length of rectantular rug is 7/6 its width.

rug is placed on floor with 1 foot margin all around.
area of floor is 4 times that of margin.
find the dimensions of the room.

length of rug = 7x

width of rug = 6x , (that avoids fractions)

floor is (7x+2) by (6x+2)

area of margin = (7x+2)(6x+2) - 42x^2

floor is 4 times the margin

(7x+2)(6x+2) = 4[(7x+2)(6x+2) - 42x^2 ]
42x^2 + 26x + 4 = 4[42x^2 + 26x + 4 - 42x^2]
42x^2 - 78x - 12 = 0
7x^2 - 13x - 2 = 0
(x-2)(7x+1) = 0
x = 2 or a negative

so the floor is
7x+2 by 6x+2 or
16 by 14 or 224 ft^2

and the rug is (7x)(6x) = 168 ft^2
margin = 224-168 = 56

is 4(56) = 224 ??, YES

The room is 16 ft by 14 ft

To find the dimensions of the room, we need to go step by step. Let's start by assigning variables to the given information:

Let 'x' represent the width of the rug.
The length of the rug is given as 7/6 times its width, which can be represented as (7/6) * x.
A 1-foot margin is present around the rug, so the length and width of the room would each increase by 2 feet: (x + 2) and ((7/6) * x + 2).

Next, we are told that the area of the room is four times that of the margin. Mathematically, we can write this as:

(x + 2) * ((7/6) * x + 2) = 4 * (x * (7/6))

Now, let's solve this equation step by step:

Expanding both sides of the equation:
(x + 2) * ((7/6) * x + 2) = 4 * (x * (7/6))
(x + 2) * (7/6 * x) + (x + 2) * 2 = 4 * (7/6 * x)

Simplifying:
(7/6 * x^2) + (7/6 * 2x) + 2x + 4 = 4 * (7/6 * x)

Multiplying everything by 6 to eliminate fractions:
7x^2 + 14x + 12x + 24 = 28x

Combining like terms:
7x^2 + 26x + 24 = 28x

Moving all terms to one side:
7x^2 - 2x - 24 = 0

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. I'll use factoring in this example:

Factoring the quadratic equation:
(7x - 12)(x + 2) = 0

Setting each factor to zero and solving for 'x':
7x - 12 = 0 or x + 2 = 0
7x = 12 or x = -2
x = 12/7 or x = -2

Since dimensions cannot be negative, we can ignore the solution x = -2.

Therefore, the width of the rug (and the room) is 12/7 foot.

Now, we can plug this value into the expression for the length of the room to find its dimensions:

Length of the rug = (7/6) * x = (7/6) * (12/7) = 2 feet
Length of the room = 2 + 2 = 4 feet

So, the dimensions of the room are 4 feet by 12/7 feet.