A merchant wants to sell the existing stock of a sought-after item at one and a half times its current price. To achieve this, they plan a stepwise price increase. They will first implement a certain price increase, and then, when two-thirds of the stock has been sold, they will apply another price increase of the same ratio. This price will remain in effect until the entire stock is sold out. What percentage price increases will achieve the desired goal?
Suppose there were $x$ items in stock $(x>0)$, and they were sold at a unit price of $y$ Ft ( $y>0$ ); thus, the total stock was originally worth $x y$ Ft. The trader wants to increase this amount to 1.5 times its original value. First, a $p \%$ increase was implemented, meaning the unit price became $\left(y+\frac{y p}{100}\right)$ Ft, and this price was used to sell $2 / 3$ of the stock. Then, the price of the remaining $1 / 3$ of the stock was increased by $p \%$ again, and this is when the desired profit was achieved.

That is,

$$
\frac{2}{3} x\left(y+\frac{y p}{100}\right)+\frac{1}{3} x\left(y+\frac{y p}{100}\right)\left(1+\frac{p}{100}\right)=1.5 x y
$$

The equation can be divided by $x y$ $(x>0, y>0)$. (Thus, the magnitude of the increase is independent of the initial number of items and the price at which they were sold!)

By rearranging the equation and substituting $z=1+\frac{p}{100}$, we obtain a quadratic equation:

$$
z^{2}+2 z-\frac{9}{2}=0
$$

Solving for the positive root (since we are dealing with a price increase), we get $p \approx 34.5 \%$.