A rectangle has a Length : Width ratio of 3 : 1. From the table below,

length(l)=3width(w)
perimeter= 2(l+b)
=8b

Area = b x 3b = 3b^2

The area must be divisible by 3 and the quotient should be a perfect square. Here 48(3x16) satisfies the condition. from here we can get b as 4. Substituting this in Perimeter we get Perimeter as 32.

Solution:

As we know Area of rectangular = Length * Breadth and the Perimeter of Rectangular = 2(Length + Breadth)

It is given in the question stem that Length: Width ratio is 3:1

Let's bring that above knowledge into an equation

Equation 1 ----> Area = 3x * x = \(3x^{2}\)

Equation 2-----> Perimeter = 2(3x + x) = 2(4x) = 8x

Now, look for the values that fit into both the equations, and the only values that match up for Area is 48 and Perimeter is 32.

1. We're asked to find a working combination for the perimeter and area of the rectangle.

2. Since we're only given the ratio between sides and no other information, we will need to consider just plugging in and testing values.

3. Let the width be equal to x. Then the length of the rectangle will be 3x.

4. \(Perimeter = 2 * Length + 2 * Width = 2 * 3x + 2 * x = 8x\) and \(Area = Length * Width = 3x * x = 3 * x^2\).

5. Now let's plug in some values of x.

- x = 1. \(Width = 8 * 1 = 8\) and \(Length = 3 * (1)^2 = 3\). This doesn't work.
- x = 2. \(Width = 8 * 2 = 16\) and \(Length = 3 * (2)^2 = 12\). This doesn't work.
- x = 3. \(Width = 8 * 3 = 24\) and \(Length = 3 * (3)^2 = 27\). This doesn't work.
- x = 4. \(Width = 8 * 4 = 32\) and \(Length = 3 * (4)^2 = 48\). This works.

6. Our answer will be: Perimeter - 32 and Area - 48.

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(5.) requires an assumption that depends on x being an integer. Another way is to test out the answer options as perimeters and areas to figure out x and compare.

Let the ratios be 3x & 1x
Perimeter = 2(L+B) = 2(3x+x) = 2*4x= 8x
Area = L*B = 3x*x

To calculate area answer should be multiple of 3 so - A,B,C are eliminated, the remaining D,E,F should be a perfect square after dividing the value by 3
D = 36/3= 13 not possible
E = 48/3 = 16
F = 72/3 = 24 not possible

So area is 48 and value for x = 4 so perimeter is 8*4 = 32

A rectangle has a Length : Width ratio of 3 : 1. From the table below, select a combination of Perimeter and Area that satisfies such a relationship.
Perimeter of a Rectangle = 2(l+b)
We are given Lenght/Breadth = 3/1; Length(l) = 3* Breadth (b)
Perimeter = 2(4 b) = 8 b
Area = l*b = 3 b^2

From the options, Perimeter = 8 (4) = 32; Area = 3 * 16 = 48

it still does, we need an area divisible by 16 and multiple of 3, 3*16 = 48.
now area = l*b ~ 2*24/3*16/4*12/6*8
hence perimeter = 52/38/32/28
option has 32