3. Given a real number $x$, the symbol $\lfloor x\rfloor$ denotes its integer part (i.e., the largest integer less than or equal to $x$) and $\{x\}$ its fractional part (i.e., $x-\lfloor x\rfloor$). Let $x, y, z$ be three positive real numbers that satisfy the following system:

$$
\left\{\begin{array}{l}
3\lfloor x\rfloor-\{y\}+\{z\}=20.3 \\
3\lfloor y\rfloor+5\lfloor z\rfloor-\{x\}=15.1 \\
\{y\}+\{z\}=0.9
\end{array}\right.
$$

What is the value of $x+y+z$?
(A) 10.8
(B) 11.1
(C) 11.6
(D) 12.8
(E) 13
3. The answer is (D). Starting 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 $\lfloor x\rfloor=7$. Now consider the second equation: since $15<3\lfloor y\rfloor+5\lfloor z\rfloor=15.1+\{x\}<17$ we get $15.1+\{x\}=16$, since $3\lfloor y\rfloor+5\lfloor z\rfloor$ is an integer, from which $\{x\}=0.9$. From the equality $3\lfloor y\rfloor+5\lfloor z\rfloor=16$ we can derive that $\lfloor z\rfloor<4$. However, for $\lfloor z\rfloor=0,1,3$ the value $3\lfloor y\rfloor=16-5\lfloor z\rfloor$ is not a multiple of three, so necessarily $\lfloor z\rfloor=2$ from which $\lfloor y\rfloor=\frac{16-5\lfloor z\rfloor}{3}=2$.

We can then conclude that

$$
x+y+z=\lfloor x\rfloor+\lfloor y\rfloor+\lfloor z\rfloor+\{x\}+\{y\}+\{z\}=7+2+2+0.9+0.9=12.8 \text{. }
$$

A final observation: combining the third equation with the relation $\{y\}-\{z\}=0.7$ that is 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 calculate the sum of the three numbers.