manalq8
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