.
Let "x" be the uniform width of the strip.
Then the dimensions of the rug are (8-2x) and (19-2x).
Thus the area of the rug is (8-2x)*(19-2x).
According to the condition, it must be equal to 102. It gives you an equation
(8-2x)*(19-2x) = 102
= 102
= 0
= 0
The discriminant d = = 529.
= = .
There are two roots.
= /4 = 12.5 is, obviously, too big (more than the wide of the room) and, therefore, does not work as the solution to the problem.
= = 1 perfectly suits as the solution.
So, the wide of the strip is 1 ft and the dimensions of the rug are 8 - 2*1 = 6 ft wide and 19 - 2*1 = 17 ft long.
Check. (8-2*1)*(19-2*1) = 6*17 = 102.
Answer. The rug should be 6 by 17 feets.
Solved.
It is classic algebraic solution.
You can solve the problem MENTALLY, too.
The difference between the room dimensions is 19-8 = 11 ft.
Since the strip is of uniform width, the dimensions of the rug must have the same difference of 11 ft.
So, you will solve the problem, if you find the decomposition of the number of 102 into the product of two factors,
whose difference is 11.
There are, actually, only one such a possibility 102 = 6*17, which you can find in a minute (if not in 3 seconds).
At this point your mental solution is completed.
Thank you for your patience reading my solution from the beginning to the end.
----------------
You can find many other similar solved problems in the lesson
- Problems on the area and the dimensions of a rectangle surrounded by a strip
- Problems on a circular pool and a walkway around it
in this site.
Also, you have this free of charge online textbook in ALGEBRA-I in this site
- ALGEBRA-I - YOUR ONLINE TEXTBOOK.
The referred lesson is the part of this online textbook under the topic
"Dimensions and the area of rectangles and circles and their elements".
Save the link to this online textbook together with its description
Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson
to your archive and use it when it is needed.
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The tutor @MathTherapy just explained to you that the original solution by @josgarithmetic was wrong.
If you are permanent visitor of this forum, you, probably, just know, that @josgarithmetic is a person who solve incorrectly
every second or every third problem here in the forum.
If you are a newcomer to the forum, I simply inform you about it.
Therefore, when you get the solution from him, for your safety, do not hurry to accept it: there is high probability it is WRONG.
Wait for reaction/solutions from other tutors.