If x and y are Positive, Is x < y? GMAT Data Sufficiency

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Question: If x and y are positive, is x < y?
1. \(\sqrt{x}<\sqrt{y}\)
2. \((x-3)^2<(y-3)^2\)

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. EACH statement ALONE is sufficient.
  5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Approach Solution (1):

(1)\(\sqrt{x}<\sqrt{y}\). Since both sides of the inequality are positive (the square root from a positive number is positive), then we can safely square: x < y. Directly answers the question
Sufficient
(2)\((x-3)^2<(y-3)^2\). If x = 3 and\(y\not=3\), the inequality will hold true: the left hand side will be 0, while the right hand side will be more than 0. Thus, if x = 3, y can be less than 3, giving a NO answer to the question, as well as more than 3, giving an YES answer to the question
Not sufficient
Correct option: A

Approach Solution (2):
(1)\(\sqrt{x}<\sqrt{y}\)
We can say that answer is YES
But let’s solve it algebraically too
Is x < y can be written as x – y < 0
\(\sqrt{x}^2-\sqrt{y}^2<0\)
\((\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})<0\)
\((\sqrt{x}+\sqrt{y})<0\)
(As x and y are positive)
So we have to find:
If\((\sqrt{x}-\sqrt{y})<0 or \sqrt{x}<\sqrt{y}\)
Statement (1) tells us exactly this
Sufficient
(2)\((x-3)^2<(y-3)^2\)
It holds in many cases: two cases
a) If x = 1 and y = 7… y > x
b) If x = 4 and y = 1… y < x
Two different answers
Insufficient
Correct option: A

Approach Solution (3):
(1)\(\sqrt{x}<\sqrt{y}\)
Since we know that x and y are both positive, squaring or square rooting those values will not change the order of them. Even if you are dealing with positive fractions, then order will not change
Thus, the answer to the question is always YES
Sufficient
(2)\((x-3)^2<(y-3)^2\)
While x and y both are positive, we could end up with an (x – 3) or (y – 3) that is negative though…. And that will impact the answer to the question
If x = 2, y = 10… then (-1)^2 is less than (7)^2 and the answer to the question is YES
If x = 2, y = 1… then (-1)^2 is less than (-2)^2 and the answer to the question is NO
Insufficient
Correct option: A

“If x and y are positive, is x < y?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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