Is xy≤1/2?
(1) x2+y2=1
(2) x2-y2=0
here also I chose E but the answer is A
Explanation given is (1) x2+y2=1. Recall that (x-y)2≥0(x-y)2≥0 (square of any number is more than or equal to zero). Expand: x2-2xy+y2≥0 and since x2+y2=1 then: 1-2xy≥0. So, xy≤1/2. Sufficient.
(2) x2-y2=0. Re-arrange and take the square root from both sides: |x|=|y|. Clearly insufficient.
Please help
Number Properties- help on this DS question
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Hi riteshpatnaik,
You've stated that you chose Answer E, but you did not explain WHY you chose that answer. If you explain your approach/logic or show your work, then I can help you to figure out what you did incorrectly.
We're asked if (X)(Y) is ≤ 1/2? This is a YES/NO question.
1) X^2 + Y^2 = 1
Since the sum of these two squares is 1, the individual values of X and Y have to be relatively small.
IF... X=0, Y=1, then (X)(Y) = 0 and the answer to the question is YES.
As the values of X and Y get closer and closer together, we still end up with a fraction that is less than or equal to 1/2...
IF... X=(root 1/2), Y=(root 1/2), then (X)(Y) = 1/2 and the answer to the question is YES.
The answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT
2) X^2 - Y^2 = 0
IF... X=(root 1/2), Y=(root 1/2), then (X)(Y) = 1/2 and the answer to the question is YES.
IF... X=2, Y=2, then (X)(Y) = 4 and the answer to the question is NO.
Fact 2 is INSUFFICIENT
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
You've stated that you chose Answer E, but you did not explain WHY you chose that answer. If you explain your approach/logic or show your work, then I can help you to figure out what you did incorrectly.
We're asked if (X)(Y) is ≤ 1/2? This is a YES/NO question.
1) X^2 + Y^2 = 1
Since the sum of these two squares is 1, the individual values of X and Y have to be relatively small.
IF... X=0, Y=1, then (X)(Y) = 0 and the answer to the question is YES.
As the values of X and Y get closer and closer together, we still end up with a fraction that is less than or equal to 1/2...
IF... X=(root 1/2), Y=(root 1/2), then (X)(Y) = 1/2 and the answer to the question is YES.
The answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT
2) X^2 - Y^2 = 0
IF... X=(root 1/2), Y=(root 1/2), then (X)(Y) = 1/2 and the answer to the question is YES.
IF... X=2, Y=2, then (X)(Y) = 4 and the answer to the question is NO.
Fact 2 is INSUFFICIENT
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
GMAT/MBA Expert
- Jay@ManhattanReview
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I think that you have a challenge in following the explanation of statement 1. Let us redo the same.riteshpatnaik wrote:
Explanation given is (1) x2+y2=1. Recall that (x-y)2≥0(x-y)2≥0 (square of any number is more than or equal to zero). Expand: x2-2xy+y2≥0 and since x2+y2=1 then: 1-2xy≥0. So, xy≤1/2. Sufficient.
Please help
S1: x^2 + y^2 = 1.
Subtracting and adding 2xy, we get,
x^2 + y^2 - 2xy + 2xy = 1
=> (x-y)^2 + 2xy = 1
=> (x-y)^2 = 1- 2xy
Since (x-y)^2 is a non-negative number, (x-y)^2 ≥ 0.
=> 1 - 2xy ≥ 0
=> 1 ≥ 2xy
=> 1/2 ≥ xy
Sufficient!
Hope this helps!
-Jay
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