A large square region is paved with $n^2$ gray square tiles, each measuring $s$ inches on a side. A border $d$ inches wide surrounds each tile. The figure below shows the case for $n=3$. When $n=9$, the $81$ gray tiles cover $81\%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this value of $n$?