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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
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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.

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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.

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Re: A number can be considered “purple” if it halfway between two perfect &nbs [#permalink] 22 Sep 2018, 22:32
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