The rectangle has side lengths in the ratio $2: 5$. If we extend all its sides by $9 \mathrm{~cm}$, we get a rectangle whose side lengths are in the ratio $3: 7$.

In what ratio will the side lengths of the rectangle be if we extend all its sides by another $9 \mathrm{~cm}$?

(M. Petrová)
The lengths of the sides of the original rectangle are in the ratio $2: 5$. This means that their lengths (in centimeters) can be denoted as $2 x$ and $5 x$. After the first extension of the sides, the rectangle has dimensions $2 x+9$ and $5 x+9$. Since the lengths of the sides of this rectangle are in the ratio $3: 7$, the following must hold:

$$
\frac{2 x+9}{5 x+9}=\frac{3}{7}
$$

Solving the equation, we get $x=36(\mathrm{~cm})$. The dimensions of the original rectangle are $2 \cdot 36=$ $=72(\mathrm{~cm})$ and $5 \cdot 36=180(\mathrm{~cm})$.

After the second extension of the sides, the rectangle has dimensions $72+18=90(\mathrm{~cm})$ and $180+18=$ $=198(\mathrm{~cm})$. The ratio of the lengths of the sides of this rectangle is therefore

$$
90: 198=5: 11
$$

Scoring suggestion. 2 points for labeling the dimensions of the original rectangle and expressing their change; 2 points for setting up and solving the equation; 2 points for calculating the final dimensions of the rectangle and their ratio.