In a group of 85 children playing sports (badminton or table tennis or both), the total number of girls equals 70% of the number of boys. Among the boys, the count playing only badminton is 50% of all boys, and the total number of boys playing badminton (including those who also play table tennis) is 60% of all boys. Overall, 40% of all children play only table tennis, and 12 children play both games. How many girls play only badminton?

Introduction / Context:
This problem combines percentages with a two-set (badminton, table tennis) count. We translate the text into equations for boys and girls across three disjoint categories: only badminton, only table tennis, and both. Careful book-keeping is key to avoid double counting.

Given Data / Assumptions:

• Total children = 85.
• Let boys = B, girls = G; G = 0.70 * B.
• Boys playing only badminton = 50% of B.
• Total boys playing badminton (only or both) = 60% of B.
• Only table tennis (boys + girls) = 40% of 85 = 34.
• Both games (boys + girls) = 12.

Concept / Approach:
Convert the gender relation to counts, then use category totals to solve for each sub-group. The equation G = 0.70B and total B + G = 85 determine B and G. Next, the badminton details determine how many boys are in ”both”, which in turn yields girls in ”both”. Finally, use the only-table-tennis count to finish the table and extract girls playing only badminton.

Step-by-Step Solution:

Verification / Alternative check:
Sum across categories equals gender totals and overall total 85, confirming consistency.

Why Other Options Are Wrong:
16 and 17 arise from arithmetic slips in distributing only-table-tennis or both; ”Data inadequate” is incorrect because the provided constraints uniquely determine the counts.

Common Pitfalls:
Forgetting that ”total boys playing badminton” includes those who also play table tennis, or misapplying the 40% to only one gender instead of all children.

Final Answer:
14