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.