Question 1208105
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Note that, grammatically, the question is stated incorrectly. "How many boys are there in both clubs?" means how many boys are in both the tennis club and the badminton club.  There is no information given that allows us to answer that.<br>
Clearly, the intent of the question is to ask for the total number of boys in the two clubs.<br>
Working with that....<br>
The boys in the tennis club are 6/11 of the total number of members, so there are 6x boys and 5x girls.<br>
The girls in the badminton club are 4/9 of the total number of members, so there are 4y girls and 5y boys.<br>
The total number of members in the two clubs is 109:<br>
{{{6x+5x+4y+5y=109}}}
{{{11x+9y=109}}}<br>
This is a Diophantine equation -- a single equation in two unknowns that has a finite number of solutions because of the restriction that both unknowns are positive integers.<br>
To find the set of solutions, solve the equation for one unknown in terms of the other and use the fact that both unknowns are positive integers.<br>
{{{11x+9y=109}}}
{{{9y=109-11x}}}
{{{y=(109-11x)/9}}}<br>
To make it easier to find the solution(s), perform the division as quotient plus remainder:<br>
{{{y=((108-9x)+(1-2x))/9}}}
{{{y=(12-x)+(1-2x)/9}}}<br>
x is an integer, so 12-x is an integer; and y must be an integer.  That means (1-2x)/9 must be an integer.<br>
One solution is with x=5, which gives us {{{y=(12-5)+(1-10)/9=12-5-1=6}}}<br>
So we have the solution x=5 and y=6; to look for other solutions, we can go back to the original equation {{{11x+9y=109}}} and see that we can get other solutions by increasing x by 9 and decreasing y by 11, or by decreasing x by 9 and increasing y by 11.  But neither of those gives us solutions in positive integers, so the solution we have is the unique solution.<br>
The total number of boys in the two clubs is 6x+5y = 6(5)+5(6) = 30+30 = 60.<br>
ANSWER: 60<br>