Question 465766
I'm going to work with the rug first, finding the dimensions and the area.  Based on that, I'll be able to solve for x (uniform width).


Rug's dimensions: If you draw a rug in the middle of a floor, you'll have bare floor on all sides.  That means that if you consider the floor's width of 8 m, there is bare floor on EACH side of the rug--that's 2x, not just x.  The width of the rug would be the total width minus 2x: 8-2x.  Similarly, the length is 12-2x.


Rug's area: The total floor area is {{{8*12=96m^2}}}.  The rug's area is 2/3 of that, or {{{(2/3)*96=64m^2}}}.


Length * Width = Area: {{{(12-2x)(8-2x)=64}}}.  Simplify the left side (you may know this as FOIL) to get {{{96 - 24x - 16x + 4x^2 = 64}}}.  Simplify more and put it in a prettier order: {{{4x^2 - 40x + 96 = 64}}}.  We tend to like quadratics to equal 0, so subtract 64 on both sides: {{{4x^2 - 40x + 32 = 0}}}.  Before moving on, I'd like to divide through by 4 (it's the GCF, and as long as you divide EVERYTHING by 4, it's totally legal): {{{x^2 - 10x + 8 = 0}}}.


Unfortunately, this does not factor.  Fortunately, we have the quadratic formula!  {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}  Here, a = 1, b = -10, and c = 8.  {{{x = (10 +- sqrt( (-10)^2-4*1*8))/(2*1) }}} 
-->  {{{x = (10 +- sqrt(100-32))/2}}} 
-->  {{{x = (10 +- sqrt(68))/2}}}
I don't like rounding, but at this point I'm going to round and say {{{sqrt(68)=8.246}}}
-->  {{{x = (10 +- 8.246)/2}}}
Split that into two separate equations to finish simplifying:
{{{x = (10 + 8.246)/2 = 18.246/2 = 9.123}}} --> x = 9.123
{{{x = (10 - 8.246)/2 = 1.754/2 = 0.877}}} --> x = 0.877


The first answer, x = 9.123, makes no sense, since that means the bare floor would be longer than the width of the floor (8 m)...that answer is wrong.


Hopefully the other answer, x = 0.877, is right.  To check, I'm going to find the dimensions of the rug: {{{8 - 2*0.877 = 6.246}}} and {{{12 - 2*0.877 = 10.246}}}.  Finally, {{{6.246*10.246=63.9965}}}, which (considering that I rounded) is pretty close to 64 :)