{Dec}

{x} = x - [x] , where [.] is the floor function, i.e. the greatest integer lesser than or equal to x.

$x-[x]=\frac {x}{5}$

$[x]=\frac {4x}{5}$

Since, the LHS of the above equation is always an integer, $x$ will have to be of the form $\frac {5n}{4}$ , where $n$ is any integer. Fortunately, $n$ cannot be greater than 3, as that would make $x$ an integer, which would result in {x} = 0. Also note that $x≥0$ , since {x}≥0

All 4 values of $n$ (i.e. 0,1,2,3) satisfy the given equation. Therefore, sum of all possible $x= 0+\frac {5}{4}+ \frac {10}{4} + \frac {15}{4}=7.5$

$\{x\}:=x-\lfloor x\rfloor$ $\therefore x-\lfloor x\rfloor = \dfrac{x}{5}$ $\Rightarrow \dfrac{4x}{5}=\lfloor x\rfloor$ $\Rightarrow {x}=5\times\dfrac{\lfloor x\rfloor}{4}.........[1]$ $Note\space that:\space 0\leq\{x\}<1\Rightarrow 0\leq\dfrac{x}{5}<1\Rightarrow 0\leq x<5\Rightarrow 0\leq\lfloor x\rfloor<5$ $\Rightarrow x\in\{\dfrac{5}{4},\dfrac{10}{4},\dfrac{15}{4}\}\space\blue{by\space putting\space each\space value\space of\space \lfloor x\rfloor\space in\space [1]}$ $\Rightarrow Answer=\dfrac{5}{4}+\dfrac{10}{4}+\dfrac{15}{4}=\dfrac{30}{4}=7.5$