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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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29 Jun 2012, 13:14
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Frame.png [ 2.69 KiB | Viewed 23748 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. $$\frac {9}{2}$$
[Reveal] Spoiler: OA

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Last edited by Bunuel on 14 Sep 2017, 06:05, edited 4 times in total.

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

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29 Jun 2012, 15:06
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Stiv wrote:
Attachment:

Frame.png [ 2.69 KiB | Viewed 19343 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 $$xy=9*15$$ --> $$x*(\frac{5}{6}x)=9*15$$ --> $$x^2=2*81$$ --> $$x=9\sqrt{2}$$.

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

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01 Jul 2012, 01:56
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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 rectangular frame [#permalink]

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28 Jul 2013, 01:04
l-length of the pic and w-width

18*15-LW - Sof the frame
LW - S of the pic

18/15=L/W --> W= 15L/18

according to the statement 18*15-LW=LW --> 18*15-L*15L/18=L*15L/18 --> L=9\sqrt{2}

A

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

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11 Jan 2014, 22:36
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 rectangular frame [#permalink]

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12 Jan 2014, 05:01
b00gigi wrote:
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?

To point out why you are getting a different answer you have to show your work. Anyway, yes, the ratio is the same but we need to find the length of the picture, which can be done as explained here: the-shaded-region-in-the-figure-above-represents-a-135095.html#p1100419

Hope this helps.
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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12 Jan 2014, 06:56
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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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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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25 Jul 2017, 13:38
Hi Math Experts,

I am unable to understand the reason to divide by 2 as stated in the reply, can somebody help?

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

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

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25 Jul 2017, 19:59
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joepc wrote:
Hi Math Experts,

I am unable to understand the reason to divide by 2 as stated in the reply, can somebody help?

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 combined area of black and white is 18*15 (black + white = 18*15). The area of black = the area of white, so black + black = 18*15 --> black =18*15/2.

Hope it's clear.
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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09 Aug 2017, 12:36
Stiv wrote:
Attachment:
Frame.png
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. $$\frac {9}{2}$$

We see that the total area of the frame and the picture is 18 x 15 = 270. Since we know that the length and width of the picture have the same ratio as the length and width of the frame, let’s denote the length of the picture by 18k and the width of the picture by 15k, where k is some positive constant.

Then, the area of the picture is (18k)(15k) = 270k^2

The area of the frame can be found by subtracting the area of the picture from the total area of the frame and the picture: 270 - 270k^2

Since the area of the frame is equal to the area of the picture, we have:

270 - 270k^2 = 270k^2

270(1 - k^2) = 270k^2

1 - k^2 = k^2

2k^2 = 1

k^2 = 1/2

k = 1/√2

Since the length of the picture was represented by 18k, the length is 18(1/√2) = 18/√2 = (18/√2)*√2/√2= 18√2/2 = 9√2.

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

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14 Sep 2017, 06:03
Confounding language - from where to more such difficult language questions...?
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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14 Sep 2017, 06:10
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DAakash7 wrote:
Confounding language - from where to more such difficult language questions...?

Check:
Our Questions' Banks
Shaded Region Problems from our Special Questions Directory.

For more:
ALL YOU NEED FOR QUANT ! ! !

Hope it helps.
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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18 Sep 2017, 13:26
Bunuel wrote:
Stiv wrote:
Attachment:
Frame.png
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 $$xy=9*15$$ --> $$x*(\frac{5}{6}x)=9*15$$ --> $$x^2=2*81$$ --> $$x=9\sqrt{2}$$.

bunuel why do place x in place of y? x∗(5/6x)
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]

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18 Sep 2017, 20:32
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SandhyAvinash wrote:
Bunuel wrote:
Stiv wrote:
Attachment:
Frame.png
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 $$xy=9*15$$ --> $$x*(\frac{5}{6}x)=9*15$$ --> $$x^2=2*81$$ --> $$x=9\sqrt{2}$$.

bunuel why do place x in place of y? x∗(5/6x)

In $$xy=9*15$$, we substitute y in terms of x, which we found above (check the highlighted part) to get $$x*(\frac{5}{6}x)=9*15$$. This allows us to get an equation with only one variable x, and solve it.
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Re: The shaded region in the figure above represents a rectangular frame   [#permalink] 18 Sep 2017, 20:32
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