In a class there were 28 boys and some girls. In a test the mean score obtained by the boys was 12.5 while that obtained by the girls was 14.5. If the overall average was 13.1. what was the total number of students in the class?

Options

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

Answer

To solve the problem step by step, we need to find the total number of students in the class, which includes both boys and girls.

### Step 1: Define Variables
Let:
- Number of boys = 28
- Number of girls = x (unknown)
- Mean score of boys = 12.5
- Mean score of girls = 14.5
- Overall mean score = 13.1

### Step 2: Calculate Total Scores
Calculate the total score of boys:
\[ \text{Total score of boys} = \text{Number of boys} \times \text{Mean score of boys} = 28 \times 12.5 = 350 \]

Calculate the total score of girls:
\[ \text{Total score of girls} = \text{Number of girls} \times \text{Mean score of girls} = x \times 14.5 \]

### Step 3: Calculate Overall Total Score
The overall total score for all students can be expressed as:
\[ \text{Total score} = \text{Total score of boys} + \text{Total score of girls} = 350 + 14.5x \]

### Step 4: Calculate Total Number of Students
The total number of students in the class is:
\[ \text{Total students} = \text{Number of boys} + \text{Number of girls} = 28 + x \]

### Step 5: Set Up the Equation for Overall Average
Using the overall mean score, we can set up the equation:
\[ \text{Overall average} = \frac{\text{Total score}}{\text{Total students}} \]
Substituting the known values:
\[ 13.1 = \frac{350 + 14.5x}{28 + x} \]

### Step 6: Cross-Multiply to Eliminate the Fraction
Cross-multiply to simplify the equation:
\[ 13.1(28 + x) = 350 + 14.5x \]

### Step 7: Distribute and Rearrange
Distributing on the left side:
\[ 366.8 + 13.1x = 350 + 14.5x \]
Now, rearranging the equation:
\[ 366.8 - 350 = 14.5x - 13.1x \]
\[ 16.8 = 1.4x \]

### Step 8: Solve for x
Now, solve for x:
\[ x = \frac{16.8}{1.4} = 12 \]

### Step 9: Calculate Total Number of Students
Now that we have the number of girls, we can find the total number of students:
\[ \text{Total students} = 28 + x = 28 + 12 = 40 \]

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

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