Two pipes $A$ and $B$ can fill a cistern in $12$ min and $16$ min respectively. Both the pipes are opened together for a certain time but due to some obstruction, the flow of water was restricted to $\displaystyle\frac{5}{6}$ of the full in pipe $B$ the obstruction, is removed after some time and the tank is how filled in $3$ min from that moment for how many minutes was the obstruction there?