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M06-08

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Bunuel
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M06-08 [#permalink] New post   16 Sep 2014, 00:27
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46% (02:43) correct 54% (02:44) wrong based on 647 sessions
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If \(q^2 + pr = 10\) and \(r^2 + pq = 10\), 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
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C
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Bunuel
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Re M06-08 [#permalink] New post   16 Sep 2014, 00:27
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Official Solution:

If \(q^2 + pr = 10\) and \(r^2 + pq = 10\), 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


Given:

(1): \(q^2 + pr = 10\);

(2): \(r^2 + pq = 10\)

Subtract (2) from (1):

\(q^2 + pr - r^2 - pq= 0\)

\(q^2 - r^2 + pr - pq= 0\)

\((q - r)(q + r) - p(q - r)= 0\)

\((q-r)(q+ r - p)= 0\)

Since it is given that \(q - r \ne 0\), we can safely divide the equation by it to obtain: \(q+r - p = 0\), which can be rewritten as \(p = q+r\)

Now, sum (1) and (2):

\(q^2 + pr + r^2 + pq= 20\)

\(q^2 + r^2 + p(r + q) = 20\)

Since we found that \(p = q+ r\), then:

\(r^2 + q^2 + p(r + q) = 20\)

\(r^2 + q^2 + p^2= 20\)


Answer: C
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Senthil7
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Re: M06-08 [#permalink] New post   05 Aug 2016, 03:48
I think this is a high-quality question and I agree with explanation. Brilliant Algebraic manipulation question!
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anuragpokhariyal
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Re: M06-08 [#permalink] New post   31 Dec 2016, 00:07
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I think this is a high-quality question and I agree with explanation. real good question
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Re: M06-08 [#permalink] New post   09 May 2023, 03:04
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Re: M06-08 [#permalink] New post   21 May 2023, 01:35
Bunuel wrote:
I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Bunuel

i wasn't able to figure out this question under time limit.is there any type of standard process to deal this type of question or has to go with manipulation.if not any tips to solve this type of problems.
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Re: M06-08 [#permalink] New post   21 May 2023, 03:16
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saby1410 wrote:
Bunuel wrote:
I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Bunuel

i wasn't able to figure out this question under time limit.is there any type of standard process to deal this type of question or has to go with manipulation.if not any tips to solve this type of problems.


7. Algebra



For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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Re: M06-08 [#permalink] New post   07 Jul 2023, 21:29
But how would you even think of doing these manipulative operations in a time constrained environment? Unless of course you come across such questions prior to the exam.

Beautiful question nevertheless!
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Re M06-08 [#permalink] New post   19 Jul 2023, 04:14
I think this is a high-quality question and I agree with explanation. Hello i would like to know how can we say that Since it is given that q−r≠0 ? I don't understand this passage please enlighten my thoughts on it ?
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Re: M06-08 [#permalink] New post   19 Jul 2023, 04:41
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Yann1402 wrote:
I think this is a high-quality question and I agree with explanation. Hello i would like to know how can we say that Since it is given that q−r≠0 ? I don't understand this passage please enlighten my thoughts on it ?


The stem says that q ≠ r, which implies that q − r ≠ 0.

Next, we have (q - r)(q + r - p) = 0, which implies that either q - r = 0 or q + r - p = 0. However, since we know that q − r ≠ 0, then it must be that q + r - p = 0.

Hope it's clear.
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Re: M06-08 [#permalink] New post   14 Apr 2024, 00:24
I'm having trouble understanding how you went from
this (q−r)(q+r)−p(q−r)=0
to this (q−r)(q+r−p)=0
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Re: M06-08 [#permalink] New post   14 Apr 2024, 02:02
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ceceyu97 wrote:
I'm having trouble understanding how you went from
this (q−r)(q+r)−p(q−r)=0
to this (q−r)(q+r−p)=0

­
We simply factor out (q - r) from both terms in (q - r)(q + r) - p(q - r) = 0 to get (q - r)(q + r - p) = 0:

(q - r)(q + r) - p(q - r) = 0 

(q - r)(q + r - p) = 0

Hope it's clear.
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Re: M06-08 [#permalink]
14 Apr 2024, 02:02
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