DS Geometry

A square countertop has a square tile inlay in the center, leaving a non-tiled strip of uniform width around the tile. If the ratio of the tiled area to the non-tiled area is 36 to 45, which of the following could be the width, in inches, of the strip?

I. 1
II. 2.5
III. 3

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

Odd question. (But very similar to a question in the Quant supplement. That question is also odd.) Any value is possible. Not just the three listed, but any number.

Say a large square has a side of 'x,' and small square has a side of 'y.' The area of the untiled strip would be x^2 - y^2. We know that y^2/(x^2 - y^2) = 36/45.

We'll cross-multiply y^2/(x^2 - y^2) = 36/45, to get 45y^2 = 36x^2 - 36y^2 ---> 81y^2 = 36x^2 ---> 81/36 = x^2/y^2 ---> 9/6 = x/y ----> 3/2 = x/y. All we have is a ratio. We're not told that x and y have to be integers, so we can create any difference between them we want.

To see why, try one example. We're looking for the length of the strip, which would be (x-y)/2. So we could set (x-y)/2 to any of the values above. If (x-y)/2 = 1, for instance, then x - y = 2, and x = y+ 2. Thus 3/2 = (y+2)/y --> 3y = 2y + 4 --> y = 4; plug this back in to get x = 6. So, sure, it's possible. But there's no need to do this. If we really wanted to, we could set (x-y)/2 = 2.5 and (x-y)/2 = 3, but clearly we'll get something. Any positive number is possible.

A square countertop has a square tile inlay in the center, leaving a non-tiled strip of uniform width around the tile. If the ratio of the tiled area to the non-tiled area is 36 to 45, which of the following could be the width, in inches, of the strip?

I. 1
II. 2.5
III. 3

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

Ratio of the tiled area to the non-tiled area = 36:45 = 4:5.

There is no math to be done here; just use common sense.
Why couldn't the width of the strip be any value?
We could choose a width for the strip, then shrink or expand the inlay until the ratio of inlay area to strip area = 4:5.

The correct answer is E.

prachi18oct wrote:A square countertop has a square tile inlay in the center, leaving a non-tiled strip of uniform width around the tile. If the ratio of the tiled area to the non-tiled area is 36 to 45, which of the following could be the width, in inches, of the strip?

I. 1
II. 2.5
III. 3

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

tiled : untiled = 36:45 so tiled : total countertop = 36:81 = 6^2 : 9^2 so the width can be 9-6 = 3/2 = 1.5
How to proceed now? Please explain

This can take up any value.
We just need to keep the tiled : untiled = 36:45 ratio intact.
So, the answer will be infinite number of values.

Here, we are given these three, so all three can be the answer.
Hence E is the correct option.

GMATGuruNY wrote:

A square countertop has a square tile inlay in the center, leaving a non-tiled strip of uniform width around the tile. If the ratio of the tiled area to the non-tiled area is 36 to 45, which of the following could be the width, in inches, of the strip?

I. 1
II. 2.5
III. 3

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

Ratio of the tiled area to the non-tiled area = 36:45 = 4:5.

There is no math to be done here; just use common sense.
Why couldn't the width of the strip be any value?
We could choose a width for the strip, then shrink or expand the inlay until the ratio of inlay area to strip area = 4:5.

The correct answer is E.

Earlier I believe it to any integer value.
But Solving the Question, I realized.

a^2 is are of tiled area
s^2 is are of countertop area
s^2 -a^2 is area of strip non tiled area

a^2 /(s^2-a^2) = 36/45 - Diving by 9 & cross multiplying
5 a^2 = 4 (s^2 - a^2)
9 a^2 = 4 s^2
3a = 2s

Taking different possible values of s & a

a=2 s= 3 s-a = 1
a=4 s= 6 s-a = 2
a=6 s= 9 s-a = 3
a=8 s= 12 s-a = 4

a=5 s=7.5 3*5=2*7.5 = 15 s-a =2.5

any value is possible but I still think it should have 2 & 3 as common Multiple.

solving further
3a = 2s
s= 3a/2

s-a = 3a/2 -s = 1a/2

1a/2 can be any value.

Answer is E. GMATGuruNY is right as always.