In a class there were 18 boys and some girls. In a test the mean score obtained by the boys was 12 while that obtained by the girls was 14. If the overall average was 13.1, what was the total number of students in the class?

Options

A. 38
B. 44
C. 42
D. 40

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Define Variables
Let the number of girls in the class be \( X \).

### Step 2: Calculate the Total Score of Boys
The mean score of the boys is 12, and there are 18 boys. Therefore, the total score of the boys can be calculated as:
\[ \text{Total score of boys} = \text{Mean score} \times \text{Number of boys} = 12 \times 18 = 216 \]

### Step 3: Calculate the Total Score of Girls
The mean score of the girls is 14. Therefore, the total score of the girls can be calculated as:
\[ \text{Total score of girls} = \text{Mean score} \times \text{Number of girls} = 14 \times X \]

### Step 4: Calculate the Overall Average
The overall average score of the class is given as 13.1. The total number of students in the class is \( 18 + X \). Therefore, the total score of the class can be expressed as:
\[ \text{Total score of class} = \text{Total score of boys} + \text{Total score of girls} = 216 + 14X \]
The overall average can also be expressed as:
\[ \text{Overall average} = \frac{\text{Total score of class}}{\text{Total number of students}} = \frac{216 + 14X}{18 + X} \]
Setting this equal to the overall average of 13.1 gives us the equation:
\[ \frac{216 + 14X}{18 + X} = 13.1 \]

### Step 5: Cross-Multiply to Solve for X
Cross-multiplying gives:
\[ 216 + 14X = 13.1(18 + X) \]
Expanding the right side:
\[ 216 + 14X = 13.1 \times 18 + 13.1X \]
Calculating \( 13.1 \times 18 \):
\[ 13.1 \times 18 = 235.8 \]
So the equation becomes:
\[ 216 + 14X = 235.8 + 13.1X \]

### Step 6: Rearranging the Equation
Rearranging the equation to isolate \( X \):
\[ 216 - 235.8 = 13.1X - 14X \]
This simplifies to:
\[ -19.8 = -0.9X \]
Dividing both sides by -0.9 gives:
\[ X = \frac{19.8}{0.9} = 22 \]

### Step 7: Calculate Total Number of Students
Now that we know the number of girls \( X = 22 \), we can find the total number of students in the class:
\[ \text{Total number of students} = 18 + X = 18 + 22 = 40 \]

### Final Answer
The total number of students in the class is **40**.

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