floor and ceiling help

First, we note that r must be positive, since otherwise $\lfloor r \rfloor + r$ is nonpositive. Next, because $\lfloor r \rfloor + r$ is an integer and $\lfloor r \rfloor + r=12.2$, the decimal part of r must be 0.5. Therefore, r=n+0.5 for some integer n, so that $\lfloor r \rfloor + r$ and $\lfloor r \rfloor + r = 2n+0.5 =16.5$. Therefore, n=8, and the only value of r that satisfies the equation is $\boxed{r=8.5}$.