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 can't be right, you entered the number backwards, you switched the order of two digits, I remember it cost less than 50 reais last week!" The cashier replies: I'm sorry, but all of our items had a $20\%$ price increase yesterday. What is the new price of the recorder?

## List 6
The old price was less than 50 reais and increased by $20 \%$. Therefore, the new price is still a two-digit number. Let's represent it as $a b$, where $a$ is the tens digit and $b$ is the units digit. Thus, the new price is $b a$, and we have:

$$
10 b+a=1.2(10 a+b) \Rightarrow 10 b-1.2 b=12 a-a
$$

Therefore:

$$
8.8 b=11 a \quad \Rightarrow \quad a=\frac{8}{10} \times b=\frac{4}{5} \times b
$$

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