jmc

Let the original number of people in the room be $x$. Half of them left, so $\frac{x}{2}$ of them are left in the room. After that, one third of this group is dancing, so $\frac{x}{2}-\frac{1}{3}\left(\frac{x}{2}\right)=\frac{x}{3}$ people are not dancing. This is given to be $12$, so $$\frac{x}{3}=12\Rightarrow x=36$$ $\boxed{36}$