imc

You can create the equation $\frac{x+1}{y+1}=\frac{11x}{10y}$ Cross multiplying and combining like terms gives $xy+11x-10y=0$. This can be factored into $(x-10)(y+11)=-110$. $x$ and $y$ must be positive, so $x>0$ and $y>0$, so $x-10>-10$ and $y+11>11$. Using the factors of 110, we can get the factor pairs: $(-1,110),$ $(-2,55),$ and $(-5,22).$ But we can't stop here because $x$ and $y$ must be relatively prime. $(-1,110)$ gives $x=9$ and $y=99$. $9$ and $99$ are not relatively prime, so this doesn't work. $(-2,55)$ gives $x=8$ and $y=44$. This doesn't work. $(-5,22)$ gives $x=5$ and $y=11$. This does work. We found one valid solution so the answer is $\boxed{1}$.