It is currently 17 Nov 2017, 23:17

### 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

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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# M06 #08

Author Message
SVP
Joined: 17 Jun 2008
Posts: 1534

Kudos [?]: 280 [10], given: 0

### Show Tags

04 Nov 2008, 01:59
10
KUDOS
17
This post was
BOOKMARKED
If $$P^2 - QR = 10$$ , $$Q^2 + PR = 10$$ , $$R^2 + PQ = 10$$ , and $$R \ne Q$$ , what is the value of $$P^2 + Q^2 + R^2?$$

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

[Reveal] Spoiler: OA
C

Source: GMAT Club Tests - hardest GMAT questions

The hint in the question is R <>Q.

Hence, we need to form an equation where (R-Q) or (Q-R) becomes a factor.

Subtracting equation 3 from 2,
Q^2 - R^2 = PQ - PR
or, (Q+R)(Q-R) = P(Q-R)
and since, Q <>R,
hence, P = Q + R.

Now, equation 1 is still left out, hence, let us use this equation now.
P^2 - QR = 10
or, P^2 - (P-R)R = 10
or, P^2 + R^2 = 10 + PR.
or, P^2 + R^2 = 10 + 10 - Q^2
or, P^2 + Q^2 + R^2 = 20.

Kudos [?]: 280 [10], given: 0

SVP
Joined: 07 Nov 2007
Posts: 1791

Kudos [?]: 1088 [6], given: 5

Location: New York

### Show Tags

13 May 2009, 04:24
6
KUDOS
millhouse wrote:
If $$P^2 - QR = 10$$ , $$Q^2 + PR = 10$$ , $$R^2 + PQ = 10$$ , and $$R \ne Q$$ , what is the value of $$P^2 + Q^2 + R^2?$$

could someone please tell me how am i suppose to id the fastest way to solve this problem?

thanks,
millhouse

I will solve this with intelligent substitution.

say p=sqrt(10) q= sqrt(10) r=0

clearly satisfies the all equations.

Ans = 10+10+0 =20
_________________

Smiling wins more friends than frowning

Kudos [?]: 1088 [6], given: 5

TOEFL Forum Moderator
Joined: 16 Nov 2010
Posts: 1603

Kudos [?]: 600 [4], given: 40

Location: United States (IN)
Concentration: Strategy, Technology

### Show Tags

01 Apr 2011, 05:02
4
KUDOS
P^2 - QR = Q^2 + PR

=> -PR -QR = Q^2 - P^2

=> -R(Q + P) = (Q+P)(Q-P)

=> Q - P = -R

Now P^2 + Q^2 + R^2 - QR + PR + PQ = 30

=> P^2 + Q^2 + R^2 - R(Q - P) + PQ = 30

=> P^2 + Q^2 + R^2 + R^2 + 10 - R^2 = 30

=> P^2 + Q^2 + R^2 = 20

_________________

Formula of Life -> Achievement/Potential = k * Happiness (where k is a constant)

GMAT Club Premium Membership - big benefits and savings

Kudos [?]: 600 [4], given: 40

Director
Joined: 01 Feb 2011
Posts: 726

Kudos [?]: 146 [4], given: 42

### Show Tags

08 May 2011, 18:04
4
KUDOS
$$P^2-QR = 10$$----equation 1
$$Q^2+PR = 10$$----equation 2
$$R^2+PQ = 10$$----equation 3

Q#R

so pick equations whereever we see q , r alone and solve them. this happens to be equations 2 and 3

$$Q^2+PR = 10$$ = $$R^2+PQ = 10$$
=>$$Q^2-R^2 = P(Q-R)$$
=>P =Q+R-- equation 4

now substituting 4 in 1 we have

$$Q^2+R^2+QR = 10$$

adding P^2 on both sides we have

$$P^2+Q^2+R^2+QR = P^2+10$$

=>$$P^2+Q^2+R^2 = P^2+10-QR$$

using 1 RHS deduces to 10+10
=>$$P^2+Q^2+R^2 = 20$$

Kudos [?]: 146 [4], given: 42

Manager
Joined: 27 Apr 2010
Posts: 119

Kudos [?]: 129 [2], given: 61

### Show Tags

05 Apr 2012, 21:44
2
KUDOS
i really like this problem... uses a bunch of substitution to solve.

here's how i did it:

$$P^2 + Q^2 + R^2=?$$

Steps:
1) $$P^2 - QR = 10$$ , $$Q^2 + PR = 10$$ , $$R^2 + PQ = 10$$ , and $$R \ne Q$$
2) $$P^2-QR=Q^2+PR$$
3) $$P^2-Q^2=PR+QR$$
4) $$(P-Q)^2=PR+QR$$
5) $$(P+Q)(P-Q)=R(P+Q)$$
6) $$P-Q=R$$ or $$P=R+Q$$ or $$Q=P-R$$
7) $$R^2+(R+Q)Q=10$$
8) $$R^2+RQ+Q^2=10$$
9) $$R^2+Q^2=10-RQ$$
10) $$R^2+Q^2=10+10+P^2$$
11) $$P^2+Q^2+R^2=20$$

Kudos [?]: 129 [2], given: 61

SVP
Joined: 17 Jun 2008
Posts: 1534

Kudos [?]: 280 [1], given: 0

### Show Tags

04 Nov 2008, 09:28
1
KUDOS
HG wrote:
Scthakur

I'm new to the forum and just started preparing. I have seen your quite a few replies and I'm impressed with your crisp logic. It's short and sweet. How did you learn? What books etc?

Thx

Hi HG,

To be frank, it is the active participation in the forum that has helped me learn the logic.

Kudos [?]: 280 [1], given: 0

Manager
Joined: 14 Dec 2009
Posts: 76

Kudos [?]: 37 [1], given: 20

### Show Tags

26 Jan 2010, 04:10
1
KUDOS
The OE says to start from subtracting equation 2 from equation 3.
I started from adding all the equations in hope to reduce some variables later but came to nowhere. It was too difficult and time consuming for me to play with 3 equations until I got to the right solution.

Kudos [?]: 37 [1], given: 20

Manager
Joined: 02 Nov 2008
Posts: 58

Kudos [?]: 2 [0], given: 0

### Show Tags

04 Nov 2008, 09:05
Scthakur

I'm new to the forum and just started preparing. I have seen your quite a few replies and I'm impressed with your crisp logic. It's short and sweet. How did you learn? What books etc?

Thx

Kudos [?]: 2 [0], given: 0

Intern
Joined: 02 Feb 2009
Posts: 36

Kudos [?]: 5 [0], given: 2

### Show Tags

11 May 2009, 13:17
If $$P^2 - QR = 10$$ , $$Q^2 + PR = 10$$ , $$R^2 + PQ = 10$$ , and $$R \ne Q$$ , what is the value of $$P^2 + Q^2 + R^2?$$

could someone please tell me how am i suppose to id the fastest way to solve this problem?

thanks,
millhouse

Kudos [?]: 5 [0], given: 2

CIO
Joined: 02 Oct 2007
Posts: 1216

Kudos [?]: 986 [0], given: 334

### Show Tags

13 May 2009, 04:12
Didn't you like the Official Explanation for this question?
_________________

Welcome to GMAT Club!

Want to solve GMAT questions on the go? GMAT Club iPhone app will help.
Result correlation between real GMAT and GMAT Club Tests
Are GMAT Club Test sets ordered in any way?

Take 15 free tests with questions from GMAT Club, Knewton, Manhattan GMAT, and Veritas.

GMAT Club Premium Membership - big benefits and savings

Kudos [?]: 986 [0], given: 334

Director
Joined: 27 Jun 2008
Posts: 540

Kudos [?]: 69 [0], given: 92

WE 1: Investment Banking - 6yrs

### Show Tags

13 May 2009, 06:48
x2suresh wrote:
millhouse wrote:
If $$P^2 - QR = 10$$ , $$Q^2 + PR = 10$$ , $$R^2 + PQ = 10$$ , and $$R \ne Q$$ , what is the value of $$P^2 + Q^2 + R^2?$$

could someone please tell me how am i suppose to id the fastest way to solve this problem?

thanks,
millhouse

I will solve this with intelligent substitution.

say p=sqrt(10) q= sqrt(10) r=0

clearly satisfies the all equations.

Ans = 10+10+0 =20

Can you please elaborate? I didn't understand how you calculated.

I'm confused with this one, I dont know how to proceed...besides is this Q really GMAT equivalent?
The least I could do,

p^2 = 10+QR
q^2 = 10-PR
r^2 = 10-PQ
or
p^2-QR = q^2+PR, r^2 +PQ = 10

dont know where to go from here!

Kudos [?]: 69 [0], given: 92

Intern
Joined: 02 Feb 2009
Posts: 36

Kudos [?]: 5 [0], given: 2

### Show Tags

14 May 2009, 21:15
dzyubam wrote:
Didn't you like the Official Explanation for this question?

explanation is fine. My problem is that I wouldnt instinctively tackle the problem in that way. And the ways I did led me no where, even without a 2 minute time limit.

I think ill stick with the intelligent substitution idea.

thanks x2suresh, +1.

Kudos [?]: 5 [0], given: 2

CIO
Joined: 02 Oct 2007
Posts: 1216

Kudos [?]: 986 [0], given: 334

### Show Tags

26 Jan 2010, 04:33
The question is tough. x2suresh proposed a really nice alternative solution for it. While his approach won't necessarily work for all similar questions, it's very good for this one. Equation 2 was subtracted from equation 3 because further simplifications were possible after doing so.
Igor010 wrote:
The OE says to start from subtracting equation 2 from equation 3.
I started from adding all the equations in hope to reduce some variables later but came to nowhere. It was too difficult and time consuming for me to play with 3 equations until I got to the right solution.

_________________

Welcome to GMAT Club!

Want to solve GMAT questions on the go? GMAT Club iPhone app will help.
Result correlation between real GMAT and GMAT Club Tests
Are GMAT Club Test sets ordered in any way?

Take 15 free tests with questions from GMAT Club, Knewton, Manhattan GMAT, and Veritas.

GMAT Club Premium Membership - big benefits and savings

Kudos [?]: 986 [0], given: 334

Manager
Joined: 14 Dec 2009
Posts: 76

Kudos [?]: 37 [0], given: 20

### Show Tags

26 Jan 2010, 05:32
dzyubam wrote:
The question is tough. x2suresh proposed a really nice alternative solution for it. While his approach won't necessarily work for all similar questions, it's very good for this one. Equation 2 was subtracted from equation 3 because further simplifications were possible after doing so.
Igor010 wrote:
The OE says to start from subtracting equation 2 from equation 3.
I started from adding all the equations in hope to reduce some variables later but came to nowhere. It was too difficult and time consuming for me to play with 3 equations until I got to the right solution.

Thank you dzyubam!

Kudos [?]: 37 [0], given: 20

Intern
Joined: 12 Jul 2009
Posts: 12

Kudos [?]: 3 [0], given: 2

### Show Tags

30 Mar 2010, 09:44
Awesome Question.

Kudos [?]: 3 [0], given: 2

Intern
Joined: 23 Feb 2010
Posts: 9

Kudos [?]: 2 [0], given: 6

### Show Tags

30 Mar 2010, 13:03
I was totally stumped looking at the question. Thanks for posting this.

Kudos [?]: 2 [0], given: 6

Manager
Joined: 04 Dec 2009
Posts: 69

Kudos [?]: 9 [0], given: 4

Location: INDIA

### Show Tags

30 Mar 2010, 20:25
i am also in total blank state, spent more then 25 min and use complex logic but

GMAT like simplicity. great Question.
_________________

MBA (Mind , Body and Attitude )

Kudos [?]: 9 [0], given: 4

Director
Joined: 25 Aug 2007
Posts: 926

Kudos [?]: 1543 [0], given: 40

WE 1: 3.5 yrs IT
WE 2: 2.5 yrs Retail chain

### Show Tags

28 May 2010, 06:24
I second this approach. Even I tried to add all the three equations and tried to figure out, but no use.

I found that except eq (1), rest are having addition of variables and Q!= R. So, I took P = R. Adding all the equations and using P=R and using some numbers, I hit on the C.

x2suresh wrote:
millhouse wrote:
If $$P^2 - QR = 10$$ , $$Q^2 + PR = 10$$ , $$R^2 + PQ = 10$$ , and $$R \ne Q$$ , what is the value of $$P^2 + Q^2 + R^2?$$

could someone please tell me how am i suppose to id the fastest way to solve this problem?

thanks,
millhouse

I will solve this with intelligent substitution.

say p=sqrt(10) q= sqrt(10) r=0

clearly satisfies the all equations.

Ans = 10+10+0 =20

_________________

Tricky Quant problems: http://gmatclub.com/forum/50-tricky-questions-92834.html
Important Grammer Fundamentals: http://gmatclub.com/forum/key-fundamentals-of-grammer-our-crucial-learnings-on-sc-93659.html

Kudos [?]: 1543 [0], given: 40

Intern
Joined: 23 Apr 2010
Posts: 10

Kudos [?]: 1 [0], given: 2

### Show Tags

12 Jun 2010, 23:12
I totally support the number substitution method on this qtn. I attempted to do it through simplification, and it took almost 15 minutes to get to the answer. I'm guessing if I'm doing far too elaborate calculations, chances are there's a faster way of solving it within 2 mins, otherwise it wouldn't be on the GMAT.

Thanks to Suresh for showing a fast way!

Kudos [?]: 1 [0], given: 2

Manager
Joined: 20 Jan 2011
Posts: 63

Kudos [?]: 1 [0], given: 8

### Show Tags

01 Apr 2011, 18:52
Good question. Time consuming though.

Kudos [?]: 1 [0], given: 8

Re: M06 #08   [#permalink] 01 Apr 2011, 18:52

Go to page    1   2    Next  [ 27 posts ]

Display posts from previous: Sort by

# M06 #08

Moderator: Bunuel

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.