Márcia is in a store buying a recorder that she has wanted for a long time. When the cashier registers the price, she exclaims: "That's not possible, you must have entered the number backwards and switched the order of two digits, because I remember that last week it cost less than 50 reais!" The cashier replies: I'm sorry, but yesterday all our items were increased by 20%. What is the new price of this recorder?
The old price was less than 50 reais and underwent an increase of $20\%$, resulting in a new price that is still a two-digit number, which we represent as $ab$, where $a$ is the tens digit and $b$ is the units digit, i.e., $ab = 10a + b$. The new price is the old price $ba$ with an increase of $20\%$, i.e.,

$$
10a + b = ab = (1.2)ba = 1.2(10b + a) = 12b + 1.2a
$$

Thus, $10a - 1.2a = 12b - b$, or $8.8a = 11b$. Therefore,

$$
b = \frac{8.8}{11} \times a = \frac{4}{5} \times a
$$

Since $a$ and $b$ are digits, we can only have $a = 5$ and $b = 4$, so the new price is $\mathrm{R} \$ 54.00$.