Aggregating participation fractions: In a school, one-fourth of the boys and three-eighths of the girls participated in the annual sports. Without the ratio of boys to girls, can we find what fraction of the total student population participated?

Introduction / Context:
When different fractions apply to subgroups (boys and girls), the overall fraction of participation depends on how large each subgroup is. Without the relative sizes (or ratio) of the subgroups, the combined fraction cannot be uniquely determined.

Given Data / Assumptions:

• Participating boys = 1/4 of boys.
• Participating girls = 3/8 of girls.
• No information on the ratio (or counts) of boys to girls.

Concept / Approach:
Overall participation fraction = (participating boys + participating girls) / total students. This equals ( (1/4)*B + (3/8)*G ) / (B + G ). The expression depends on B and G separately, not just their sum, so B : G is required to get a single number.

Step-by-Step Solution:

Verification / Alternative check:
Example 1: If B = G, f = ( (1/4) + (3/8) ) / 2 per capita = (1/4 + 3/8)/(1 + 1) in ratio terms, leading to one value. Example 2: If B is much larger than G, the overall fraction shifts toward 1/4; if G dominates, it shifts toward 3/8. Hence no unique answer.

Why Other Options Are Wrong:
4/12 and 8/12 assume equal subgroup sizes without stating it; 5/8 is impossible as it exceeds both contributing subgroup fractions unless weights are extreme and still cannot be justified uniquely; 11/24 is a specific number requiring a specific B : G not given.

Common Pitfalls:
Averaging 1/4 and 3/8 directly (arithmetic mean) or adding numerators/denominators—both are invalid without weights.

Final Answer:
Data inadequate