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The shaded region in the figure above represents a

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The shaded region in the figure above represents a [#permalink] New post 29 Jun 2012, 13:14
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The shaded region in the figure above represents a rectangular frame with length 18 inches and width 15 inches. The frame encloses a rectangular picture that has the same area as the frame itself. If the length and width of the picture have the same ratio as the lenght and width of the frame, what is the length of the picture, in inches?

A. 9\sqrt2

B. \frac {3}{2}

C. \frac {9}{\sqrt2}

D. 15 ( 1 - \frac {1}{\sqrt2})

E. \frac {9}{2}
[Reveal] Spoiler: OA

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Last edited by Bunuel on 06 Jan 2014, 03:24, edited 3 times in total.
Added the diagram.
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Re: The shaded region in the figure above represents a [#permalink] New post 29 Jun 2012, 15:06
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Stiv wrote:
Attachment:
Frame.png
Frame.png [ 2.69 KiB | Viewed 4894 times ]
The shaded region in the figure above represents a rectangular frame with length 18 inches and width 15 inches. The frame encloses a rectangular picture that has the same area as the frame itself. If the length and width of the picture have the same ratio as the lenght and width of the frame, what is the length of the picture, in inches?

A. 9\sqrt2
B. \frac {3}{2}
C. \frac {9}{\sqrt2}
D. 15 ( 1 - \frac {1}{\sqrt2}
E. \frac {9}{2}


Say the length and the width of the picture are x and y respectively. Since they have the same ratio as the lenght and width of the frame, then \frac{x}{y}=\frac{18}{15} --> y=\frac{5}{6}x.

Next, since the frame encloses a rectangular picture that has the same area as the frame itself and the whole area is 18*15, then the areas of the frame (shaded region) and the picture (inner region) are \frac{18*15}{2}=9*15 each.

The area of the picture is xy=9*15 --> x*(\frac{5}{6}x)=9*15 --> x^2=2*81 --> x=9\sqrt{2}.

Answer: A.

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Re: The shaded region in the figure above represents a [#permalink] New post 01 Jul 2012, 01:56
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The total area is 15*18=270, the area of the picture is half of the whole area = 135. the ration of the width and length of the picture is the same as the frames 15/18 or 5/6. We need to find the length of the picture 5x*6x=135, 30x^2=135, x^2=135/30, x=3/sqrt2, so the length = 6*3/sqrt2=9sqrt2

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Re: The shaded region in the figure above represents a [#permalink] New post 14 Jul 2012, 00:22
Yes, I understand now. Thank you for the explanation!

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Re: The shaded region in the figure above represents a [#permalink] New post 28 Jul 2013, 01:04
l-length of the pic and w-width

18*15-LW - Sof the frame
LW - S of the pic

18/15=L/W --> W= 15L/18

according to the statement 18*15-LW=LW --> 18*15-L*15L/18=L*15L/18 --> L=9\sqrt{2}


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Re: The shaded region in the figure above represents a [#permalink] New post 11 Jan 2014, 22:36
Why do I get a different answer if I just left at l/w=18/15 than simplifying to l/w=6/5 ?? Aren't the ratios the same?
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Re: The shaded region in the figure above represents a [#permalink] New post 12 Jan 2014, 05:01
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b00gigi wrote:
Why do I get a different answer if I just left at l/w=18/15 than simplifying to l/w=6/5 ?? Aren't the ratios the same?


To point out why you are getting a different answer you have to show your work. Anyway, yes, the ratio is the same but we need to find the length of the picture, which can be done as explained here: the-shaded-region-in-the-figure-above-represents-a-135095.html#p1100419

Hope this helps.

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Re: The shaded region in the figure above represents a [#permalink] New post 12 Jan 2014, 06:56
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Total area of the given figure= 18*15 = 270
Area of frame = Area of the picture => We need to divide the total area into two parts, 270/2 = 135. The frame and picture have 135 inch^2 area each.
l(pic) l(frame)
----- = ---------- = 6/5 ==> Area of picture = 135= 6k * 5k ==> 30k^2=135 ==> k =3/sqrt(2). So, l(pic)= 6* 3/sqrt(2) = 9*sqrt(2)
w(pic) w(frame)
Re: The shaded region in the figure above represents a   [#permalink] 12 Jan 2014, 06:56
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