$ $因为$\frac {1}{2}=\frac {172}{344},$
$\frac {199}{344}>\frac {172}{344},$
所以$\frac {199}{344}>\frac {1}{2}。$
$ $又因为$\frac {1}{2}=\frac {144}{288},$
$\frac {111}{288}<\frac {144}{288},$
所以$\frac {111}{288}<\frac {1}{2},$
所以$\frac {199}{344}>\frac {111}{288}。$