解：$(1)$设$0.\dot {2} \dot {7}$为$x$，
由$0.\dot {2} \dot {7} =0.2727...$，可知$100x=27.2727...$，
所以$100x-x=27$
解得$x=\frac {27}{99}=\frac {3}{11}.$
所以$0.\dot {2} \dot {7} =\frac {3}{11} .$
$(2)$设$0.\dot {2}1 \dot {6} =x$，
由$0.\dot {2}1 \dot {6} =0.216216...$，可知$1000x=216.216216... $，
所以$1000x-x=216$
解得$x=\frac {216}{999}=\frac {8}{37} .$
所以$0.\dot {2}1 \dot {6} =\frac {8}{37} .$
