Ludvík noticed for a certain division example that if he doubles the dividend and increases the divisor by 12, he gets his favorite number as the result. He would get the same number if he reduced the original dividend by 42 and halved the original divisor.

Determine Ludvík's favorite number.

(M. Petrová)
If we denote the actors in the original example as $a: b$, then Ludvík's observation can be written as:

$$
2 a:(b+12)=(a-42): \frac{b}{2} .
$$

This is a double expression of Ludvík's favorite number, which we will denote as $\ell$ for subsequent modifications.

From the expression on the left, we get

$$
2 a=b \ell+12 \ell,
$$

from the expression on the right, we get

$$
\begin{aligned}
a-42 & =\frac{b}{2} \cdot \ell \\
2 a & =b \ell+84
\end{aligned}
$$

By comparing these two expressions, we find that $12 \ell=84$, so Ludvík's favorite number is $\ell=7$.

Evaluation. 3 points for writing down the relationships from the problem and auxiliary modifications; 3 points for the calculation, result, and quality of the commentary.

Notes. There is an infinite number of pairs of numbers $a$ and $b$ that satisfy Ludvík's equation. Randomly discovered possibilities leading to the correct result (e.g., $a=49$ and $b=2$) cannot be considered a complete solution to the problem - such processing should be evaluated no higher than 3 points.

Without additional notation of the ratio, we can modify the initial equality:

$$
\begin{aligned}
2 a \cdot \frac{b}{2} & =(a-42)(b+12), \\
a b & =a b-42 b+12 a-12 \cdot 42, \\
42(b+12) & =12 a .
\end{aligned}
$$

From this, after simplification, we get $2 a=7(b+12)$, so the desired ratio is $2 a:(b+12)=7$.