6. One side of the square was increased by 5 times, and the other side was reduced by 2 times, resulting in a rectangle with an area of $160 \mathrm{~cm}^{2}$. What is the side length of the square
6. First method of solution.

![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-082.jpg?height=357&width=922&top_left_y=1872&top_left_x=567)

1) $160: 5=32\left(\mathrm{~cm}^{2}\right)$ - area of half the square;
2) $32 \times 2=64\left(\mathrm{~cm}^{2}\right)$ - area of the original square;
3) $64: 8=8$ (cm) - side of the square.

Second method of solution. The algebraic solution of this problem is as follows: let $x$ be the side of the square, then $x / 2$ and $5 x$ are the sides of the rectangle. Its area is 160 sq.cm.

$$
5 x \times x / 2=160 \text { or } 5 x \times x=320, x \times x=64, x=8(\text { cm). }
$$

Answer: the side of the square is 8 cm.