The perimeter of the rectangle is 30 cm, its sides are 2: 3. Determine its area.

We denote the lengths of the sides of the rectangle through x and y.
According to the condition of the problem, the perimeter of this rectangle is 30 cm, therefore, the following relation holds:
2 * (x + y) = 30.
It is also known that the sides of a given rectangle are 2: 3, therefore, the following ratio takes place:
x / y = 2/3.
We solve the resulting system of equations. Substituting in the first equation the value x = (2/3) * y from the second equation, we get:
2 * ((2/3) * y + y) = 30.
We solve the resulting equation:
2 * (5/3) * y = 30;
(10/3) * y = 30;
y = 30 / (10/3);
y = 30 * (3/10);
y = 9 cm.
Knowing y, we find x:
x = (2/3) * y = (2/3) * 9 = 6 cm.
Find the area S of this rectangle:
S = x * y = 6 * 9 = 54 cm².
Answer: the area of ​​this rectangle is 54 cm².