square countertop

This is a trick question.

You see the values and maybe get the impression that you have to calculate something, but here's the thing.

For any value of the width of the strip, you can create a strip and an inner square with the right dimensions to create the ratio of areas 25 to 39.

Basically what you have are an inner square and and outer square. The ratio of the areas of the inner square and outer square is 25:(25 + 39) or 25:64

So what you need to create are two squares such that the ratio of their sides is √25:√64 = 5:8

Any two squares such that their sides are 5x and 8x will work.

Sides of the smaller square: 5x
Sides of the larger square: 5x + 2(the width of the strip) or 8x

Using the choices given, we can create the following sets of lengths of sides.

I. 5x + 2(1) = 8x
II. 5x + 2(3) = 8x
III. 5x + 2(4) = 8x

You can plug in any positive number for width of the strip and come up with a corresponding value for x.

So any positive width is possible, and the correct answer is E.

I think the key here is that the width need not be an integer.

We know that the ratio of Tile : Total = 25 : 64. Hence we could think of the tile as having a width of length x, and the entire countertop as having a width of (8/5)x. That gives us a strip of width (3/5)x.

(3/5)x = 1, (3/5)x = 3, and (3/5)x = 4 all give positive values of x, so all are possible.