Request for an intuitive way of doing this problem. The answer in the book seems very cumbersome.

OG QR #175: A square countertop has a square tile inlay in the center, leaving an untiled strip of uniform width around the tile. If the ratio of the tiled area to the untiled area is 25 to 39, which of the following could be the width, in inches, of the strip?

I. 1
II. 3
III. 4

A) I only
B) II only
C) I an II only
D) I and III only
E) I, II, and III

Answer is E.

This is exactly the kind of question that the vast majority of people should skip on the actual test. It is incredibly difficult to do this question in 2 minutes, even if you completely understand ratios. I'll tell you how to do so below, but just FYI - if you're seeing something like this for the first time, most people will not be able to figure it out in the allotted time.

Notice first, it's a Roman Numeral question. These are always more time-consuming that regular PS questions because you essentially have to solve it 3 times. That's your first warning sign. Then, the complexity of the question itself also clues you in that this is probably a good "educated guess and move on" question.

Now, here's the "intuitive math" way. First sketch a picture so you can keep straight what's going on in the problem.

The given ratio of tiled:untiled is 25:39.
Write this ratio as:
tiled area:untiled area:total area
25x:39x:64x (the x represents the unknown multiplier for this ratio)
because this is a square, we can also determine that
tiled side: xxxxx :total side (note: I'm ignoring the untiled part because there isn't an "untiled side")
5SQRTx: xxx : 8SQRTx

So the ratio for the side of the inner square to the side of the larger square is 5:8 and this is algebraically represented as 5SQRTx:8SQRTx, where x represents the unknown multiplier.

The problem gave no constraints for either the unknown multiplier or for any of the values in the problem (that is, we could have a fractional side or a fractional multiplier). The multiplier, x, can therefore be anything.

The width of the untiled part will be equal to (8SQRTx - 5SQRTx)/2 (because there is a width on each side of the inner square - consult your picture to see this). If x can be anything, then so can the width... so all three possible roman numerals work. E.

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

Thanks Stacy. I understand your methodology for answering the question and your rationale for why an educated guess may be sufficient on this problem type.

much appreciated,

Jake