The sides of the rectangle are 2 to 6, and its perimeter is 32 cm. Find the area of this rectangle.

This is a task in parts. If the sides relate like 2: 6, then this means that one side contains 2 parts, and the other – 6 of the same parts.
Let the length of one part be x cm, then the length of one side containing 2 parts is 2x cm, and the length of the second side containing 6 parts is 6x cm.The perimeter of a rectangle is equal to the sum of the lengths of its sides, or 2 (2x + 6x) cm or 32 see Let’s make an equation and solve it.
2 (2x + 6x) = 32;
2 * 8x = 32;
16x = 32;
x = 32: 16;
x = 2 (cm) – length of one part;
2x = 2 * 2 = 4 (cm) – the length of one side;
6x = 6 * 2 = 12 (cm) – the length of the second side.
Answer. 4 cm, 12 cm.