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A rectangular rug covers half of a rectangular floor that is 9 feet

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A rectangular rug covers half of a rectangular floor that is 9 feet  [#permalink]

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New post 19 Sep 2017, 22:36
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Question Stats:

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A rectangular rug covers half of a rectangular floor that is 9 feet wide and 12 feet long. If the dimensions of the rug are in the same ratio as those of the floor, how many feet long
is the rug?

(A) 6
(B) 21/2
(C) 2√7
(D) 6√2
(E) 4√6

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Re: A rectangular rug covers half of a rectangular floor that is 9 feet  [#permalink]

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New post 20 Sep 2017, 11:17
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Bunuel wrote:
A rectangular rug covers half of a rectangular floor that is 9 feet wide and 12 feet long. If the dimensions of the rug are in the same ratio as those of the floor, how many feet long
is the rug?

(A) 6
(B) 21/2
(C) 2√7
(D) 6√2
(E) 4√6


The rug covers half of the floor i.e area of the rug = 1/2 of the rectangular floor
=12x9/2
Let the length and width of the rug be l,b respectively.
lxb=12x9/2
Since the dimensions of the rug are in the same ratio as those of the floor, \(\frac{l}{12}\) x \(\frac{b}{9}\) = \(\frac{1}{2}\) which is \(\frac{1}{\sqrt{2}}\) x \(\frac{1}{\sqrt{2}}\)
Hence, l=\(\frac{12}{\sqrt{2}}\) = \(6\sqrt{2}\)

Answer D.

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A rectangular rug covers half of a rectangular floor that is 9 feet  [#permalink]

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New post 20 Sep 2017, 14:13
4
Bunuel wrote:
A rectangular rug covers half of a rectangular floor that is 9 feet wide and 12 feet long. If the dimensions of the rug are in the same ratio as those of the floor, how many feet long
is the rug?

(A) 6
(B) 21/2
(C) 2√7
(D) 6√2
(E) 4√6

Great question. Find ratio of width to length from floor dimensions, and take 1/2 area of floor set equal to L*W to find rug length.

The dimensions of the rug are in the same ratio as the floor. Express W in terms of L (to have only variable L in the area equation). That ratio will hold for the rug.

\(\frac{W}{L}=
\frac{9}{12} = \frac{3}{4}\)

\(W = \frac{3}{4}L\)

Area of floor = 12 * 9 = 108
Area of rug is half that: 54

\(L * W\) = rug area, substitute for \(W\)
\(L *
\frac{3}{4}L = 54\)
\((\frac{3}{4})L^2 =
54\)
\(L^2 =
(54)(\frac{4}{3})\)
\(L^2 = 72\)
\(L=
\sqrt{72}=\sqrt{36*2}\)
\(L = 6\sqrt{2}\)

Answer D
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Re: A rectangular rug covers half of a rectangular floor that is 9 feet  [#permalink]

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New post 11 Oct 2018, 20:35
1
Bunuel wrote:
A rectangular rug covers half of a rectangular floor that is 9 feet wide and 12 feet long. If the dimensions of the rug are in the same ratio as those of the floor, how many feet long
is the rug?

(A) 6
(B) 21/2
(C) 2√7
(D) 6√2
(E) 4√6


Alternatively, we have learnt in similar triangles that when the sides of the triangles are in the ratio a:b, the ratio of areas will be \(a^2 : b^2\).
The same concept will apply to the rectangle too since the sides are in the same ratio. Since ratio of areas is 1:2, ratio of sides will be \(1:\sqrt{2}\).

So length of the rug will be \(12/\sqrt{2} = 6\sqrt{2}\)

Answer (D)
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Re: A rectangular rug covers half of a rectangular floor that is 9 feet   [#permalink] 11 Oct 2018, 20:35
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