Sixteen children are divided into two groups, A and B, with 10 children in group A and 6 children in group B. The average percentage marks of group A is 75, and the overall average percentage marks of all 16 children is 76. What is the average percentage marks of the children in group B?

Introduction / Context:
This question is a classic weighted average problem involving two subgroups within a larger group. We know the average marks of one subgroup and the overall average, and we must use this information to find the average of the remaining subgroup. Such questions are very common in aptitude tests and help you understand how averages behave when combining or splitting groups.

Given Data / Assumptions:

• Total number of children = 16.
• Group A has 10 children with average percentage marks = 75.
• Group B has 6 children with unknown average percentage marks.
• The overall average percentage marks of all 16 children = 76.
• We assume percentage marks are computed out of the same maximum and can be added normally.

Concept / Approach:
The key idea is that the total marks for the whole group is the sum of the total marks for each subgroup. The average of a group is defined as total marks divided by number of children. So we first compute the total marks of all 16 children using the overall average. Then we compute the total marks of group A using its average. The difference between these totals gives the total marks of group B. Finally, we divide by the number of children in group B to find their average.

Step-by-Step Solution:
Step 1: Compute total marks of all 16 children using the overall average. Total marks (all) = 16 * 76 = 1216. Step 2: Compute total marks of group A using its average. Total marks (A) = 10 * 75 = 750. Step 3: Compute total marks of group B. Total marks (B) = Total marks (all) - Total marks (A) = 1216 - 750 = 466. Step 4: Number of children in group B = 6. Step 5: Average marks of group B = Total marks (B) / Number of children in B = 466 / 6. Step 6: Simplify 466 / 6 by dividing numerator and denominator by 2: 466 / 6 = 233 / 3. Step 7: Thus the average percentage marks of group B is 233/3, which is approximately 77.67 percent.

Verification / Alternative check:
We can verify by substituting the average of group B back into the weighted average formula. Total marks from group B is (233 / 3) * 6 = 233 * 2 = 466. Combined with group A total of 750, we get 750 + 466 = 1216. Divide by the total number of children 16 to get 1216 / 16 = 76, which matches the given overall average. This confirms that our computed average for group B is correct.

Why Other Options Are Wrong:
The value 243/4 corresponds to 60.75 percent, which would bring the overall average below 76. Similarly, 254/5 equals 50.8 percent, also too low. The fraction 345/7 is approximately 49.29 percent, again inconsistent with a higher overall average. Simply using 76 (the overall average) for group B is not justified because group A already has a lower average (75), so group B must be higher than 76 to pull the combined average to 76. Only 233/3 satisfies the weighted average relationship exactly.

Common Pitfalls:
A frequent mistake is to average the averages directly, for example taking (75 + 76) / 2, which is incorrect because the groups have different sizes. Another common error is to forget to multiply the average by the number of children to obtain total marks. Some students also try to solve the problem using guesswork instead of a clear algebraic approach, which can be risky with fractional answers. Always remember that average = total / number, and that total is additive across groups.

Final Answer:
The average percentage marks of the children in group B is 233/3.