Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\%$?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{infinitely many}$
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 $\boxed{\textbf{(B) }1}$.