In a class there are 9 boys and some girls. In a test the mean score of the boys is 12 and the mean score of the girls is 14. If the overall average score of all the students is 13.1, what is the total number of students in the class?

Introduction / Context:
This question checks understanding of weighted averages where two groups have different averages and sizes. You must infer the number of girls from the overall class average and then find the total strength of the class.

Given Data / Assumptions:
- Number of boys = 9. - Average score of boys = 12. - Let number of girls be g. - Average score of girls = 14. - Overall average of all students = 13.1.

Concept / Approach:
Average equals total score divided by number of students. We find the total score of boys and of girls in terms of g. The sum of these gives total class score. Dividing that sum by total students (9 + g) and equating it to 13.1 leads to a solvable linear equation for g. Then we add boys and girls to find the total number of students.

Step-by-Step Solution:
Step 1: Total score of boys = 9 * 12 = 108. Step 2: Total score of girls = 14g. Step 3: Total students = 9 + g. Step 4: Overall average = (108 + 14g) / (9 + g) = 13.1. Step 5: Write equation (108 + 14g) = 13.1(9 + g). Step 6: Expand right side: 108 + 14g = 117.9 + 13.1g. Step 7: Rearrange: 14g - 13.1g = 117.9 - 108 gives 0.9g = 9.9, so g = 11. Step 8: Total students = 9 + 11 = 20.

Verification / Alternative check:
With 9 boys and 11 girls, total score = 108 + 14 * 11 = 108 + 154 = 262. Overall average = 262 / 20 = 13.1, which matches the given value. Thus the solution is consistent with all data in the question.

Why Other Options Are Wrong:
- 19, 21, 22 and 24 all correspond to different numbers of girls and lead to overall averages that are not 13.1. - Quick mental guesses usually ignore the weighted nature of the average, so they do not satisfy the precise equation.

Common Pitfalls:
Learners sometimes take the simple average of 12 and 14, which is 13, and try to match 13.1 by trial and error without equations. Others forget that more girls pull the average closer to 14 while more boys pull it toward 12. Writing an equation using totals is the most reliable method.

Final Answer:
The total number of students in the class is 20.