Solution:

The answer is (D). Let us start from the first equation of the system: we have $19 < 20,3 + \{y\} - \{z\} < 22$, so the value of $3\lfloor x\rfloor$, which is an integer multiple of $3$, must necessarily be $21$, from which $\lfloor x\rfloor = 7$.

Let us now consider the second equation: since $15 < 3\lfloor y\rfloor + 5\lfloor z\rfloor = 15,1 + \{x\} < 17$ we obtain $15,1 + \{x\} = 16$, since $3\lfloor y\rfloor + 5\lfloor z\rfloor$ is an integer, from which $\{x\} = 0,9$.

Again from the equality $3\lfloor y\rfloor + 5\lfloor z\rfloor = 16$ we can deduce that $\lfloor z\rfloor < 4$. However for $\lfloor z\rfloor = 0, 1, 3$ the value $3\lfloor y\rfloor = 16 - 5\lfloor z\rfloor$ does not turn out to be a multiple of three, hence necessarily $\lfloor z\rfloor = 2$ from which $\lfloor y\rfloor = \frac{16 - 5\lfloor z\rfloor}{3} = 2$.

We can then conclude that
$$x + y + z = \lfloor x\rfloor + \lfloor y\rfloor + \lfloor z\rfloor + \{x\} + \{y\} + \{z\} = 7 + 2 + 2 + 0,9 + 0,9 = 12,8 \text{.}$$

One final remark: combining the third equation with the relation $\{y\} - \{z\} = 0,7$ derived from the first equation, it is possible to obtain $\{y\} = 0,8$ from which $\{z\} = 0,1$, but this information is not necessary to compute the sum of the three numbers.