Out of a total 85 children playing badminton or table tennis or both, total number of girls in the group is 70% of the total number of boys…

Let us consider the number of boys to be x.

Number of girls = 70% of x

Therefore, x + 70% of x = 85

$(\begin{array}{l} \Rightarrow {\rm{}}x + \frac{{70 \times x}}{{100}} = 85\\ \Rightarrow {\rm{}}x + \frac{{7x}}{{10}} = 85 \end{array})$

⇒ 10x + 7x = 850

⇒ 17x = 850

⇒ x = 850/17

⇒ x = 50

Therefore, number of boys = 50 and the number of girls are (85 – 50) = 35

Number of boys playing only badminton = 50% of boys $( = \frac{{50}}{{100}} \times 50{\rm{}} = {\rm{}}25{\rm{}})$

∴ Total number of boys playing Table Tennis = 50 – 25 = 25

Total number of boys playing badminton = 60% of boys = $(\frac{{60}}{{100}} \times 50 = {\rm{}}30)$

∴ Total number of boys playing both table tennis and badminton = 30 – 25 = 5

∴ Total number of boys playing only Table Tennis = 25 – 5 = 20

Number of children playing only table tennis = 40% of all children $( = {\rm{}}\frac{{40}}{{100}} \times 85 = {\rm{}}34)$

∴ Number of girls playing only Table tennis = 34 – 20 = 14

Total number of children playing badminton and tennis both = 12

∴ total number of girls playing both badminton and table tennis both = 12 – 5 = 7

Therefore, number of girls playing only badminton = 35 – (14 + 7) = 14