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

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Math Expert
Joined: 02 Sep 2009
Posts: 64249
A rectangular rug covers half of a rectangular floor that is 9 feet  [#permalink]

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19 Sep 2017, 22:36
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Difficulty:

55% (hard)

Question Stats:

66% (02:33) correct 34% (02:15) wrong based on 83 sessions

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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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Posts: 3862
A rectangular rug covers half of a rectangular floor that is 9 feet  [#permalink]

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20 Sep 2017, 14:13
5
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 the ratio of width to length from floor dimensions, and then 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 (in order 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 $$W$$
$$L * \frac{3}{4}L = 54$$
$$(\frac{3}{4})L^2 = 54$$
$$L^2 = (54)(\frac{4}{3})$$
$$L^2 = 72$$
$$L= \sqrt{72}=\sqrt{36*2}$$
$$L = 6\sqrt{2}$$

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

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20 Sep 2017, 11:17
3
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=$$\frac{12}{\sqrt{2}}$$ = $$6\sqrt{2}$$

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

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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 $$12/\sqrt{2} = 6\sqrt{2}$$

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

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01 Dec 2019, 07:46
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Re: A rectangular rug covers half of a rectangular floor that is 9 feet   [#permalink] 01 Dec 2019, 07:46