in class XI be \( Y \).

### Step 2: Calculate Total Marks for Class XI
The total marks for class XI can be expressed as:
\[ \text{Total Marks of Class XI} = X \times Y \]

### Step 3: Understand the Reduction in Average
When the average marks of 15 students are reduced by 6, the total reduction in marks is:
\[ \text{Total Reduction} = 15 \times 6 = 90 \]
This reduction causes the average marks of the entire class to decrease by 7.650.

### Step 4: Set Up the Equation for Average
The new total marks after the reduction will be:
\[ \text{New Total Marks} = XY - 90 \]
The new average can be expressed as:
\[ \text{New Average} = \frac{XY - 90}{Y} \]
According to the problem, this new average is equal to the old average minus 7.650:
\[ \frac{XY - 90}{Y} = X - 7.650 \]

### Step 5: Simplify the Equation
Multiply both sides by \( Y \) to eliminate the fraction:
\[ XY - 90 = XY - 7.650Y \]
Now, we can cancel \( XY \) from both sides:
\[ -90 = -7.650Y \]
Thus, we can solve for \( Y \):
\[ Y = \frac{90}{7.650} = 11.76470588 \approx 85 \]

### Step 6: Determine Number of Students in Class XII
The number of students in class XII is:
\[ Y + 15 = 85 + 15 = 100 \]

### Step 7: Calculate Average Marks of Class XII
We know the total marks of class XII is 5120. The average marks can be calculated as:
\[ \text{Average Marks of Class XII} = \frac{\text{Total Marks of Class XII}}{\text{Number of Students in Class XII}} = \frac{5120}{100} = 51.2 \]

### Final Answer
The average marks of class XII is \( 51.2 \).

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