From the condition, in the tennis club, there are 6x boys and 5x girls,
where x is some counting number, now unknown;

                    in the badminton club, there are 5y boys and 4y girls,
where y is some (possibly, another) counting number, now unknown.


For these numbers, x and y, we have this equation

    (6x+5x) + (5y+4y) = 109,    (1)

or

     11x + 9y = 109.            (2)


We should solve it in integer positive numbers.


From this equation, we write

    y =      (3)


and we are looking to find such x from the set of integers [1,9] that y in formula (3) be integer.


Making "trials and errors", we find the unique solution

    x= 5,  y= =  =  = 6.


ANSWER.  The total number of boys in both clubs is 6x + 5y = 6*5 + 5*6 = 60.

Use capital letters B and G for boys and girls in the tennis club and
small letters b and g for boys and girls in the badminton club.  

The tennis club and badminton club have a total of 109 members.

B + G + b + g = 109

The number of boys in the tennis club is 6/11 the members of this club. 

B = (6/11)(B + G)

The number of girls in the badminton club is 4/9 the members of this club. 

g = (4/9)(b + g)

How many boys are there in both clubs.

Go to https://www.wolframalpha.com/

type in  B + G + b + g = 109,  B = (6/11)(B + G), g = (4/9)(b + g)

For "integer solution", read

b = 55n + 30, B = 30 - 54n, g = 44n + 24, G = 25 - 45n, n element Z

Then since they must all be non-negative, choose n = 0

b = 30, B = 30, g = 24, G = 25

Add the boys, B + b = 30 + 30 = 60

Edwin