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.

Suggested GMAT Data Sufficiency Questions:

• What is the Sum of Five Numbers? GMAT Data Sufficiency
• In an Isosceles Triangle PQR, How Many Degrees is ∠ P GMAT Data Sufficiency
• A company makes and sells two products, P and Q. the costs per unit of GMAT Data Sufficiency
• When 120 is Divided by Positive Single-Digit Integer M the Remainder GMAT Data Sufficiency
• There are twenty students in a class, all of whom scored between 0 and 100 GMAT Data Sufficiency
• There are 8 red tegu lizards in the reptile house at a local zoo GMAT Data Sufficiency
• A certain fruit stand sold a total of 76 oranges to 19 customers GMAT Data Sufficiency
• Three dice, each with faces numbered 1 through 6, were tossed onto a game board GMAT Data Sufficiency
• Jeff plans to lose weight by reducing his daily calorie intake each we GMAT data sufficiency
• Bonnie Can Paint a Stolen Car in x hours, and Clyde Can Paint GMAT Data Sufficiency
• Alice has $15, which is enough to buy 11 muffins and GMAT Data Sufficiency
• The product of three distinct positive integers is equal to the GMAT Data Sufficiency
• If at Least One Astronaut does not Listen to Bach at Solaris Space Station GMAT Data Sufficiency
• Set A Consists of k Distinct Numbers GMAT Data Sufficiency
• Four friends go to Macy’s for shopping and buy a top each GMAT Data Sufficiency
• If 3x – 2y = 7, what is the value of x? x and y are positive integer GMAT Data Sufficiency
• Of the 58 Patients of Vertigo Hospital, 45 have Arachnophobia GMAT Data Sufficiency
• If the Lengths of the Legs of a Right Triangle are Integers GMAT Data Sufficiency
• If x is a Positive Integer, is√x an Integer GMAT Data Sufficiency
• If Bob's Age is Three Times David's Age, What is Bob's Age? GMAT Data Sufficiency
• A Circle is Inscribed in a Quadrilateral ABCD Such a Way that it Touches GMAT Data Sufficiency
• What is the Area of a Rectangular Field? GMAT Data Sufficiency
• If x and y are negative integers, then what is the value of xy GMAT Data Sufficiency
• If x is a non-zero integer, what is the value of x GMAT Data Sufficiency
• In the Figure above, FGHI is Inscribed in the Circle with the Centre J GMAT Data Sufficiency
• In the xy- plane, region R consists of all the points (x, y) GMAT Data Sufficiency
• Is 2ab > ab? GMAT Data Sufficiency
• What is the Probability That it will Rain Tomorrow GMAT Data Sufficiency
• If r > 0 and s > 0, is r/s < s/r? GMAT Data Sufficiency
• How many integers n are there such that r < n < s? GMAT Data Sufficiency
• By How Many Dollars was The Price of a Certain Portable Tape Recorder GMAT Data Sufficiency
• At a Certain Bookstore, Each Notepad Costs x Dollars GMAT Data Sufficiency
• A Train Traveled from Station A to Station B at an Average Speed of 80 GMAT Data Sufficiency
• A Teacher Distributed a Number of Candies, Cookies, and Toffees GMAT Data Sufficiency
• What is the Perimeter of Quadrilateral Q? GMAT Data Sufficiency
• What is the Perimeter of Rectangle ABCD GMAT Data Sufficiency
• Mr. Alex Usually Starts at 9:00am and Reaches His Office Just GMAT Data Sufficiency
• If x and y Represent Number on The Number Line, is x + y = 0 GMAT Data Sufficiency
• Is 9 the HCF of p and q? GMAT Data Sufficiency
• Is√(x−3)2=3−x?GMAT Data Sufficiency
• What is the volume of a certain rectangular solid? GMAT data sufficiency
• If abc 0, is a < b < c? (1) 1/c < 1/b < 1/a (2) c > a GMAT Data Sufficiency
• A Recent Lunch Meeting at a Certain Club was Attended by Members GMAT Data Sufficiency
• w, x, y, and z are integers. If w > x > y > z > 0, is y a common GMAT Data Sufficiency
• What is the sum of 3 consecutive integers? GMAT Data Sufficiency
• Two cars A and B start from diametrically opposite points of GMAT Data Sufficiency
• What Is The Value Of X + Y In The Figure Above? GMAT Data Sufficiency
• The Length, Width, and Height of a Rectangular Box, in Centimeters are L, W, and H GMAT Data Sufficiency
• Is the perimeter of a rectangle greater than 8 inches GMAT Data Sufficiency
• Is the Positive Integer n Divisible by 8? (1) n is Divisible by the Product of Three GMAT Data Sufficiency