Mirka is thinking of a two-digit number. If she reverses the order of the digits, she gets a number that differs from the original number by $75 \%$. What number could she have in mind?

(Bednářová)
Let the original number be $10a + b$, so the new number is $10b + a$. (2 points)

First, assume that the new number is 75% larger. From this, we get the following equation:

$$
1.75(10a + b) = 10b + a
$$

which simplifies to $2a = b$. Now, it is easy to calculate that the numbers that satisfy this are: 12, 24, 36, and 48.

Next, assume that the new number is 75% smaller. From this, we get the following equation:

$$
0.25(10a + b) = 10b + a
$$

which simplifies to $a = \frac{13}{2}b$. In this case, we do not get any solutions.

Therefore, Mirka could have thought of the numbers 12, 24, 36, or 48. (4 points)

Note: 100% is taken as the original number. If we took 100% as the new number, we would get the numbers 21, 42, 63, 84.