A rectangle and a square have the same perimeter. One side-length of the rectangle is 25% longer than the other. What is the ratio between the areas of

Let's pick a square with side lengths of 100    perimeter = 400

    area = 100x 100 = 10, 000

  then the rectangle will have side lengths     of 88.88888 and 111.11111     Perimeter = 400

     area = 88.88888... x 111.11111...= 9876.54321

ratio of rectangle to square areas   =    9876.45321/10000 = ~ 80/81

Let  one side of the rectangle = x     ....and the other side  =  1.25x

Then the  perimeter  =  2 [ 1x + 1.25x]  = 2 [2.25x] = 4.5x

So the side of the square  =  [4.5 x ] / 4  

And the area of the rectangle  =  x * 1.25x  = 1.25x^2    (1)

And the area of the square  = ( [4.5 x ] / 4)^2  = [ 20.25/16] x^2 = [81/64]x^2    (2)

So the ratio  of  (1) to (2)   =   1.25x^2 / [ (81/64)x^2 ] =  1.25 / (81/64)  = [5/4] / [ 81/64]  =

[5/4] * [ 64/81]  =     [64/4] * [ 5/81]  =  [16*5] / 81  =  80/81    as EP  found!!!