The arithmetic mean of the scores of a group of students in a test was 52. The brightest 20% of them secured a mean score of 80 and the dullest 25% a mean score of 31. The mean score of remaining 55% is :

Options

A. 45
B. 50
C. 51.4 approx.
D. 54.6approx.

Answer

To solve the problem step by step, we will use the information provided about the mean scores of different groups of students and set up an equation to find the mean score of the remaining 55% of students.

### Step-by-Step Solution:

1. **Identify the total number of students**:
Let's assume the total number of students is 100 (this simplifies calculations since we are working with percentages).

2. **Calculate the total score of all students**:
The arithmetic mean of the scores of the entire group is 52. Therefore, the total score of all students can be calculated as:
\[ \text{Total Score} = \text{Mean} \times \text{Number of Students} = 52 \times 100 = 5200 \]

3. **Calculate the total score of the brightest 20% of students**:
The brightest 20% of students is 20 students (20% of 100). Their mean score is 80, so their total score is:
\[ \text{Total Score of Brightest} = 80 \times 20 = 1600 \]

4. **Calculate the total score of the dullest 25% of students**:
The dullest 25% of students is 25 students (25% of 100). Their mean score is 31, so their total score is:
\[ \text{Total Score of Dullest} = 31 \times 25 = 775 \]

5. **Calculate the total score of the remaining 55% of students**:
The remaining 55% of students is 55 students (100 - 20 - 25 = 55). Let the mean score of these students be \( A \). Therefore, their total score can be expressed as:
\[ \text{Total Score of Remaining} = A \times 55 \]

6. **Set up the equation**:
The total score of all students is the sum of the scores of the brightest, dullest, and remaining students:
\[ 1600 + 775 + 55A = 5200 \]

7. **Combine and simplify the equation**:
\[ 2375 + 55A = 5200 \]

8. **Isolate \( A \)**:
Subtract 2375 from both sides:
\[ 55A = 5200 - 2375 \]
\[ 55A = 2825 \]

9. **Solve for \( A \)**:
Divide both sides by 55:
\[ A = \frac{2825}{55} \approx 51.3636 \]

10. **Final Answer**:
Rounding to one decimal place, the mean score of the remaining 55% of students is approximately:
\[ A \approx 51.4 \]