Question 1208105
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The tennis club and badminton club have a total of 109 members. 
The number of boys in the tennis club is 6/11 the members of this club. 
The number of girls in the badminton club is 4/9 the members of this club. 
How many boys are there in both clubs.
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<pre>
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 = {{{(109 - 11x)/9}}}     (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= {{{(109 - 11*5)/9}}}= {{{(109-55)/9}}} = {{{54/9}}} = 6.


<U>ANSWER</U>.  The total number of boys in both clubs is 6x + 5y = 6*5 + 5*6 = 60.
</pre>

Solved.


On the way, I explained to you how to solve Diophantine equation (2) in integer positive numbers.


Notice that we managed to solve a problem with only one equation in two unknowns.


It was possible because the other restriction was that the solution must be in integer (whole) numbers.