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The shaded region in the figure above represents a rectangular frame

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Bunuel
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The shaded region in the figure above represents a rectangular frame [#permalink] The shaded region in the figure above represents a rectangular frame   12 Sep 2018, 01:00
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The shaded region in the figure above represents a rectangular frame with a length of 20 inches and a width of 16 inches. The frame encloses a rectangular picture that has same area as the frame itself. If the length and width of the picture have the same ratio as the length and width of the frame, what is the length of the picture, in inches?


A. \(5\sqrt{2}\)

B. \(\frac{5}{\sqrt{2}}\)

C. \(\frac{10}{\sqrt{2}}\)

D. 10

E. \(10\sqrt{2}\)


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Re: The shaded region in the figure above represents a rectangular frame [#permalink] The shaded region in the figure above represents a rectangular frame   12 Sep 2018, 02:25
Bunuel wrote:

The shaded region in the figure above represents a rectangular frame with a length of 20 inches and a width of 16 inches. The frame encloses a rectangular picture that has same area as the frame itself. If the length and width of the picture have the same ratio as the length and width of the frame, what is the length of the picture, in inches?


A. \(5\sqrt{2}\)

B. \(\frac{5}{\sqrt{2}}\)

C. \(\frac{10}{\sqrt{2}}\)

D. 10

E. \(10\sqrt{2}\)


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Attachment:
image019.gif


Given, Area of frame =Area of picture
Area of frame =Area of bigger rectangular region - Area of picture
Or, Area of picture =1/2*20*16=160

Let l and w are the length and width of picture respectively.
Given l:w=20:16=5:4
Or, w=(4/5)*l

So area of picture=l*(4/5)*l=160
Or, \(l^2\)=200
Or, l=10\(\sqrt{2}\)

Ans. (E)
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Re: The shaded region in the figure above represents a rectangular frame [#permalink] The shaded region in the figure above represents a rectangular frame   14 Sep 2018, 13:01
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Bunuel wrote:

The shaded region in the figure above represents a rectangular frame with a length of 20 inches and a width of 16 inches. The frame encloses a rectangular picture that has same area as the frame itself. If the length and width of the picture have the same ratio as the length and width of the frame, what is the length of the picture, in inches?


A. \(5\sqrt{2}\)

B. \(\frac{5}{\sqrt{2}}\)

C. \(\frac{10}{\sqrt{2}}\)

D. 10

E. \(10\sqrt{2}\)


Show Spoiler
Attachment:
image019.gif




Lw=20*16-lw
2lw=20*16=lw=160.

w/l=4/5


substitute l*l=200,

l=10root2.


Option E.
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Re: The shaded region in the figure above represents a rectangular frame [#permalink] The shaded region in the figure above represents a rectangular frame   16 Sep 2021, 13:52
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The shaded region in the figure above represents a rectangular frame with a length of 20 inches and a width of 16 inches. The frame encloses a rectangular picture that has same area as the frame itself. If the length and width of the picture have the same ratio as the length and width of the frame, what is the length of the picture, in inches?

it's given that the length and width of the picture have the same ratio as the length and width of the frame

l/b = 20/16 = 5/4

Let's assume, l = 5k and b = 4k

Its given that Area of the picture = Area of the frame (Shaded region)

Area of the picture =\( l * b = 5k*4k = 20 k^2\)

Area of the frame (Shaded region) = 20*16 - Area of the picture

Therefore,
\(20*16 - 20 k^2 = 20 k^2\)
\(20*16 = 2 * 20 k^2\)
\(k^2 = 8\)
\(k = 2\sqrt{2}\)

length of the picture =\( 5 k = 5 * 2\sqrt{2} = 10\sqrt{2}\)

Option E is the correct answer.

Hope this helps,
Clifin J Francis,
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GMAT Club Bot
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]
16 Sep 2021, 13:52
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Bunuel
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