Problem 3. If one side of a rectangle is increased by $8 \mathrm{~cm}$, a square with a perimeter of $6.8 \mathrm{~dm}$ is obtained. What is the perimeter of the rectangle expressed in centimeters?
Solution. First method. The perimeter of the square is $6.8 \mathrm{dm}=68 \mathrm{~cm}$. Let $a$ be the smaller side of the rectangle. The perimeter of the square is $L=4(a+8)$, which means that $4(a+8)=68$, i.e., $a+8=17$, from which we get $a=9 \mathrm{~cm}$. Therefore,

![](https://cdn.mathpix.com/cropped/2024_06_05_c3c23cf15e32fbc44a06g-03.jpg?height=397&width=505&top_left_y=556&top_left_x=954)

$$
L^{\prime}=2(a+a+8)=2(9+17)=52 \mathrm{~cm}
$$

Second method. The perimeter of the square is $6.8 \mathrm{dm}=68 \mathrm{~cm}$. A square with a perimeter of $68 \mathrm{~cm}$ has a side of $68: 4=17 \mathrm{~cm}$. This is one side of the rectangle, and the other side is equal to $17-8=9 \mathrm{~cm}$.

Finally, the perimeter of the rectangle is

$$
L^{\prime}=2(9+17)=52 \mathrm{~cm}
$$

Third method. The perimeter of the square is $6.8 \mathrm{dm}=68 \mathrm{~cm}$. When one side of the rectangle is increased by $8 \mathrm{~cm}$, its perimeter increases by $2 \cdot 8=16 \mathrm{~cm}$. Therefore, the perimeter of the rectangle is equal to $68-16=52 \mathrm{~cm}$.