A number can be considered “purple” if it halfway between two perfect

Question

A number can be considered “purple” if it halfway between two perfect squares. Is x purple?

(1) The product of x and 2 is equal to the sum of two squares.

(2) The average of x and the square of 2 is 7.

DavidTutorexamPAL · Answer

Instead of trying to explicitly solve, we'll use numbers to show us the logic.
This is an Alternative approach.

(1) Lets choose two squares: 1 and 4. Then 2x = 5 so x=2.5 Is x in the middle of two squares? Well, the only square smaller than x is 1 and the first square larger than x is 4. Since 2.5 = (1+4)/2) then x is in the middle between them. But wait, this is exactly how we calculated x! If 2x = square + square then x = (square + square)/2 meaning it is in the middle of 2 squares.
Sufficient.

(2) So (x + 4)/2 = 7 and x = 10. Is x in the middle of two squares? If it were in the middle of 1 and a square then 10 + (10-1) = 19 would be a square which it is not. If it were in the middle of 4 and a square then 10 + (10 - 4) = 16 would be a square, which it is. Then x is in the middle of 4 and 16 and the answer is YES.
Sufficient.

(D) is our answer.

arjunnath · Answer

To answer Yes/No for the second statement, is it not necessary to have the values of the square numbers? As in we can say that x is PURPLE for 16 and 4 but not for 9 and 16 or the square of any number for that matter?

ayush93 · Answer

Statement 1 simply means x is avg of 2 squares , which makes it the middle number no matter whatever values of squares you take. Yes Sufficient.
But, in Statement 2:
Some values gives NO, it is not the middle number of 2 squares (i.e for 2 and 16) 
and a Yes, it is a middle number (i.e for 4 and 16).

This statement gives no unique solution for whether it is a middle number or not.

So answer will be A.

arjunnath · Answer

ayush93

I feel the same but the OA for it according to Veritas is D. 
I wish someone could provide a proper explanation for this.

Bunuel  could you please help here.
Thanks in advance.

aa008 · Answer

Statement1 is not valid for 0.1 & 2
Statement2 is sufficient.

Answer should be B.

Posted from my mobile device

soapbolt · Answer

Option D is correct Answer

Statement1 says exactly what is asked in the question 
Statement 2 proves x is indeed in the middle of 2 perfect square

Natty97  in your Example --> 2 is not the perfect square.

Rahulss · Answer

I also had similar doubt, but the questions says....between two perfect squares... so i guess it means between any of the perfect squares will suffice.

ruchik · Answer

For option a) let x =1 than 2x = 2 it is sum of two perfect squares (1+1) - but than x is not purple.
The question does not mention two different perfect squares. 
 Bunuel

soapbolt · Answer

ruchik  
If we consider perfect square to be equal then question will going to become  is X= X then it is purple . Because half of 2 same number is that same number.

and this is what you have proved in your example. hence this makes statement sufficient.

also if we have to consider 2 different perfect square then also statement 1 is sufficient

GMATaspirant641 · Answer

Answer is D

Soln:

Stmt 1: Let X be (a+b)/2. For now, lets just assume that a & b are some random numbers.

(a+b)/2 * 2 = sum of 2 perfect squares

a + b = sum of 2 perfect squares. so, (a+b)/2 = x = purple

hence A is sufficient

Stmt 2: now, lets use (a+b)/2 = X here as well. Lets take 'square of 2' as 'b' and check if 'a' is also a perfect square.

((a+4)/2 + 4) / 2 = 7 => a = 16 = 4^2. so, a & b are perfect squares. X is purple

Please let me know if my approach is ok

Archit3110 · Answer

DavidTutorexamPAL 
I understood the part of stmnt 1 but in the case of stmn 2 we have determined the value of x as 10
now we know that since x is 10 so possible perfect squares between which 10 comes are
(1,16)( 4,16) ( 4,25) ( 9,16) .. so on 
we get yes that x is purple when two perfect squares are ( 4,16) 
and no that x is not purple when the two perfect squares are ( 9,16) (1,16) or any other set of perfect square
why is that even though we are getting both yes & no as the answer for stmnt 2 , we are still validating to be true as 'purple' when its not the case always..

Bunuel 
 GMATinsight 
 chetan2u 
please advise on solution for #2 ..

chetan2u · Answer

Hi

If any question, be this or any other, your statement will be almost always sufficient if you know the value of the variable.
Here any number is purple will always have some two numbers equidistant which will give answer as not purple. So purple is when even one set of number satisfies the condition

GMATinsight · Answer

Archit3110 
The language is that "the number x can be considered purple ..." which intends that either x can be (definitely yes) or x can NOT be (definitely NO) considered purple as per given scenarios in statements. Which means that question can not have an inconsistent answer for any unique value of x and that makes this question objectionable.

Any statement in such question will be insufficient only if the value of x is indeterminate with some possibility of purple values of x some non-purple values of x

ZZhang · Answer

How is statement 1 sufficient? It does not specify that the two squares must be different.

Case 1 
a = 1, a^2 = 1
b = 1, b^2 = 1
Sum of 2 squares = 1+1 = 2. 
2x = 2

Thus x = 1 = a = b, which is not between a and b, and is not purple.

Case 2 
If a = 1, a^2 = 1
b = 2, b^2 = 4
Sum of 2 squares = 1+4 = 5
2x = 5

Thus x = 2.5, which is halfway between 1 and 4, and is purple.

Statement 1 is insufficient. 
Statement 2 gives an exact value and is sufficient. B.

Pritishd · Answer

S1:  The product of x and 2 is equal to the sum of two squares  Sufficient ] 
Let the two numbers be  a  and  b , then  2x = a^2 + b^2  ->  x = \frac{a^2 + b^2 }{ 2} 
E.g.
(1)  a = 2  and  b = 3 , then  \frac{4 + 9 }{ 2} = \frac{13 }{ 2} = 6.5

(2)  a = 2  and  b = 4 , then  \frac{4 + 16 }{ 2} = \frac{20 }{ 2} = 10

S2:  The average of x and the square of 2 is 7  Sufficient ] 
This says  \frac{x + 2^2 }{ 2} = 7  ->  \frac{x + 4}{2} = 7  ->  x = 10 . We have seen the example in the previous statement that  10  is between perfect squares  2  and  4

Ans.  D

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A number can be considered “purple” if it halfway between two perfect

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