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# How many tiles of the size and shape shown above are needed to complet

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Math Expert
Joined: 02 Sep 2009
Posts: 52905
How many tiles of the size and shape shown above are needed to complet  [#permalink]

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03 Oct 2017, 23:34
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00:00

Difficulty:

35% (medium)

Question Stats:

71% (01:21) correct 29% (01:19) wrong based on 41 sessions

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How many tiles of the size and shape shown above are needed to completely cover a rectangular floor measuring 30 feet by 20 feet?

(A) 120
(B) 150
(C) 200
(D) 250
(E) 300

Attachment:

2017-10-04_1121_003.png [ 3.39 KiB | Viewed 925 times ]

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How many tiles of the size and shape shown above are needed to complet  [#permalink]

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04 Oct 2017, 07:57
Bunuel wrote:

How many tiles of the size and shape shown above are needed to completely cover a rectangular floor measuring 30 feet by 20 feet?

(A) 120
(B) 150
(C) 200
(D) 250
(E) 300

Attachment:
2017-10-04_1121_003.png

I think we must assume that there is a large square with side length = 2, and a missing piece that is also a square, with side length = 1.

We do not know if the figure has right angles. Two adjacent equal sides, without more, cannot render, definitively, these figures as squares.

The "more" needed to be certain that, e.g., the larger figure (with half the right and bottom sides grayed out) is a square (not exhaustive):

1) it is a rectangle (rectangle with equal adjacent sides is a square), or

2) it is a parallelogram with two equal adjacent sides and one right angle at a vertex; or

3) 3 of the four vertices are right angles

Am I missing something? I cannot see a way to prove these figures are squares.

Assume the large figure is a square (which the prompt seems to suggest). Its side length = 2, its uncut area is (2*2) = 4

Asssume also that the large square has a square piece cut out.

The square piece cut out has side length = 1, area = 1

Total area of the tile is 4 - 1 = 3

The tiles are all the same shape. If the tile shown were rotated 180° (or, flipped on the diagonal, had its left lower corner where its right corner is), it would interlock with itself to form a rectangle with length = 3 and width = 2.

Area of rectangle formed by two interlocked tiles =
(2) * (tile with area 3) = 6, OR

Rectangle formed by interlocked tiles has area of LW = (2*3) = 6

Floor area = LW = (30*20) = 600

Floor area / small rectangles' area:

600/6 = 100 tiles of area 6

But it takes TWO tiles to make the rectangular area of 6

So 100 rectangles * 2 tiles per rectangle = 200 tiles

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How many tiles of the size and shape shown above are needed to complet  [#permalink]

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04 Oct 2017, 08:28

From the figure drawn for a tile, we can deduce that 2 such tiles will cover an area of 6 ft.
Since we need to cover a rectangular floor measuring 30 ft by 20 ft(which is 600ft),
we would need 100 such sets.

Therefore, we need exactly 200 tiles to cover the rectangular floor(Option C)
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Re: How many tiles of the size and shape shown above are needed to complet  [#permalink]

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04 Oct 2017, 08:36
We have to find out for given tile shape, how many tiles we can fit in a rectangular floor of 30feet * 20 feet.

If we look at the tile image, we can see that by combining 2 tiles ( tile image + inverting the image) a rectangle of size 3*2 can be formed (Check the attached figure)
So we can combine 2 tiles, form a rectangle and than place it in rectangular floor

Rectangular tile size = 2*3 = 6
Area of rectangular floor = 30*20 = 600

No. of rectangular tiles that can be used on floor = 600/6 = 100

So number of give tiles = 2* number of rectangular tiles = 2* 100 = 200

Attachments

Tile combine.png [ 2.43 KiB | Viewed 758 times ]

Manager
Joined: 01 Feb 2017
Posts: 188
Re: How many tiles of the size and shape shown above are needed to complet  [#permalink]

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05 Oct 2017, 00:05
We have to combine two L-shaped tiles to form one composite unit of size 3ft x 2ft.

Now, we need 100 composite units to cover a floor of dimensions 30ft x 20ft : 10 units each along length and width respectively.

100 Composite units amount to 200 L-shaped tiles.

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Joined: 11 Sep 2013
Posts: 21
Schools: CBS '19
Re: How many tiles of the size and shape shown above are needed to complet  [#permalink]

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18 Aug 2018, 08:32
Area to be covered => 30x20=600

Area of the given (L) shape => {area of entire shape including the missing part} - {area of the missing part} => 4-1=3

Solution: $$\frac{600}{3}$$=200

Re: How many tiles of the size and shape shown above are needed to complet   [#permalink] 18 Aug 2018, 08:32
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