In a class there are 9 boys and some girls. The average score of the boys in a test is 12, while the average score of the girls is 14. If the overall average score of all the students is 13.1, how many students are there in the class?

Introduction / Context:
This is another weighted average problem similar to earlier ones, but now with different group sizes. The question tests your ability to handle averages when one group size is known and the other is unknown, yet the overall average is given. Solving such questions correctly is important in exam scenarios involving statistics and arithmetic mean.

Given Data / Assumptions:

• Number of boys = 9.
• Average score of boys = 12.
• Average score of girls = 14.
• Overall average score of the class = 13.1.
• Number of girls is unknown, let it be g.

Concept / Approach:
Again we convert averages into totals. The total score for boys is fixed. The total score for girls is 14g. The overall total must equal 13.1 times the total number of students, which is 9 + g. Setting these expressions equal allows us to solve for g and thus determine the total class size.

Step-by-Step Solution:
Step 1: Total score of boys = 9 * 12 = 108.Step 2: Total score of girls = 14 * g.Step 3: Total number of students = 9 + g.Step 4: Overall total score = 13.1 * (9 + g).Step 5: Write equation: 108 + 14g = 13.1(9 + g).Step 6: Compute right side: 13.1 * 9 = 117.9, so 108 + 14g = 117.9 + 13.1g.Step 7: Subtract 13.1g from both sides: 0.9g = 9.9, hence g = 11.Step 8: Total students = 9 + 11 = 20.

Verification / Alternative check:
Check totals: Girls total = 11 * 14 = 154. Combined total = 108 + 154 = 262. Average = 262 / 20 = 13.1, which matches the given overall average. This confirms that the computed number of students is correct.

Why Other Options Are Wrong:
Options 21, 22, or 19 would require different numbers of girls and would change the weighted average, failing to preserve the overall mean of 13.1. Option 18 is too small a total and cannot produce the given average with the specified subgroup averages. Only 20 yields a consistent solution.

Common Pitfalls:

• Using a simple average of 12 and 14 instead of using the correct weighted relation.
• Errors while distributing 13.1 or when simplifying decimal coefficients.
• Assuming equal numbers of boys and girls, which is not given in the problem.

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