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# Subject: Geom. Again From: Victor Date: Tuesday July 29,

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Eternal Intern
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Subject: Geom. Again From: Victor Date: Tuesday July 29, [#permalink]

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29 Jul 2003, 08:48
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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.

Why is it z = (x - y)/2
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Ride em cowboy

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29 Jul 2003, 10:41
Because there are two strips on the two sides of the tiled area. See the disgram. Answer is (8-5)/2 = 1.5
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29 Jul 2003, 10:49
Because there are two strips on the two sides of the tiled area. See the disgram. Answer is (8-5)/2 = 1.5
Eternal Intern
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29 Jul 2003, 12:24
Stolyar,
Good to see you. 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?

VT
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29 Jul 2003, 12:43
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

Hope it helps.
Intern
Joined: 27 Apr 2003
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29 Jul 2003, 12:45
you are considering the cg of the smaller square and the larger square a re coinciding.

as in the figure above, the difference is distributed on 2 sides hence /2
Eternal Intern
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29 Jul 2003, 13:38
you are considering the cg of the smaller square and the larger square a re coinciding.

Son, what are you talking about? "Cg" is that English?

Prakunda, you didn't quite answer it? What are the dimension(S) to get us 39 for untiled area?

Victor

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29 Jul 2003, 14:33
Hey Victor,

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
Eternal Intern
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29 Jul 2003, 16:21
I know this is really nerdy; but is it possible to find the dimensions of the untiled area to get an area of 39?

VT
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30 Jul 2003, 09:57
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
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30 Jul 2003, 21:26
Good job, this is a classic problem!

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02 Aug 2003, 05:20
Quote:
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)

Bharathi.
Eternal Intern
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02 Aug 2003, 07:14
Welcome on board ,

Why would you square the width of the pathways
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03 Aug 2003, 21:13
Curly05 wrote:
Welcome on board ,

Why would you square the width of the pathways

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.

Bharathi.
03 Aug 2003, 21:13
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