Let L be the original length of the rectangle and W be the original width of the rectangle. We know that:
(L - 6) = (W + 3) (1) (since the length is decreased by 6cm and the width is increased by 3cm, the result is a square)
The area of the rectangle is LW, and the area of the square is (L - 6)(W + 3). We also know that the area of the square is 27cm^2 smaller than the area of the rectangle. So we can write:
(L - 6)(W + 3) = LW - 27 (2)
Expanding the left side of equation (2), we get:
LW - 6W + 3L - 18 = LW - 27
Simplifying and rearranging, we get:
3L - 6W = 9
Dividing both sides by 3, we get:
L - 2W = 3 (3)
Now we have two equations with two unknowns, equations (1) and (3). We can solve this system of equations by substitution. Rearranging equation (1), we get:
L = W + 9
Substituting this into equation (3), we get:
(W + 9) - 2W = 3
Simplifying, we get:
W = 6
Substituting this value of W into equation (1), we get:
L - 6 = 9
So:
L = 15
Therefore, the area of the rectangle is:
A = LW = 15 x 6 = 90 cm^2.