How many values of $r$ are there such that $\lfloor r \rfloor + r = 15.5?$
I know that $\lfloor r \rfloor$ is the greatest integer less than or equal to $r$, so it must be an integer. That means that $r$ must have a fractional part of $0.5$, since otherwise the sum would not be a half-integer. So I can write $r = n + 0.5$, where $n$ is an integer. Then $\lfloor r \rfloor + r = n + (n + 0.5) = 15.5$. Simplifying, I get $2n + 0.5 = 15.5$, so $2n = 15$ and $n = 7.5$. But $n$ has to be an integer, so this is a contradiction. Therefore, there are no values of $r$ that satisfy the equation.

