Question

The perimeters of square region S and rectangular 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 1

Answer:

The ratio of the area of region R to the area of region S is:

                    $\dfrac{24}{25}$

Step-by-step explanation:

The sides of R are in the ratio : 2:3

Let the length of R be: 2x

and the width of R be: 3x

i.e. The perimeter of R is given by:

$Perimeter\ of\ R=2(2x+3x)$

( Since, the perimeter of a rectangle with length L and breadth or width B is given by:

$Perimeter=2(L+B)$ )

Hence, we get:

$Perimeter\ of\ R=2(5x)$

i.e.

$Perimeter\ of\ R=10x$

Also, let " s " denote the side of the square region.

We know that the perimeter of a square with side " s " is given by:

$\text{Perimeter\ of\ square}=4s$

Now, it is given that:

The perimeters of square region S and rectangular region R are equal.

i.e.

$4s=10x\\\\i.e.\\\\s=\dfrac{10x}{4}\\\\s=\dfrac{5x}{2}$

Now, we know that the area of a square is given by:

$\text{Area\ of\ square}=s^2$

and

$\text{Area\ of\ Rectangle}=L\times B$

Hence, we get:

$\text{Area\ of\ square}=(\dfrac{5x}{2})^2=\dfrac{25x^2}{4}$

and

$\text{Area\ of\ Rectangle}=2x\times 3x$

i.e.

$\text{Area\ of\ Rectangle}=6x^2$

Hence,

Ratio of the area of region R to the area of region S is:

$=\dfrac{6x^2}{\dfrac{25x^2}{4}}\\\\=\dfrac{6x^2\times 4}{25x^2}\\\\=\dfrac{24}{25}$

Answer 2

Answer with Step-by-step explanation:

the sides of rectangular region R are in the ratio 2 : 3.

i.e. If length=2x

then, breath=3x

Perimeter of rectangular region=2(length+breath)

                                    = 2(2x+3x)

                                    =10x

and Area of rectangular region=2x×3x

                                                  =length×breath

                                                  =6x²

Let s be the side length of square region S

Perimeter of square region=4s

The perimeters of square region S and rectangular region R are equal.

i.e. 4s=10x

s=$\dfrac{5}{2}x$

Area of square region=s²

                                    =$\dfrac{5}{2}x\times \dfrac{5}{2}x$

                                    =$\dfrac{25}{4}x^2$

Ratio of the area of region R to the area of region S

$=6x^2:\dfrac{25}{4}x^2 \\\\=6:\dfrac{25}{4}\\\\=24:25$

Hence, the ratio of the area of region R to the area of region S is:

24:25