Find the length of the sides of a rectangle if its perimeter is 54 cm and the sides are 6: 3.

Let us denote by x and y the required lengths (in centimeters) of the sides of the rectangle. Then the perimeter P of the rectangle will be equal to P = 2 * (x + y). According to the condition of the assignment, we have: 2 * (x + y) = 54. Dividing both sides of this equality by 2, we obtain: x + y = 27. According to another condition of the assignment, we have: x: y = 6: 3. Let’s apply the main property proportions. Then, 3 * x = 6 * y, whence x = 2 * y. Substitute this expression into the equality x + y = 27. We have: 2 * y + y = 27 or 3 * y = 27, whence y = 27: 3 = 9. Calculate x = 2 * y = 2 * 9 = 18. So Thus, the lengths of the sides of the rectangle are 18 cm and 9 cm.