7.1. Given a square and a rectangle. The longer side of the rectangle is $11\%$ greater than the side of the square, and the shorter side is $10\%$ less than the side of the square. a) Is the area of the rectangle greater or smaller compared to the area of the square, and by what percentage? b) Answer the same questions regarding the perimeters of the rectangle and the square.
Answer: a) the area of the rectangle is less by $0.1 \%$, b) the perimeter of the rectangle is greater by $0.5 \%$. Solution. Let $a$ be the side of the square, then the sides of the rectangle are $1.11 a$ and $0.9 a$. The area and perimeter of the rectangle are respectively $1.11 \cdot 0.9 a^{2}=0.999 a^{2}$ and $2(1.11+0.9) a=4.02 a$. In percentage terms, the area of the rectangle is $0.999 a^{2} / a^{2}=99.9 \%$, and the perimeter is $4.02 a / 4 a=1.005=100.5 \%$. Thus, the area of the rectangle is less than the area of the square by $0.1 \%$, and the perimeter is greater than the perimeter of the square by $0.5 \%$.