Problem Solving

The perimeters of the square region S and the rectangle region R are equal. If the sides of R are in the ratio 2:3, what is the ratio of the area of region R to the area of region S?

Answer is 24:25

Thank you for helping me to understand this question.

Let's assume that the length of the rectangle is 3x and that the width of the rectangle is 2x. We know that the perimeter of a rectangle is given by the formula 2(length)+2(width), which in this problem, would be 2(3x)+2(2x)=10x. The problem tells us that the perimeter of rectangle R is equal to the perimeter of square S. The formula for a perimeter of square is 4*s and we know that the perimeter of the square S must be 10x, so:

4s=10x and s=10x/4

Area of a square: s^2-->(10x/4)^(2)=100x^2/16
Area of a rectangle: length*width--> 3x*2x=6x^2

(6x^2)/(100x^2/16)=6x^2*16/100x^2=24/25

bronzie35 wrote:The perimeters of the square region S and the rectangle region R are equal. If the sides of R are in the ratio 2:3, what is the ratio of the area of region R to the area of region S?

Answer is 24:25

Thank you for helping me to understand this question.

It's quite straightforward once you break down the question.

As: Area of Square region = s^2 (s is side of square)
Ar: Area of Rectangle region = l*b (l is length, b is breadth)

Ratio of l:b = 2:3
=> l = (2/3) * b

Perimeters are of Square and Rectangle are Equal:
4*s = 2*(l+b)
=> 2*s = l+b = (5/3) * b
=> s = (5/6) * b
=> s^2 = (25/36) * (b^2)

Ar:As = [(25/36) * (b^2)] / [(5/3) * (b^2)]
Ar:As = 24 / 25

Hope this helps!

Here's how I did it:

I want a rectangle that is 2x3, however a square with the same perimeter would have sides of (2+3)/2. Not very nice to work with. Let's try doubling the size of the rectangle.

Technically to compare perimeters, you should add up all the sides, but if you are only concerned with creating a shape with the same perimeter, than you only need to worry about 2 sides because the opposite two sides are the same.

If we have a rectangle that is 4 x 6, then a square with the same perimeter would have sides of (4+6)/2. That's 5

Area of Rectangle is 24
Area of square is 25

(In actual fact I made my rectangle 12 x 18 and the square 15 x 15 which was totally unnecessary and I realized that as I was typing this. In that case, I expressed the ratio as a fraction: (12)(18)/(15)(15) and reduced.