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

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

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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

[Reveal] Spoiler:
Attachment:
Picture 1.png
Picture 1.png [ 6.97 KiB | Viewed 15656 times ]
[Reveal] Spoiler: OA

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

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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

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

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

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New post 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
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Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink]

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

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

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

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

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New post 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
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Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink]

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

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New post 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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Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink]

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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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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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

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

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New post 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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New post 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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Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink]

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New post 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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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
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Re: In the figure shown above, line segment QR has length 12 and rectangle [#permalink]

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New post 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   [#permalink] 30 Jan 2018, 15:51

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