12. (6 points) Given that $x$ is a simplest proper fraction, if its numerator is increased by $a$, it simplifies to $\frac{1}{3}$; if its denominator is increased by $a$, it simplifies to $\frac{1}{4}$, then $x=$ . $\qquad$
【Solution】Solution: Let the original fraction $x$ be $\frac{\mathrm{c}}{\mathrm{b}}$, then:
$$
\frac{\mathrm{c}+\mathrm{a}}{\mathrm{b}}=\frac{1}{3}
$$

Then: $b=3(c+a)=3 c+3 a$ (1)
$$
\frac{c}{b+a}=\frac{1}{4}
$$

Then: $4 c=a+b$ (2)
Substituting (1) into (2) yields:
$$
\begin{aligned}
4 c & =a+3 c+3 a \\
4 c & =4 a+3 c
\end{aligned}
$$

Then: $c=4 a$ (3)
Substituting (3) into (1) yields:
$$
b=3 c+3 a=3 \times 4 a+3 a=15 a
$$

Therefore, $\frac{\mathrm{c}}{\mathrm{b}}=\frac{4 \mathrm{a}}{15 \mathrm{a}}=\frac{4}{15}$
Thus, $x=\frac{4}{15}$.
The answer is: $\frac{4}{15}$.