Answer: $4 \%$.

Solution. Let the width of the castle be $a$, and the width of the extension be $b$. Then the original perimeter is $4 a$, and the final perimeter is $4 a+2 b$. Therefore:

$$
1.1 \cdot 4 a=4 a+2 b \Leftrightarrow b=0.2 a
$$

From this, the area of the castle becomes $a^{2}+(0.2 a)^{2}=1.04 a^{2}$, which means the area has increased by $4 \%$.

Criteria. Any correct solution: 7 points.

The side of the extension is correctly found, but the further solution is missing or incorrect: 4 points.

Only the correct answer is provided: 0 points.
1. (7 points) Percival's castle had a square shape. One day, Percival decided to expand his domain and added a square extension to the castle. As a result, the perimeter of the castle increased by $10 \%$. By what percentage did the area of the castle increase?