manalq8 wrote:

If the perimeter of square region S and the perimeter of rectangular region R are equal and the sides of R are in the ratio 2:3 then the ratio of the area of R to the area of S

A. 25:16

B. 24:25

C. 5:6

D. 4:5

E. 4:9

We are given that the perimeters of square region S and rectangular region R are equal and that the sides of R are in the ratio 2 : 3. Let’s label the sides of our figures:

Width of rectangle R = 2x

Length of rectangle R = 3x

Side of square S = s

The perimeter of rectangular region R is 2(2x) + 2(3x) = 4x + 6x = 10x.

The perimeter of square region S is 4s.

Since the two perimeters are equal we can create the following equation:

4s = 10x

2s = 5x

s = (5/2)x

Lastly, we need to determine the areas of both rectangle R and square S.

Area of rectangle R = length * width

A = (3x)(2x) = 6x^2

Since s = (5/2)x, we can use (5/2)x for the side of S.

Area of square S = side^2

A = [(5/2)x]^2

A = (25x^2)/4

We must determine the ratio of the area of region R to the area of region S.

Area of R/Area of S

6x^2/[(25x^2)/4]

6/(25/4)

24/25

Alternate Solution:

We know the sides of the rectangle have a ratio of 2:3; thus we can express the sides of this rectangle as 2x and 3x for some number x. The perimeter of the rectangle, in terms of x, is then 3x + 2x + 3x + 2x = 10x. This is also the perimeter of the square, so taking x = 2 will give us easy numbers to work with.

When x = 2, the sides of the rectangle are 4 and 6; thus the area of the rectangle is 4 x 6 = 24.

Also, when x = 2, the perimeter of the square is 10x = 20; thus a side of the square will be 5. The area of the square will then be 5 x 5 = 25.

So, the ratio of the area of the rectangle to the area of the square is 24:25.

Answer: B

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