In Lhotě, they elected a mayor. Two citizens ran for the position: Ing. Schopný and his wife Dr. Schopná. There were three polling stations in the village. In the first and second polling stations, Dr. Schopná received more votes. In the first station, the ratio of votes was $7: 5$, and in the second station, it was $5: 3$. In the third polling station, the ratio of votes was $3: 7$ in favor of Ing. Schopný. The election ended in a tie, as both candidates received the same number of votes. What was the ratio of the number of valid ballots cast in each of the polling stations, given that the same number of people cast a valid vote in the first and second stations?
Let the number of valid votes cast in the first polling station be denoted by $x$, in the second polling station also $x$, and in the third polling station by $y$. Engineer Capable received $\frac{7}{12} x$ votes in the first polling station, $\frac{5}{8} x$ votes in the second, and $\frac{3}{10} y$ votes in the third. The total number of valid votes cast was $x + x + y$, of which half were for Engineer Capable. This leads to the equation

$$
\begin{aligned}
\frac{7}{12} x + \frac{5}{8} x + \frac{3}{10} y & = \frac{1}{2}(x + x + y) \\
70 x + 75 x + 36 y & = 60(x + x + y)
\end{aligned}
$$

After simplification, we get

$$
\begin{aligned}
25 x & = 24 y \\
\frac{x}{y} & = \frac{24}{25}
\end{aligned}
$$

The number of valid voting slips cast in each polling station was in the ratio $x: x: y$, i.e., $24: 24: 25$

[3 points for setting up the equation, 2 points for solving the equation, 1 point for the correct answer]