jmc

Because the 6 empty chairs are $\frac{1}{4}$ of the chairs in the room, there are $6\times 4=24$ chairs in all. The number of seated people is $\frac{3}{4}\times 24=18$, and this is $\frac{2}{3}$ of the people present. It follows that $$\frac{18}{\text{people present}}=\frac{2}{3}.$$So there are $\boxed{27}$ people in the room.