5. Answer: by $4 \%$.

Let the side of the larger square be denoted by $a$, and the side of the smaller square by $b$. As a result of cutting out the smaller square from the perimeter of the sheet, one segment of length $b$ disappears, and three such segments appear instead, meaning the perimeter increases by $2 b$. Therefore, $2 b$ constitutes $10 \%$ of the old perimeter: $2 b=0.1 \cdot 4 a$. From this, $b=0.2 a$. Thus, the area of the sheet has decreased by $b^{2}=0.04 a^{2}$, i.e., by $4 \%$.
5. From the edge of a large square sheet, a small square was cut off, as shown in the figure, and as a result, the perimeter of the sheet increased by $10 \%$. By what percent did the area of the sheet decrease?