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The shaded region in the figure above represents a rectangular frame

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The shaded region in the figure above represents a rectangular frame [#permalink]

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New post 18 Mar 2010, 14:34
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Question Stats:

50% (02:24) correct 50% (02:57) wrong based on 32 sessions

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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. \(\frac {9}{2}\)

Additional Info on the Problem
Source: Paper Test
Test Code 28
Section 5
# 15

OPEN DISCUSSION OF THIS QUESTION IS HERE: the-shaded-region-in-the-figure-above-represents-a-135095.html


[Reveal] Spoiler:
The was I was approaching this question was that I found the ratio of 18:15, that is 6:5
and then trying to find the length with the given answer options to see if i can get the same area as 18*15=270.

I get [9(2)^1/2 *5/6] *9(2)^1/2 which is equal to 135. I dont see how this will be equal to 270. Maybe I dont understand the question properly.


[Reveal] Spoiler:
Attachment:
geometry rectangle.JPG
geometry rectangle.JPG [ 65.63 KiB | Viewed 1735 times ]
[Reveal] Spoiler: OA
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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New post 18 Mar 2010, 15:02
Let x and y are length and breath of frame.
Let a and b are length and breadth of picture

given frame area = area of picture that means 2ab=xy ---------------(1)

also a/b = x/y = 6/5 => b = 5a/6

Now put b = 5a/6 and xy = 18*15 in equation (1), you will get a = 9(2)^1/2
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New post 18 Mar 2010, 15:11
Thank you!! can you suggest any more questions to practice off?
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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New post 19 Mar 2010, 05:38
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New post 20 Mar 2010, 19:58
How can you say 2ab=xy when the area is the same?
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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New post 21 Mar 2010, 06:36
cshenba wrote:
How can you say 2ab=xy when the area is the same?


Let x and y are length and breath of frame.
Let a and b are length and breadth of picture

now total area = area of picture + area enclosed between frame and picture.

its given area of picture = area enclosed between frame and picture.

=> xy = ab + ab => xy = 2ab
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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New post 28 Sep 2015, 11:08
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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New post 28 Sep 2015, 23:11
Expert's post
Image

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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: the-shaded-region-in-the-figure-above-represents-a-135095.html
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Re: The shaded region in the figure above represents a rectangular frame   [#permalink] 28 Sep 2015, 23:11
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