Taking 10 to the power of both sides in each equation, these equations become: $10^{x}=\left(10^{y-1}+1\right) \cdot 10^{-1}$ and $10^{y}=\left(10^{x}+1\right) \cdot 10^{-1}$. Let $a=10^{x}$ and $b=10^{y}$. Our equations become: $10a=b/10+1$ and $10b=a+1$ and we are asked to compute $a/b$. Subtracting the equations gives $10a-10b=b/10-a \Longrightarrow 11a=101b/10$ giving an answer of $101/110$.