The price of the book "New Riddles" was reduced by $62.5\%$. Matěj found out that both prices (before and after the reduction) are two-digit numbers and can be expressed with the same digits, just in a different order. By how many Kč was the book discounted?

(M. Volfová)
The original price of the book in Kč will be written in the form $10a + b$, where $a$ and $b$ are unknown non-zero digits. After the discount, the price of the book was $10b + a$. The price reduction was $62.5\%$, which means it was reduced to $37.5\%$, so

$$
\frac{37.5}{100} \cdot (10a + b) = 10b + a
$$

Given the equality $\frac{37.5}{100} = \frac{75}{200} = \frac{3}{8}$, we can rewrite the previous equation as:

$$
\begin{aligned}
\frac{3}{8} \cdot (10a + b) & = 10b + a, \\
30a + 3b & = 80b + 8a, \\
22a & = 77b, \\
2a & = 7b.
\end{aligned}
$$

The only single-digit natural numbers that satisfy this equation are $a = 7$ and $b = 2$. The original price was 72 Kč, and after the discount, it was 27 Kč. The book was discounted by 45 Kč.