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The shaded region in the figure above represents a [#permalink]
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29 Jun 2012, 14:14
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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. \(\frac {9}{2}\)
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Last edited by Bunuel on 06 Jan 2014, 04:24, edited 3 times in total.
Added the diagram.



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Re: The shaded region in the figure above represents a [#permalink]
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29 Jun 2012, 16:06
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Stiv wrote: Attachment:
Frame.png [ 2.69 KiB  Viewed 14826 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]
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01 Jul 2012, 02:56
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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]
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14 Jul 2012, 01:22
Yes, I understand now. Thank you for the explanation!
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Re: The shaded region in the figure above represents a [#permalink]
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28 Jul 2013, 02:04
llength of the pic and wwidth
18*15LW  Sof the frame LW  S of the pic
18/15=L/W > W= 15L/18
according to the statement 18*15LW=LW > 18*15L*15L/18=L*15L/18 > L=9\sqrt{2}
A



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Re: The shaded region in the figure above represents a [#permalink]
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11 Jan 2014, 23:36
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]
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12 Jan 2014, 06:01



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Re: The shaded region in the figure above represents a [#permalink]
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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)



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