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# A ﬂat rectangular picture, represented by the unshaded region in the ﬁ

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Joined: 02 Sep 2009
Posts: 56307
A ﬂat rectangular picture, represented by the unshaded region in the ﬁ  [#permalink]

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08 May 2018, 01:47
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Difficulty:

45% (medium)

Question Stats:

69% (02:43) correct 31% (02:36) wrong based on 77 sessions

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A ﬂat rectangular picture, represented by the unshaded region in the ﬁgure above, is mounted in a ﬂat rectangular frame, represented by the shaded region. The frame is 1 inch wide on all sides. For what value of x, in inches, is the area of the frame equal to the area of the picture?

A. 4
B. 5
C. 6
D. 7
E. 8

Attachment:

vGqVkd7.jpg [ 7.52 KiB | Viewed 1735 times ]

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A ﬂat rectangular picture, represented by the unshaded region in the ﬁ  [#permalink]

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08 May 2018, 02:06
Bunuel wrote:

A ﬂat rectangular picture, represented by the unshaded region in the ﬁgure above, is mounted in a ﬂat rectangular frame, represented by the shaded region. The frame is 1 inch wide on all sides. For what value of x, in inches, is the area of the frame equal to the area of the picture?

A. 4
B. 5
C. 6
D. 7
E. 8

We'll show two approaches.

The first relies on using the figure without any calculations.
This is an Alternative approach.

The top and bottom part of the frame each have width x and height 1 so their area is x.
The left and right part of the frame (without the corners) each have width 1 and height x so their area is x.
So the area of the frame is 4x.
Since the picture has height x, for it to have 4x area it must have a width of 4.
Then x = 4 + 2 = 6.

(C) is our answer.

The second is an equation driven, a Precise approach.

The area of the frame is the area of the large rectangle (frame+picture) minus that of the picture.
The area of the picture is the product of its height and width.

Since the frame has width is 1 inch on all sides, the large rectangle has area x*(x+2) = x^2 +2x
The picture has height x and width x - 2 so its area is x*(x-2) = x^2 - 2x.
Then the frame has area x^2 + 2x - (x^2 - 2x) = 4x

So, the two are equal when x^2 - 2x = 4x --> x^2 - 6x = 0 --> x(x - 6) = 0 and x = 6.

(C) is our answer.
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Re: A ﬂat rectangular picture, represented by the unshaded region in the ﬁ  [#permalink]

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28 Jul 2018, 11:28
Bunuel wrote:

A ﬂat rectangular picture, represented by the unshaded region in the ﬁgure above, is mounted in a ﬂat rectangular frame, represented by the shaded region. The frame is 1 inch wide on all sides. For what value of x, in inches, is the area of the frame equal to the area of the picture?

A. 4
B. 5
C. 6
D. 7
E. 8

Attachment:
vGqVkd7.jpg

The area of the picture is x(x-2)
The area of the frame is x(x+2) - x(x-2)
x(x+2) - x(x-2) = x(x-2)
x(x+2) - 2x(x-2) = 0
x(x+ 2 - 2x + 4) = 0 ------> either x = 0 or x = x = 6

Option C
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Re: A ﬂat rectangular picture, represented by the unshaded region in the ﬁ  [#permalink]

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03 Aug 2018, 15:08
Bunuel wrote:

A ﬂat rectangular picture, represented by the unshaded region in the ﬁgure above, is mounted in a ﬂat rectangular frame, represented by the shaded region. The frame is 1 inch wide on all sides. For what value of x, in inches, is the area of the frame equal to the area of the picture?

A. 4
B. 5
C. 6
D. 7
E. 8

Attachment:
vGqVkd7.jpg

We see that the area of the picture (without the frame) is x(x - 2) and the total area of the picture and the frame is x(x + 2). Since the area of the picture is equal to the area of the frame, the total area of the picture and the frame is thus twice the area of the picture. That is, we have:

x(x + 2) = 2x(x - 2)

x^2 + 2x = 2x^2 - 4x

x^2 - 6x = 0

x(x - 6) = 0

x = 0 or x = 6

Since x can’t be 0, x must be 6.

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Re: A ﬂat rectangular picture, represented by the unshaded region in the ﬁ  [#permalink]

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08 Aug 2018, 02:18
(x+2) * x = x^2+2x (total area)
(x-2)* x = x^2-2x (area of the flat)
x^2+2x- (x^2-2x) = x^2+2x-x^2+2x=4x (area of the frame)
Now, 4x=x^2-2x
When we start plugging numbers, only 6 satisfies the equation!
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Re: A ﬂat rectangular picture, represented by the unshaded region in the ﬁ   [#permalink] 08 Aug 2018, 02:18
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# A ﬂat rectangular picture, represented by the unshaded region in the ﬁ

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