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

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A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 09 Apr 2018, 01:57
1
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A
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D
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Question Stats:

44% (02:16) correct 56% (02:31) wrong based on 174 sessions

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Re: A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 09 Apr 2018, 03:54
Bunuel wrote:
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.


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.
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Re: A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 22 Sep 2018, 06:43
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?
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Re: A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 22 Sep 2018, 11:41
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DavidTutorexamPAL wrote:
Bunuel wrote:
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.


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.


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.
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Re: A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 22 Sep 2018, 21:43
ayush93

Quote:
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.


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.
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Re: A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 22 Sep 2018, 22:32
Statement1 is not valid for 0.1 & 2
Statement2 is sufficient.

Answer should be B.

Posted from my mobile device
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Re: A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 04 Dec 2018, 05:21
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.
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Re: A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 07 Dec 2018, 01:13
Natty97 wrote:
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?




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.
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Re: A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 09 Dec 2018, 01:23
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.
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Re: A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 09 Dec 2018, 02:07
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.
Quote:
A number can be considered “purple” if it halfway between two perfect squares.


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
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Re: A number can be considered “purple” if it halfway between two perfect  [#permalink]

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New post 24 Dec 2018, 05:58
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
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Re: A number can be considered “purple” if it halfway between two perfect   [#permalink] 24 Dec 2018, 05:58
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