Inspired by Paola Ramirez

$\begin{cases} x+ \lfloor y \rfloor+ \{ z \}=1.1 & \implies \lfloor x \rfloor + \{ x \} + \lfloor y \rfloor+ \{ z \} = 1.1 & ...(1) \\ \lfloor x \rfloor+ \{ y \}+z=2.2 & \implies \lfloor x \rfloor+ \{ y \}+ \lfloor z \rfloor + \{ z \} = 2.2 & ...(2) \\ \{ x \}+y+\lfloor z \rfloor=3.3 & \implies \{ x \}+ \lfloor y \rfloor + \{ y \}+\lfloor z \rfloor = 3.3 & ...(3) \end{cases}$

$\begin{aligned} (3)-(2)-(1): \quad - 2 \lfloor x \rfloor - 2\{ z \} & = 0 \\ \implies \lfloor x \rfloor & = -\{z\} \quad \quad \small \color{#3D99F6}{\text{The only solution is}} \\ \implies \lfloor x \rfloor & = \{z\} = 0 \\ x & = \{x\} \\ z & = \lfloor z \rfloor \end{aligned}$

$\begin{aligned} (1): \quad \{ x \} + \lfloor y \rfloor & = 1.1 \quad \quad \small \color{#3D99F6}{\lfloor x \rfloor = \{z\} = 0 } \\ \implies \lfloor y \rfloor & = 1 \\ \{x\} & = 0.1 = x \\ (2): \quad \{ y \}+ \lfloor z \rfloor & = 2.2 \quad \quad \small \color{#3D99F6}{\lfloor x \rfloor = \{z\} = 0 } \\ \implies \lfloor z \rfloor & = 2 = z \\ \{y\} & = 0.2 \\ \implies y & = \lfloor y \rfloor + \{y\} = 1.2 \end{aligned}$

$\implies 100xyz = 100(0.1)(1.2)(2) = \boxed{24}$