Out of a group of 85 children who play either Badminton, Table Tennis, or both, the number of girls is 70% of the number of boys. Among the boys:
50% play only Badminton.
60% play Badminton in total (including those who play both sports).

Additionally:
40% of the total children play only Table Tennis.
A total of 12 children play both Badminton and Table Tennis.

Determine the number of girls who play only Badminton.

Question

Out of a group of 85 children who play either Badminton, Table Tennis, or both, the number of girls is 70% of the number of boys. Among the boys:  
50% play only Badminton.  
60% play Badminton in total (including those who play both sports).

Additionally:  
40% of the total children play only Table Tennis.  
A total of 12 children play both Badminton and Table Tennis.

Determine the number of girls who play only Badminton.

Accepted Answer

**Determine the number of boys and girls:**  
   - The ratio of boys to girls is given as $$10:7$$.  
   - Total children = 85, so:  
     $$
     	ext{Boys} = \frac{10}{17} 	imes 85 = 50, \quad 	ext{Girls} = 85 - 50 = 35.
     $$

**Calculate boys playing Badminton and Table Tennis:**  
   - Boys playing only Badminton = $$50\% \, 	ext{of boys} = \frac{50}{100} 	imes 50 = 25$$.  
   - Total boys playing Badminton = $$60\% \, 	ext{of boys} = \frac{60}{100} 	imes 50 = 30$$.  
   - Boys playing only Table Tennis = $$50 - 30 = 20$$.

**Calculate children playing only Table Tennis:**  
   - Children playing only Table Tennis = $$40\% \, 	ext{of total children} = \frac{40}{100} 	imes 85 = 34$$.  
   - Girls playing only Table Tennis = $$34 - 20 = 14$$.

**Calculate girls playing both sports (Badminton and Table Tennis):**  
   - Total children playing both sports = $$12$$.  
   - Girls playing both sports = $$12 - (30 - 25) = 12 - 5 = 7$$.

**Calculate girls playing only Badminton:**  
   - Girls playing only Badminton = Total girls - Girls playing only Table Tennis - Girls playing both sports:  
     $$
     	ext{Girls playing only Badminton} = 35 - 14 - 7 = 14.
     $$

Thus, the number of girls who play only Badminton is **14**.

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