M06-08

Question

If  q^2 + pr = 10  and  r^2 + pq = 10 , and  r 
e q , what is the value of  p^2 + q^2 + r^2?

A. 10
B. 15
C. 20
D. 25
E. 30

Bunuel · Accepted Answer

Official Solution:

If  q^2 + pr = 10  and  r^2 + pq = 10 , and  r 
e 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 
e 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

Senthil7 · Answer

I think this is a  high-quality  question and I agree with explanation. Brilliant Algebraic manipulation question!

anuragpokhariyal · Answer

I think this is a  high-quality  question and I agree with explanation. real good question

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

saby1410 · Answer

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.

Bunuel · Answer

7. Algebra 
     Theory   
 Math : Algebra 101 
 Algebra: Tips and hints 
 FOIL on the GMAT: Simplifying and Expanding 
 Factoring Quadratics 
 Solving Quadratic Equations  
 Using Algebra vs. Logic on GMAT Quant Questions 
     Questions   
 Tough Algebra Set 
 Algebra Ability Quiz by e-GMAT 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

Yann1402 · Answer

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 ?

Bunuel · Answer

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.

ceceyu97 · Answer

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

Bunuel · Answer

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.

satyaprakash89 · Answer

I think this is a  high-quality  question and I agree with explanation.

medhaojha · Answer

I think this is a  high-quality  question.

vn049636 · Answer

I think this is a  high-quality  question and I agree with explanation.

phmahi1997 · Answer