12. (5 points) Teacher Wang drives from home to location $A$. On the way there, the first $\frac{1}{2}$ of the journey is traveled at a speed of 50 kilometers per hour, and the remaining distance is traveled at a speed increased by $20 \%$; On the return trip, the first $\frac{1}{3}$ of the journey is traveled at a speed of 50 kilometers per hour, and the remaining distance is traveled at a speed increased by $32 \%$. As a result, the return trip takes 31 minutes less than the trip there. How far is Teacher Wang's home from location $A$ in kilometers?
12. (5 points) Teacher Wang drives from home to location $A$. On the way there, the first $\frac{1}{2}$ of the journey is traveled at 50 kilometers/hour, and the remaining distance is traveled at a speed increased by $20 \%$; On the return trip, the first $\frac{1}{3}$ of the journey is traveled at 50 kilometers/hour, and the remaining distance is traveled at a speed increased by $32 \%$. As a result, the return trip takes 31 minutes less than the trip there. The distance between Teacher Wang's home and location $A$ is 
330 
kilometers.

【Solution】Given that the speed on the way there is 50 kilometers/hour, and the speed for the remaining distance is $50 \times(1+20 \%)$ $=50$ kilometers/hour; The speed on the return trip is 50 kilometers/hour, and the speed for the remaining distance is $50 \times(1+32 \%)=$ 66 kilometers/hour.

Let the total distance be $x$ kilometers, then:
$$
\begin{array}{l}
\left(\frac{1}{2} x \times \frac{1}{50}+\frac{1}{2} x \times \frac{1}{60}\right)-\left(\frac{1}{3} x \times \frac{1}{50}+\frac{2}{3} x \times \frac{1}{66}\right)=\frac{31}{60} \\
\frac{11}{600} x-\frac{83}{4950} x=\frac{31}{60} \\
\frac{31}{19800} x=\frac{31}{60} \\
x=330
\end{array}
$$

Answer: The distance between Teacher Wang's home and location $A$ is 330 kilometers.
Therefore, the answer is: 330.