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Difficulty: 655-705 Level,    Geometry,                      
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]
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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}


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

Answer: A
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]
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 \(xy=9*15\) --> \(x*(\frac{5}{6}x)=9*15\) --> \(x^2=2*81\) --> \(x=9\sqrt{2}\).

Answer: A.


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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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 \(xy=9*15\) --> \(x*(\frac{5}{6}x)=9*15\) --> \(x^2=2*81\) --> \(x=9\sqrt{2}\).

Answer: A.


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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Stiv wrote:
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. \(\frac {9}{2}\)


From the question stem ("...the length and width of the picture have the same ratio as the length and width of the frame") we know:

The frame+picture ("big" rectangle) and the picture ("small" rectangle) are two SIMILAR rectangles. (*)

(*) From above we have proportionality on the corresponding sides. The necessary additional condition - equality in the corresponding internal angles - is guaranteed: they are all 90 degrees!

Again from the question stem we know what the examiner defines as "length" and "width" (by the dimensions associated to these words), so that our FOCUS is:

\(? = x\,\,\,\,\left[ {{\text{inches}}} \right]\,\,\,\,\,\,\left( {{\text{See}}\,\,{\text{figure}}\,\,{\text{below}}} \right)\)

From "The frame encloses a rectangular picture that has the same area as the frame itself." we know that the "big" (rectangle) has TWICE the area of the "small" (rectangle).

To avoid using the second dimension of the picture, as it was done in previous (correct) solutions, let´s remember an important geometric property:

In any two similar polygons, the ratio of their areas is equal to the square of the ratio of similarity of the polygons!

Therefore:

\(2 = \frac{{{S_{\,{\text{big}}}}}}{{{S_{\,{\text{small}}}}}} = {\left( {\frac{{18}}{x}} \right)^2}\,\,\,\,\mathop \Rightarrow \limits^{x\,\, > \,\,0} \,\,\,\,\,\sqrt 2 = \frac{{18}}{x}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x\sqrt 2 = 18\)

\(x\sqrt 2 = 18\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x\sqrt 2 \cdot \sqrt 2 = 18\sqrt 2 \,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = x = 9\sqrt 2\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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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. \(\frac {9}{2}\)

IMPORTANT: the diagrams in GMAT problem solving questions are DRAWN TO SCALE unless stated otherwise.
So, we can use this fact to solve the question by simply "eyeballing" the diagram.

See my video below on this topic as well as other assumptions we can make about diagrams on the GMAT

If you had to ESTIMATE the length of the picture, what would you say it is?
12? 13? 14? 15?

As long as you're in this range, you should be able to solve this one.

ASIDE: On test day, you should have memorized the following approximations:
√2 ≈ 1.4
√3 ≈ 1.7
√5 ≈ 2.2

Now let's check the answer choices....

A. 9√2 ≈ (9)(1.4) ≈ 13. This is within our estimated range. KEEP

B. 3/2 = 1.5. This is WAYYYY outside our estimated range. ELIMINATE

C. 9/√2 ≈ 9/1.4 ≈ 6. This is WAYYYY outside our estimated range. ELIMINATE

D. 15(1 - 1/√2) ≈ 15(1 - 0.7) ≈ (15)(0.3) ≈ 4.5. This is WAYYYY outside our estimated range. ELIMINATE

E. 9/2 = 4.5. This is WAYYYY outside our estimated range. ELIMINATE

Answer: A

RELATED VIDEO

Originally posted by BrentGMATPrepNow on 21 Sep 2018, 08:55.
Last edited by BrentGMATPrepNow on 17 May 2021, 07:31, edited 2 times in total.
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The shaded region in the figure above represents a rectangular frame [#permalink]
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Say length of the picture is x and width of the picture is y.

Info 1) length = 18, width = 15
Info 2) (length * width) — (xy) = (xy)
Info 3) 18:15 = x:y

So we can calculate this.


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832ADDED-EC87-430F-87DA-4A2EF18B86AC.jpeg
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]
Hi guys, I have a concerning question about this. The questions asks for the length, which corresponds to y. I therefore did x = 6/5y and then used y for the rest of the problem. I did not get any of the answers because of this. I understand how x can get A, but x corresponds to the width, right? Y corresponds to the length.

Bunuel

Is there an error in my understanding (likely) or in the question stem?

My work:

x = 6/5y
9*15 = xy
9*15 = (6/5y)y
from here I go awry and get the wrong answer.

Why are you guys using x instead when that doesn't relate to the length?
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]
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mzmoncada wrote:
Hi guys, I have a concerning question about this. The questions asks for the length, which corresponds to y. I therefore did x = 6/5y and then used y for the rest of the problem. I did not get any of the answers because of this. I understand how x can get A, but x corresponds to the width, right? Y corresponds to the length.

Bunuel

Is there an error in my understanding (likely) or in the question stem?

My work:

x = 6/5y
9*15 = xy
9*15 = (6/5y)y
from here I go awry and get the wrong answer.

Why are you guys using x instead when that doesn't relate to the length?


You can denote by x and y anything you want. It's not necessary that x is width and y is length. If you denote y to be length and x to be width then solution is as follows:

Say the length and the width of the picture are \(y\) and \(x\) respectively. Since they have the same ratio as the length and width of the frame, then \(\frac{y}{x}=\frac{18}{15}\) --> \(x=\frac{5}{6}y\).

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 \(xy=9*15\) --> \(y*(\frac{5}{6}y)=9*15\) --> \(y^2=2*81\) --> \(y=9\sqrt{2}\).

Answer: A.
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Re: The shaded region in the figure above represents a rectangular frame [#permalink]
Simple approximation question

Since they have the same ratio as the length and width of the frame, the length and breadth of the pic can only be 6 and 5 or 12 and 10

Only option A is closest to 12
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Stiv wrote:

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

PS35461.01

Attachment:
Frame.png


WHY are we doing math on this question?!?! :thumbdown: :thumbdown:

Just LOOK at the figure. The length of the white rectangle surely can't be less than 9. B, C, D, and E are ridiculous!!

Answer choice A.


Note that this is an Official geometry question rated on this forum as 700-level. ZERO math skills required and it shouldn't take longer than 20 seconds. LOOK at geometry questions before you go head-first into mathy mcmath math land.

Edited to add: I just noticed that the average time spent is over 3 minutes. Yikes! But what makes things really bad is that 34% of people are missing it, with C and D each accounting for 11%. If you spend 3 minutes doing a bunch of work to arrive at an answer choice, you should at least take the extra second at the end to make sure that it makes sense. Look back at the figure. Is the length of that white rectangle really 6.4 (answer choice C) or 4.3 (answer choice D)? Not even close. If you're going to spend 3 minutes (even when you don't have to ;) ), you might as well make sure you're not completely wasting those three minutes!


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

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. \(9\sqrt2\)
B. \(\frac {3}{2}\)
C. \(\frac {9}{\sqrt2}\)
D. \(15 ( 1 - \frac {1}{\sqrt2})\)
E. \(\frac {9}{2}\)
PS35461.01
Attachment:
Frame.png


Since the length and width of the picture have the same ratio as the length and width of the frame, i.e. in the ratio 18 : 15, let us assume the the length and width of the picture = 18k and 15k respectively
Area of the total rectangle = 18 * 15 = 270
Area of the picture = 18k * 15k = 270 * k^2

Since the area of the shaded part is equal to the area of the picture, the area of the picture is equal to half the area of the whole rectangle
=> 270 * k^2 = 270/2
=> k^2 = 1/2
=> k = 1/√2
=> Length = 18k = 18 * 1/√2 = 9√2

Answer A
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