Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

It is currently 23 Jul 2019, 23:42

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

The shaded region in the figure above represents a rectangular frame

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Manager
Manager
avatar
Joined: 16 Feb 2012
Posts: 169
Concentration: Finance, Economics
GMAT ToolKit User
The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post Updated on: 14 Sep 2017, 07:05
6
1
57
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

61% (03:04) correct 39% (03:26) wrong based on 450 sessions

HideShow timer Statistics


Attachment:
Frame.png
Frame.png [ 2.69 KiB | Viewed 35353 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. \(\frac {9}{2}\)

_________________
Kudos if you like the post!

Failing to plan is planning to fail.

Originally posted by Stiv on 29 Jun 2012, 14:14.
Last edited by Bunuel on 14 Sep 2017, 07:05, edited 4 times in total.
Added the diagram.
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56376
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 29 Jun 2012, 16:06
11
7
Stiv wrote:
Attachment:
Frame.png
Frame.png [ 2.69 KiB | Viewed 28952 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 \(xy=9*15\) --> \(x*(\frac{5}{6}x)=9*15\) --> \(x^2=2*81\) --> \(x=9\sqrt{2}\).

Answer: A.
_________________
Most Helpful Community Reply
Manager
Manager
avatar
Joined: 28 Feb 2012
Posts: 106
Concentration: Strategy, International Business
Schools: INSEAD Jan '13
GPA: 3.9
WE: Marketing (Other)
GMAT ToolKit User
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 01 Jul 2012, 02:56
4
1
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
_________________
If you found my post useful and/or interesting - you are welcome to give kudos!
General Discussion
Manager
Manager
User avatar
B
Joined: 16 Jan 2011
Posts: 96
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 28 Jul 2013, 02:04
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
Intern
Intern
avatar
Joined: 29 Aug 2013
Posts: 12
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 11 Jan 2014, 23:36
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?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56376
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 12 Jan 2014, 06:01
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.
_________________
Intern
Intern
avatar
Joined: 11 Nov 2013
Posts: 1
Location: India
Concentration: Operations, General Management
GPA: 4
WE: Information Technology (Computer Software)
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 12 Jan 2014, 07:56
1
1
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)
Intern
Intern
avatar
B
Joined: 28 Sep 2016
Posts: 18
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 25 Jul 2017, 14:38
1
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
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56376
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 25 Jul 2017, 20:59
2
1
1
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.
_________________
Target Test Prep Representative
User avatar
G
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2821
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 09 Aug 2017, 13:36
1
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
_________________

Jeffrey Miller

Head of GMAT Instruction

Jeff@TargetTestPrep.com
TTP - Target Test Prep Logo
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Manager
Manager
User avatar
S
Joined: 30 Jul 2014
Posts: 121
GPA: 3.72
Reviews Badge
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 14 Sep 2017, 07:03
Confounding language - from where to more such difficult language questions...?
_________________
A lot needs to be learned from all of you.
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56376
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 14 Sep 2017, 07:10
1
Manager
Manager
avatar
G
Joined: 05 Oct 2016
Posts: 90
Location: United States (OH)
GPA: 3.58
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 18 Sep 2017, 14:26
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 \(xy=9*15\) --> \(x*(\frac{5}{6}x)=9*15\) --> \(x^2=2*81\) --> \(x=9\sqrt{2}\).

Answer: A.


bunuel why do place x in place of y? x∗(5/6x)
_________________
Kudos APPRECIATED!
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56376
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 18 Sep 2017, 21:32
1
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.
_________________
GMATH Teacher
User avatar
P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 937
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 20 Sep 2018, 15:11
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.
Attachments

20Set18_8h.gif
20Set18_8h.gif [ 6.86 KiB | Viewed 5654 times ]


_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
CEO
CEO
User avatar
V
Joined: 12 Sep 2015
Posts: 3859
Location: Canada
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 21 Sep 2018, 08:55
2
Top Contributor
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 our 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 FROM OUR COURSE

_________________
Test confidently with gmatprepnow.com
Image
Senior Manager
Senior Manager
User avatar
P
Status: Gathering chakra
Joined: 05 Feb 2018
Posts: 372
Premium Member CAT Tests
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

Show Tags

New post 25 Feb 2019, 13:13
+1 for checking the answers method:

Since we know that the area of the small rectangle is half the big area, the small length should be a bit more than half of 18, so we can estimate the answers using √2 ≈ 1.4 (that L has to be >9).

Only A) fits this criteria.
GMAT Club Bot
Re: The shaded region in the figure above represents a rectangular frame   [#permalink] 25 Feb 2019, 13:13
Display posts from previous: Sort by

The shaded region in the figure above represents a rectangular frame

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne