$\frac{3}{2^{2}}\times \frac{8}{3^{2}}\times \frac{15}{4^{2}}\times ...\times \frac{99}{10^{2}}$

$= \frac{1\times 3}{2^{2}}\times \frac{2\times 4}{3^{2}}\times \frac{3\times 5}{4^{2}}\times ...\times \frac{9\times 11}{10^{2}}$

$= \frac{1}{2}\times \frac{3}{2}\times \frac{2}{3}\times \frac{4}{3}\times \frac{3}{4}\times \frac{5}{4}\times ...\times \frac{8}{9}\times \frac{10}{9}\times \frac{9}{10}\times \frac{11}{10}$

$= \frac{1}{2}\times \frac{11}{10}$

$=\frac{11}{20}$.