jmc

First, we note that $r$ must be positive, since otherwise $\lfloor r\rfloor +r$ is nonpositive. Next, because $\lfloor r\rfloor$ is an integer and $\lfloor r\rfloor +r=12.2$, the decimal part of $r$ must be $0.2$. Therefore, $r=n+0.2$ for some integer $n$, so that $\lfloor r\rfloor =n$ and $\lfloor r\rfloor +r=2n+0.2=12.2$. Therefore, $n=6$, and the only value of $r$ that satisfies the equation is $\boxed{r=6.2}$.