Question 174134
Let the width of the patio be {{{w}}}, then the length of the patio is {{{2w}}}.  Since the garden is 1.5 meters wide, and the garden goes all the way around, the width of the rectangle formed by the outside edge of the garden must be {{{w + 1.5 + 1.5=w + 3}}}.  Similarly, the length of the outside rectangle must be {{{2w + 3}}}.


The area of the brick patio is the length times the width or {{{2w^2}}} and the area of the entire rectangle including the garden and the patio is again length times width or {{{(2w + 3)(w + 3) = 2w^2 + 9w + 9}}}.


We know the area of the garden part is 54 square meters and the area of the patio part is {{{2w^2}}}, so the sum of these two quantities must be the overall area.  Now that we have two expressions for the overall area, we can set them equal:


{{{2w^2 + 9w + 9 = 2w^2 + 54}}}


Add {{{-2w^2}}} to both sides:


{{{9w + 9 = 54}}}


Add {{{-9}}} to both sides:


{{{9w=45}}}


Finally, multiply both sides by {{{1/9}}}


{{{w = 5}}} meaning the width of the patio is 5 meters.


Since the length of the patio is {{{2w}}}, the length must be 10 meters.


Check:


Overall area:  {{{2w^2 + 9w + 9=2(5)^2+9(5)+9=50+45+9=104}}}


Area of the patio:  {{{2w^2=2(5)^2=50}}}


Overall area minus the area of the patio must be equal to the area of the garden.  {{{104-50=54}}}, and 54 square meters was the given area of the garden.  Answer checks.