Is xy<1? GMAT Data Sufficiency

Question: Is xy<1?

1)x^2 + y^2 < 1
2)x + y < 1

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.

Correct Answer: A

Solution and Explanation
Approach Solution 1:
Given in the question, two statements are given and it is asked to find out if the two statements are sufficient to get the answer.
It is asked if x*y > 1 or not.

The given two statements are –

1. \(x^2\) +\(y^2\)< 1
2. x+y < 1

First we’ll check the first statement,
\(x^2\)+ \(y^2\)< 1
It should be noted that x^2 and y^2 will always be positive.
Therefore the value of \(x^2\)and \(y^2\)should be less than 1
\(x^2\)<1 and \(y^2\)< 1
-1 < x < 1 & -1 < y < 1
Therefore the product xy will also lie in the region (-1,1)
Xy < 1
And x < 1 and y < 1
Therefore this statement is sufficient to get the answer.
Now checking the second statement,
Given,
x+y < 1
Here we cannot say anything about the values of x and y
Let x = 0.2 and y = 0.4
The sum x + y < 1, so these values satisfy the equation.

The product x.y is also less than 1.
But if x = -2 and y = -3
X + y < 1
This equation is satisfied but the product x.y > 1
Therefore this statement is not sufficient to get the answer.
So the correct option will be option A.

Approach Solution 2:
Given in the question, two statements are given and it is asked to find out if the two statements are sufficient to get the answer.
It is asked if x*y > 1 or not.
The given two statements are –

1. \(x^2\)+\(y^2\) < 1
2. x+y < 1​

First we’ll check the first statement,
\(x^2\)+ \(y^2\)< 1
We know that \(x^2\)+ \(y^2\)- 2xy = \((x-y)^2\)
\(x^2\)+ \(y^2\)= \((x-y)^2\) + 2xy
\((x-y)^2\) + 2xy <1
\((x-y)^2\) >=0
2xy < 1
Xy < 1
This statement is sufficient to get the answer.
From second statement,
X + y < 1
Let x = -3 and y = -2
The sum is less than 1 but the product is greater than 1.
Therefore this statement is not sufficient to get the answer.
The correct option is option A.

Approach Solution 3:
a) x^2 & y^2 are always positive
So for (x^2 + y^2 )<1 both x & y should be less than 1 and greater than -1 (-1 Therefore, xy<1 (x<1 & y<1)
Sufficient.

b) x+y<1
( Possible x=0.2,y=0.3: therefore xy<1) but (if x=-2 & y=-4 : xy>1)
Not Sufficient.

“Is xy<1?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide". 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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