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Subject: Geom. Again From: Victor Date: Tuesday July 29, [#permalink]
29 Jul 2003, 07:48

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Subject: Geom. Again
From: Victor
Date: Tuesday July 29, 2003 at 08:43:10

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) 1.5, 3, 4.5

But, more importantly I have some questions about squares.
Let's call x- the length of the side of the countertop
y- the length of the side of the tiled area
Shouldn't the length of the untiled strip just be z= x - y.

Prakunda, Is there a difference between the length of an untiled strip and the length of the untiled area?
So what are the dimensions to get us 39?

See, because the countertop and the tiled inlay both are square, the untiled strip area is arounf the tiled area - right?

Now area of the inlay is 25 which implies the sides of the tiled area is 5.
Now the area of the untiled area only is 39. So area of the countertop as a whole is 25 + 39 = 64.

That implies the sides of the countertop = 8
So, the difference between the length of the countertop and the tiled area is 8-5 = 3

Because, the untiled area is on both the sides of the tile, and the tile is centered, we are dividing the difference (i.e., 3) by 2 and getting 1.5

You are right. The answer is all 3 (1.5,3,4.5). Here is why:

"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.? "

Ratio of tiled area / untiled area = 25/39

Because the tiled area is a square, we have to assume that the side is 5x.
Hence the area is 25x^2.

Now the untiled area = 39x^2. [Their ratio is still 25:39]

So the area of the countertop = 25x^2 + 39x^2 = 64x^2
=> Sides of the countertop = 8x

Now, looking at any direction the inner square (tiled) is in the center of the outer square (countertop). So, the untiled space is the same on the 2 sides of the smaller square.

This length will be = (8x - 5x)/2 = 1.5x

If the multiplying factor x has a value 1, 1.5x will be 1.5
If x = 2, 1.5x will be 3
if, x=3, 1.5x will be 4.5

See the attched .doc which show this in a tabular format. Thanks

Victor,
Only when the countertop has sides =8 and the tiled area has sides=5, the untiled area will be exactly 39. See the attached file for those dimentions. Thanks

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) 1.5, 3, 4.5

Let a be the side of the tiled square and b be the width of the pathway.

The ratios are equated as a^2 : 4(b^2 + ab) = 25 : 39

When we take a as 5, we get b as 1.5. Taking a as 10, 15 we get b to be 3 and 4.5. So, the width could be any of 1.5, 3 and 4.5 !!! (depends on the value of a)

The area of the inner square is a^2
The length of the outer square is (a+2b) and its area is (a+2b)^2. The area of the untiled portion is hence 4b^2 + 4ab.