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The shaded region in the figure above represents a rectangul

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

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Question Stats:

71% (03:22) correct 29% (03:43) wrong based on 48 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: OA

Last edited by Bunuel on 28 Sep 2015, 22:10, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Re: The shaded region in the figure above represents a rectangul [#permalink]

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niheil wrote:
This is a tough one. Can anyone help me out with this:

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 length and width of the frame, what is the length of the picture, in inches?

(A) \(9\sqrt{2}\)

(B) \(\frac{3}{2}\)

(C) \(\frac{9}{\sqrt{2}}\)

(D) \(15(1-\frac{1}{\sqrt{2}})\)

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

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



Let the length of the picture be x. Since its length and width are in the same ratio as that of the frame, the width must be (5/6)x.

Area of frame = 18*15 - Area of picture.

But we know area of picture = area of frame, hence :

Area of picture = 18*15/2
x * (5/6)x = 9*15
x^2 = 27*6 = 81*2
Hence x = 9 * sqrt(2)

Answer is (a)
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New post 17 Oct 2010, 15:52
I'm so sorry for forgetting to include the diagram or mentioning that the question came with one. Thanks for the help shrouded1.
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New post 17 Oct 2010, 19:26
hi, if its not much of a problem can you please include the diagram. thanx
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New post 18 Oct 2010, 13:23
18*15-X*5/6*X=X*5/6*
X=3\sqrt{2}
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New post 14 Apr 2011, 17:28
The explanation is not proper. Can someone else explain please.

The answer is coming down to \(3/\sqrt{2}\) and not the OA \(9/\sqrt{2}\).
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New post 14 Apr 2011, 22:05
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shrouded1 wrote:
niheil wrote:
This is a tough one. Can anyone help me out with this:

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 length and width of the frame, what is the length of the picture, in inches?

(A) \(9\sqrt{2}\)

(B) \(\frac{3}{2}\)

(C) \(\frac{9}{\sqrt{2}}\)

(D) \(15(1-\frac{1}{\sqrt{2}})\)

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

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



Let the length of the picture be x. Since its length and width are in the same ratio as that of the frame, the width must be (5/6)x.

Area of frame = 18*15 - Area of picture.

But we know area of picture = area of frame, hence :

Area of picture = 18*15/2
x * (5/6)x = 9*15
x^2 = 27/6 = 9/2
Hence x = sqrt(9/2)


x^2 = 27*6=9*9*2

So, x = 9\(\sqrt{2}\)


Answer is (a)



The approach is right but there is a calculation mistake (in red above).

If you make correct calculations (in blue above), you will get right answer as A.
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New post 14 Apr 2011, 22:23
agreed, small calc mistake there ... willl correct
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niheil wrote:
This is a tough one. Can anyone help me out with this:

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 length and width of the frame, what is the length of the picture, in inches?

(A) \(9\sqrt{2}\)

(B) \(\frac{3}{2}\)

(C) \(\frac{9}{\sqrt{2}}\)

(D) \(15(1-\frac{1}{\sqrt{2}})\)

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


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


Attachment:
rectangular_picture_frame.PNG
rectangular_picture_frame.PNG [ 10.34 KiB | Viewed 10199 times ]


Given:
Length of the frame(outer side of the black portion) = 18 inches
Width of the frame(outer side of the black portion) = 15 inches

Total Area of the frame(black portion) and picture(orange portion) = length*width = 18*15
\(A_t=18*15\)

Let the length of the picture(orange portion) be "l", we need to find this.
Let the width of the picture(orange portion) be "w"
Area of the picture(orange portion) = l*w
\(A_p=l*w\)

Area of the frame(black portion) = Total Area of the frame(black) and picture(orange) - Area of the picture(orange)
\(A_f=A_t-A_p\)

"The frame encloses a rectangular picture that has the same area as the frame itself"
\(A_p=A_f\)

\(A_p=A_t-A_p\)
\(A_t=2A_p\)
\(2A_p=18*15\)

\(A_p=\frac{18*15}{2}\)

\(l*w=\frac{18*15}{2}\) -----------------------1

"length and width of the picture(orange) have the same ratio as the length and width of the frame(black)"
\(\frac{l}{w}=\frac{18}{15}\)

\(w=\frac{15}{18}*l\) --------------------2

Substituting "w" from 2 in 1:

\(l*\frac{15}{18}*l=\frac{18*15}{2}\)
\(l^2=\frac{(18)^2}{2}\)

Taking the square root on both sides:
\(l=\frac{18}{\sqrt{2}}\)

\(l=\frac{2*9}{\sqrt{2}}\)

\(l=\frac{\sqrt{2}*\sqrt{2}*9}{\sqrt{2}}\)

\(l=9\sqrt{2}\)

Ans: "A"
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New post 06 Jan 2014, 03:15
Is this one really supposed to be easy?
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New post 06 Jan 2014, 03:23
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FirstScore350 wrote:
Is this one really supposed to be easy?


It's ~650 level question.

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 rectangul   [#permalink] 06 Jan 2014, 03:23
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