The perimeter of a square equals the perimeter of a rectangle. If the lengths of the sides of the rectangle are in a ratio of 2:5, what is the ratio of the square's area to the rectangle's area?

Step 1: Let's denote the side of the square as 'a'. Therefore, the perimeter of the square is 4a.

Step 2: Let's denote the sides of the rectangle as 2x and 5x (since they are in the ratio 2:5). Therefore, the perimeter of the rectangle is 2(2x + 5x) = 14x.

Step 3: Since the perimeters of the square and the rectangle are equal, we can set up the equation 4a = 14x. Solving for 'a' in terms of 'x', we get a = 3.5x.

Step 4: The area of the square is a^2 = (3.5x)^2 = 12.25x^2.

Step 5: The area of the rectangle is (2x)(5x) = 10x^2.

Step 6: The ratio of the square's area to the rectangle's area is 12.25x^2 : 10x^2 = 1.225 : 1.

Therefore, the ratio of the square's area to the rectangle's area is 1.225 : 1.