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# In the figure shown above, line segment QR has length 12 and rectangle

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Manager
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In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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Updated on: 30 Jan 2018, 21:03
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Question Stats:

72% (02:20) correct 28% (02:12) wrong based on 621 sessions

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In the figure shown above, line segment QR has length 12, and rectangle MPQT is a square. If the area of rectangular region MPRS is 540, what is the area of rectangular region TQRS?

(A) 144
(B) 216
(C) 324
(D) 360
(E) 396

Attachment:

Picture 1.png [ 6.97 KiB | Viewed 17669 times ]

Originally posted by oss198 on 05 Jan 2014, 16:49.
Last edited by Bunuel on 30 Jan 2018, 21:03, edited 2 times in total.
Edited the question.
##### Most Helpful Expert Reply
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Joined: 16 Oct 2010
Posts: 8195
Location: Pune, India
Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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06 Jan 2014, 22:03
9
5
oss198 wrote:
Thank you for your answer, but still : how do you find 48 quickly? you don't find the root of 12^2-4*540 so easily, do you?

ps : your signature is awesome

It's easy to solve this equation without using the formula (which is cumbersome in my opinion)
x^2 + 12x - 540 = 0

We try to split 540 into two factors such that their difference is 12
540 = 54*10 = 18*3*10
You can see immediately that the 2 factors will be 18 and 30.

x^2 + 30x - 18x -540 = 0
(x + 30)(x - 18) = 0
x = 18

Here is a post discussing how to split the middle term quickly: http://www.veritasprep.com/blog/2013/12 ... equations/
_________________

Karishma
Veritas Prep GMAT Instructor

Save up to $1,000 on GMAT prep through 8/20! Learn more here > GMAT self-study has never been more personalized or more fun. Try ORION Free! ##### Most Helpful Community Reply Manager Joined: 20 Dec 2013 Posts: 123 Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink] ### Show Tags 06 Jan 2014, 00:22 9 5 oss198 wrote: Attachment: Picture 1.png In the figure shown above, line segment QR has length 12, and rectangle MPQT is a square. If the area of rectangular region MPRS is 540, what is the area of rectangular region TQRS? (A) 144 (B) 216 (C) 324 (D) 360 (E) 396 A very good contender for back solving, lets start with option C (Always start with C) Area of rectangle.........Side QT.......Area of square.....Total area of the figure (A) 144 (B) 216 (C) 324....................324/12=27.... 27*27 =729........... 729 + 324 = 1053 (D) 360 (E) 396 The total area was supposed to be 540 however it is 1053 hence we move up (go for smaller values). Now we know that answer is either A or B so we try any of them. Area of rectangle.........Side QT.......Area of square.....Total area of the figure (A) 144 (B) 216....................216/12 =18.....18*18 =324.......... 324 + 216 =540 (BINGO!) (C) 324....................324/12=27.... 27*27 =729........... 729 + 324 = 1053 (D) 360 (E) 396 Answer is B. _________________ 76000 Subscribers, 7 million minutes of learning delivered and 5.6 million video views Perfect Scores http://perfectscores.org http://www.youtube.com/perfectscores ##### General Discussion Math Expert Joined: 02 Sep 2009 Posts: 48037 Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink] ### Show Tags 05 Jan 2014, 17:01 1 2 In the figure shown above, line segment QR has length 12, and rectangle MPQT is a square. If the area of rectangular region MPRS is 540, what is the area of rectangular region TQRS? (A) 144 (B) 216 (C) 324 (D) 360 (E) 396 The area of MPRS = the area of MPQT + the area of TQRS. 540 = x^2 + 12x --> x = 18. The area of TQRS = 12*18 = 216. Answer: B. _________________ Manager Joined: 18 Jul 2013 Posts: 69 Location: Italy GMAT 1: 600 Q42 V31 GMAT 2: 700 Q48 V38 Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink] ### Show Tags 05 Jan 2014, 17:12 Bunuel wrote: The area of MPRS = the area of MPQT + the area of TQRS. 540 = x^2 + 12x --> x = 18. The area of TQRS = 12*18 = 216. Answer: B. Hi Bunuel, thank you for the answer, can you explain your technique to find the x quickly? It takes me a lot of time.. Director Joined: 25 Apr 2012 Posts: 701 Location: India GPA: 3.21 WE: Business Development (Other) Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink] ### Show Tags 06 Jan 2014, 00:12 oss198 wrote: Bunuel wrote: The area of MPRS = the area of MPQT + the area of TQRS. 540 = x^2 + 12x --> x = 18. The area of TQRS = 12*18 = 216. Answer: B. Hi Bunuel, thank you for the answer, can you explain your technique to find the x quickly? It takes me a lot of time.. If I may, To find the roots of the equation ax^2+bx+c=0 by the formulae This equation has 2 solutions given by (-b+/-$$\sqrt{b^2-4ac}$$)/2a------> (-12+/-$$\sqrt{12^2-4*540}$$)/2 So Roots are (-12 +/-48 )/2------> -30 and 18 since length cannot be negative so X=18 and therefore the area =18*12----->216 _________________ “If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.” Manager Joined: 18 Jul 2013 Posts: 69 Location: Italy GMAT 1: 600 Q42 V31 GMAT 2: 700 Q48 V38 Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink] ### Show Tags 06 Jan 2014, 16:42 1 WoundedTiger wrote: oss198 wrote: Bunuel wrote: The area of MPRS = the area of MPQT + the area of TQRS. 540 = x^2 + 12x --> x = 18. The area of TQRS = 12*18 = 216. Answer: B. Hi Bunuel, thank you for the answer, can you explain your technique to find the x quickly? It takes me a lot of time.. If I may, To find the roots of the equation ax^2+bx+c=0 by the formulae This equation has 2 solutions given by (-b+/-$$\sqrt{b^2-4ac}$$)/2a------> (-12+/-$$\sqrt{12^2-4*540}$$)/2 So Roots are (-12 +/-48 )/2------> -30 and 18 since length cannot be negative so X=18 and therefore the area =18*12----->216 Thank you for your answer, but still : how do you find 48 quickly? you don't find the root of 12^2-4*540 so easily, do you? ps : your signature is awesome SVP Status: The Best Or Nothing Joined: 27 Dec 2012 Posts: 1835 Location: India Concentration: General Management, Technology WE: Information Technology (Computer Software) Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink] ### Show Tags 23 Feb 2014, 22:06 2 Factorize 540 & pick up the best fit 540 = 2 x 270 = 2 x 2 x 135 = 2 x 2 x 3 x 45 = 2 x 2 x 3 x 3 x 3 x 5 Rearranging: = (2 x 3 x 5) x (2 x 3 x 3) No. in above brackets add upto as 30 & 18; there multiplication is 540 & substraction is 12, so they best fit in the equation _________________ Kindly press "+1 Kudos" to appreciate Intern Joined: 25 May 2014 Posts: 22 GPA: 3.55 Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink] ### Show Tags 17 Jul 2014, 16:55 VeritasPrepKarishma wrote: oss198 wrote: Thank you for your answer, but still : how do you find 48 quickly? you don't find the root of 12^2-4*540 so easily, do you? ps : your signature is awesome It's easy to solve this equation without using the formula (which is cumbersome in my opinion) x^2 + 12x - 540 = 0 We try to split 540 into two factors such that their difference is 12 540 = 54*10 = 18*3*10 You can see immediately that the 2 factors will be 18 and 30. x^2 + 30x - 18x -540 = 0 (x + 30)(x - 18) = 0 x = 18 Here is a post discussing how to split the middle term quickly: http://www.veritasprep.com/blog/2013/12 ... equations/ Hi Karishma, obviously another awesome post! The blog post helped me a lot. Would you say though that this technique would be even faster than simply back-solving when it comes to quadratic equations? Thanks Franco Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8195 Location: Pune, India Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink] ### Show Tags 17 Jul 2014, 22:02 1 francoimps wrote: Would you say though that this technique would be even faster than simply back-solving when it comes to quadratic equations? Thanks Franco Backsolving is a useful technique but its relevance in GMAT is decreasing. The options given are such that it is harder to back solve now. For example, here you will need to divide the option by 12 (the length of QR) and then try to plug in what you get in the equation. Notice that all options are divisible by 12 (which will be true for all good options) so it might be some time before you arrive at the answer using backsolving. During the exam, use whatever comes to your mind - you wouldn't have the time to consciously decide to use one method over another. _________________ Karishma Veritas Prep GMAT Instructor Save up to$1,000 on GMAT prep through 8/20! Learn more here >

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Manager
Joined: 26 Mar 2017
Posts: 145
Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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13 May 2017, 01:25
well I found the correct equation within seconds but then I could not find the roots fast enough

Is there any shortcut/trick to find the roots ?
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Joined: 02 Sep 2009
Posts: 48037
Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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13 May 2017, 01:36
daviddaviddavid wrote:
well I found the correct equation within seconds but then I could not find the roots fast enough

Is there any shortcut/trick to find the roots ?

Check the links below:

Factoring Quadratics: http://www.purplemath.com/modules/factquad.htm
Solving Quadratic Equations: http://www.purplemath.com/modules/solvquad.htm

Theory on Algebra: http://gmatclub.com/forum/algebra-101576.html
Algebra - Tips and hints: http://gmatclub.com/forum/algebra-tips- ... 75003.html

DS Algebra Questions to practice: http://gmatclub.com/forum/search.php?se ... &tag_id=29
PS Algebra Questions to practice: http://gmatclub.com/forum/search.php?se ... &tag_id=50

Hope it helps.
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Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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13 May 2017, 01:54
Bunuel wrote:
daviddaviddavid wrote:
well I found the correct equation within seconds but then I could not find the roots fast enough

Is there any shortcut/trick to find the roots ?

Check the links below:

Factoring Quadratics: http://www.purplemath.com/modules/factquad.htm
Solving Quadratic Equations: http://www.purplemath.com/modules/solvquad.htm

Theory on Algebra: http://gmatclub.com/forum/algebra-101576.html
Algebra - Tips and hints: http://gmatclub.com/forum/algebra-tips- ... 75003.html

DS Algebra Questions to practice: http://gmatclub.com/forum/search.php?se ... &tag_id=29
PS Algebra Questions to practice: http://gmatclub.com/forum/search.php?se ... &tag_id=50

Hope it helps.

thx Bunuel

ill check it

I guess my problem is just the high number of 540 but unfortunately the GMAT often uses such high number in quadratic equations related to these kind of geometric problems
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Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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25 May 2017, 10:35
1
oss198 wrote:
Attachment:
Picture 1.png
In the figure shown above, line segment QR has length 12, and rectangle MPQT is a square. If the area of rectangular region MPRS is 540, what is the area of rectangular region TQRS?

(A) 144
(B) 216
(C) 324
(D) 360
(E) 396

First re-draw the image on the page and replace QR with 12 and PQ with x. We know that PM is also x, so the total area is (12+x)*x = 540. It turns into the quadratic 12x + x^2 = 540 or -540 + x^2 + 12x. The toughest part of the problem is figuring out which factor roots to use. The best method is to realize 540 is 9*60 and double 9 while halving 60 to get 18*30. These are separated by 12, the distance we need for the middle term of the quadratic (12x).

If x = 18 then the area of TQRS is 18*12 = 216.
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Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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05 Jul 2017, 21:45
I was scratching my head when I saw 540 but when I wrote it as 54*10 and then 9*6*5*2, a light bulb lit up.
18 and 30. Now looking at the LHS to get 12 we need a larger positive number. This means 18 is the only option that stays non negative when factored.
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Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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02 Oct 2017, 09:54
PerfectScores wrote:
oss198 wrote:
Attachment:
Picture 1.png
In the figure shown above, line segment QR has length 12, and rectangle MPQT is a square. If the area of rectangular region MPRS is 540, what is the area of rectangular region TQRS?

(A) 144
(B) 216
(C) 324
(D) 360
(E) 396

A very good contender for back solving, lets start with option C (Always start with C)

Area of rectangle.........Side QT.......Area of square.....Total area of the figure
(A) 144
(B) 216
(C) 324....................324/12=27.... 27*27 =729........... 729 + 324 = 1053
(D) 360
(E) 396

The total area was supposed to be 540 however it is 1053 hence we move up (go for smaller values). Now we know that answer is either A or B so we try any of them.

Area of rectangle.........Side QT.......Area of square.....Total area of the figure
(A) 144
(B) 216....................216/12 =18.....18*18 =324.......... 324 + 216 =540 (BINGO!)
(C) 324....................324/12=27.... 27*27 =729........... 729 + 324 = 1053
(D) 360
(E) 396

Answer is B.

why do you always start with C? Also can you please explain your strategy? Thanks.
Manager
Joined: 21 Jun 2017
Posts: 85
Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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07 Oct 2017, 09:43
oss198 wrote:
Attachment:
Picture 1.png
In the figure shown above, line segment QR has length 12, and rectangle MPQT is a square. If the area of rectangular region MPRS is 540, what is the area of rectangular region TQRS?

(A) 144
(B) 216
(C) 324
(D) 360
(E) 396

Solved it using quadratic expression.
(12 + x) = length = PR
x = length of PQ
MP = PQ because square
therefore, no need for second variable, just use x again
(12 + x) (x) = 540
x^2 + 12x - 540 = 0
(x - 18 ) (x + 30) = 0
x = 18, -30

x= 18
18 x 12 = 216
Therefore the answer is (B)
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Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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07 Oct 2017, 14:07
ayas7 wrote:
PerfectScores wrote:
oss198 wrote:
Attachment:
Picture 1.png
In the figure shown above, line segment QR has length 12, and rectangle MPQT is a square. If the area of rectangular region MPRS is 540, what is the area of rectangular region TQRS?

(A) 144
(B) 216
(C) 324
(D) 360
(E) 396

A very good contender for back solving, lets start with option C (Always start with C)

Area of rectangle.........Side QT.......Area of square.....Total area of the figure
(A) 144
(B) 216
(C) 324....................324/12=27.... 27*27 =729........... 729 + 324 = 1053
(D) 360
(E) 396

The total area was supposed to be 540 however it is 1053 hence we move up (go for smaller values). Now we know that answer is either A or B so we try any of them.

Area of rectangle.........Side QT.......Area of square.....Total area of the figure
(A) 144
(B) 216....................216/12 =18.....18*18 =324.......... 324 + 216 =540 (BINGO!)
(C) 324....................324/12=27.... 27*27 =729........... 729 + 324 = 1053
(D) 360
(E) 396

Answer is B.

why do you always start with C? Also can you please explain your strategy? Thanks.

ayas7 , a comment and links that might help.

For questions where the answer choices are listed in ascending order, and we want to backsolve, we start with C because it is the middle value. C gives a benchmark.

If C yields an answer that is too large? Toss out Answers D and E, which will be greater than C. Then we only have to pick between Answers A and B.

There are two really good explanations for starting with C, given by a GMATclub expert, HERE
and HERE

THIS POST
by Bunuel includes the link I gave above, and a lot more. Scroll down slightly to "2. Strategies and Tactics."

One more. The post immediately above, which I just found rather accidentally, is part of what looks to be a phenomenal collection, Ultimate GMAT Quantitative Megathread, by Bunuel, composed of GMAT Quant . . . everything. MEGATHREAD HERE

Hope that helps.
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that within me there lay an invincible summer.

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Joined: 02 Sep 2009
Posts: 48037
Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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08 Oct 2017, 04:13
genxer123 wrote:
There are two really good explanations for starting with C, given by a GMATclub expert, HERE
and HERE

THIS POST
by Bunuel includes the link I gave above, and a lot more. Scroll down slightly to "2. Strategies and Tactics."

One more. The post immediately above, which I just found rather accidentally, is part of what looks to be a phenomenal collection, Ultimate GMAT Quantitative Megathread, by Bunuel, composed of GMAT Quant . . . everything. MEGATHREAD HERE

Hope that helps.

Here are the links:
Backsolving on GMAT Math
How to Plug in Numbers on GMAT Math Questions
Why Approximate?
GMAT Math Strategies — Estimation, Rounding and other Shortcuts
The Power of Estimation for GMAT Quant
The 4 Math Strategies Everyone Must Master, Part 1 (1. Test Cases and 2. Choose Smart Numbers.)
The 4 Math Strategies Everyone Must Master, part 2 (3. Work Backwards and 4. Estimate)
Intelligent Guessing on GMAT
How to Avoid Tedious Calculations on the Quantitative Section of the GMAT
GMAT Tip of the Week: No Calculator? No Problem.
The Importance of Sorting Answer Choices on the GMAT
Identifying the Correct Answer on GMAT Quant Questions
How to Manage Unmanageable Numbers on the GMAT
Should You Double Check Your Answer Choices on the GMAT?
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Re: In the figure shown above, line segment QR has length 12 and rectangle  [#permalink]

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30 Jan 2018, 15:51
Some good explanations here. I got 95% of the way there: set up the quadratic and took the prime factor of 540, but it never clicked that it was 30 and 18. I'm not sure how I would solve such an equation on the actual examination.
Re: In the figure shown above, line segment QR has length 12 and rectangle &nbs [#permalink] 30 Jan 2018, 15:51

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# In the figure shown above, line segment QR has length 12 and rectangle

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