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In the diagram above, the fourteen rectangular tiles are all  [#permalink]

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In the diagram above, the fourteen rectangular tiles are all identical. What percent of the area of rectangle ABCD is covered by the tiles?

(1) ABCD is a square.

(2) EFGH is a square.

Originally posted by nave on 10 Mar 2013, 15:55.
Last edited by Bunuel on 04 Feb 2014, 03:22, edited 1 time in total.
Edited the question.
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Re: In the diagram above, the fourteen rectangular tiles are all  [#permalink]

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nave81 wrote:
In the diagram above, the fourteen rectangular tiles are all identical. What percent of the area of rectangle ABCD is covered by the tiles?

(1) ABCD is a square.

(2) EFGH is a square.

Figure attached
Suppose the dimention of the tiles are l*b

STAT1
Let side of Square ABCD = a
then if you consider side AB then
AB = a = 3l + 2b

if you consider side AD then

so, 4l = 3l + 2b
or, l =2b

Also, we know that a = 4l
so, l = a/4
b = l/2 = a/8
So, we can find the area of the tile in terms of "a"
Area of all the tiles = (constant)* a^2
So, we can find the percentage of area occupied by tile = ((constant)* a^2 / a^2 ) * 100
So, SUFFICIENT

STAT2
Let side of EFGH = c
then if you consider EF then
EF = c = 3l

if you consider EH then
EH = c = 2l + l-b + l-b = 4l-2b

=> 3l = 4l-2b
l = 2b

And we have c = 3l
=> l = c/3
b = l/2 = c/6

AB = 4l = 4c/3

So, we can find area of ABCD in terms of C And we can find area of the tiles in terms of c
So, we can find the percentage of area occupied by tiles
So, SUFFICIENT

Hope it helps!
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Re: In the diagram above, the fourteen rectangular tiles are all  [#permalink]

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BUMP! Any quicker ways to solve this?
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manimgoindowndown wrote:
BUMP! Any quicker ways to solve this?

Statement 1 :
AB = BC as the figure is square
Hence 3l+2b = 4l ==> l=2b and 4l = S (S = side of the square)

% = (14*l*b) *100/S^2 ==> as you know all the relationships, you can make this one variable equation and get the answer.

Statement 2 :
EF = FG as the figure is square
Hence 3l = 2l + 2b ==> l=2b and 4l = S (S = side of the square)

% = (14*l*b) *100/S^2 ==> as you know all the relationships, you can make this one variable equation and get the answer.

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Re: In the diagram above, the fourteen rectangular tiles are all  [#permalink]

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(1) SUFFICIENT: Let the length (longer dimension) of each rectangular tile be called L, and the width (shorter dimension) of each tile W. Then each horizontal side of square ABCD has total length 2W + 3L, and each vertical side has total length 4L.

Because ABCD is a square, these total lengths must be equal: 2W + 3L = 4L, which reduces to L = 2W. Therefore, each side of square ABCD is equal to 4L = 8W, and the total area of square ABCD is (8W)(8W) = 64W2.

The total area of the tiles is 14(L × W) = 14(2W × W) = 28W2. The desired fraction is thus (28W2)/(64W2) = 28/64. There is no need to reduce this fraction; the statement is sufficient.

(2) SUFFICIENT: Let the length (longer dimension) of each rectangular tile be called L, and the width (shorter dimension) of each tile W. Then each horizontal side of square ABCD has total length 3L, and each vertical side has total length 4L – 2W.

Because EFGH is a square, these total lengths must be equal: 3L = 4L – 2W, which reduces to L = 2W. Therefore, each side of square ABCD is equal to 3L = 6W.

In turn, ABCD must also be a square, since each of its sides is 2W longer than the corresponding side of EFGH (i.e., longer by W on each side). Therefore, each side of ABCD is equal to 6W + 2W = 8W, and the total area of square ABCD is (8W)(8W) = 64W2.

The total area of the tiles is 14(L × W) = 14(2W × W) = 28W2. The desired fraction is thus (28W2)/(64W2) = 28/64. There is no need to reduce this fraction; the statement is sufficient.

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Re: In the diagram above, the fourteen rectangular tiles are all  [#permalink]

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Hey Bunuel,
thanks..
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Re: In the diagram above, the fourteen rectangular tiles are all  [#permalink]

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Okay, think of it like a real life problem. I have actually sort of seen this in floor tile designs:

n the diagram above, the fourteen rectangular tiles are all identical. What percent of the area of rectangle ABCD is covered by the tiles?

(1) ABCD is a square. - obvious, inner area is also a square due to identical difference between outer and inner square. As each side has 4 division, we get x/4; the smaller part of tile being x/8 (half of x/4). Suff

(2) EFGH is a square. - so outer part has to be a square. Again we reach the x/4 and x/8 combination

Sufficient.

Ans D

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Re: In the diagram above, the fourteen rectangular tiles are all  [#permalink]

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Top Contributor
nave wrote:
Attachment:
1.jpg
In the diagram above, the fourteen rectangular tiles are all identical. What percent of the area of rectangle ABCD is covered by the tiles?

(1) ABCD is a square.

(2) EFGH is a square.

Target question: What percent of the area of rectangle ABCD is covered by the tiles?

Statement 1: ABCD is a square

IMPORTANT: Once we know that ABCD is a square, we also know that EFGH is a square. Notice that if you take square ABCD and "slice" off the same amount (i.e., the width of each rectangle) from the four sides, we get another square (EFGH).

Let L = length of one rectangle.

Side AD has length 4L, which means all four sides of square ABCD have length 4L.
So, the area of ABCD = (4L)(4L) = 16L²
Side EF has length 3L, which means all four sides of square EFGH have length 3L.
So, the area of EFGH = (3L)(3L) = 9L²
From this, we can conclude that the total area of the rectangles = 16L² - 9L² = 7L²
So, the fraction of square ABCD taken up by tiles = (7L²)/(16L²) = 7/16
Since we could convert 7/16 to a percent, we could determine the percent of the area of rectangle ABCD is covered by the tiles.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: EFGH is a square
Using the same logic that we used above, we know that ABCD must also be a square.
From here, if we follow the same steps as above, we can answer the target question with certainty.
So statement 2 is SUFFICIENT

Cheers,
Brent
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Re: In the diagram above, the fourteen rectangular tiles are all  [#permalink]

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In the diagram above, the fourteen rectangular tiles are all identical. What percent of the area of rectangle ABCD is covered by the tiles?

(1) ABCD is a square.

(2) EFGH is a square.

Hi Bunuel, I did this using a different approach. Please correct me if I'm wrong.

St 1. The figure ABCD is a square. Hence, with the same tile measurement, the square ABCD can be divided into 32 figures.
So the percentage should be (14/32)*100

St 2. As EFGH is a square, we can derive that ABCD is also a square. Using the same approach as st 1, we can find the percentage.

Hence, D it is.
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In the diagram above, the fourteen rectangular tiles are all  [#permalink]

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I tried to solve this problem intuitively. Since the question asked the percentage of area occupied by the tiles, the actual dimensions of the ABCD does not matter. What matters is the relative dimension of the tile area EFGH with respect to the ABCD, i.e., Is it possible to get different dimensions of EFGH, within the constraints of ABCD being square and 14 tiles of same measurement can be fit into it. If you carefully see, had ABCD been a rectangle, it would have been possible to get different measurements for EFGH. Since ABCD is square, there is no way to increase or reduce the dimensions of EFGH without changing the number of tiles that can be fit into it.

Hence ABCD being square is sufficient.
Hence EFGH is also sufficient. In the diagram above, the fourteen rectangular tiles are all   [#permalink] 22 Sep 2018, 07:54
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