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Are x and y both positive?

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by VJesus12 » Mon May 28, 2018 2:05 am

Timer

00:00

Answers

A

B

C

D

E

Stats

Difficulty—

Are x and y both positive?

(1) 2x - 2y = 1

(2) x/y > 1

The OA is the option C.

Why is C? I think it should be E. I'd appreciate a clarification here. Please.
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Source: — Data Sufficiency |
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by GMATGuruNY » Mon May 28, 2018 2:36 am
VJesus12 wrote:Are x and y both positive?

(1) 2x - 2y = 1

(2) x/y > 1
Statement 1: 2x-2y = 1.
2(x-y) = 1.
x-y = 1/2.
x = y + 1/2.
It's possible that y=1/2 and x=1.
It's possible that y=0 and x=1/2.
Since in the first case x and y are both positive and in the second case x and y are not both positive, insufficient.

Statement 2: x/y > 1.
It's possible that x=2 and y=1, since 2/1 > 1.
It's possible that x=-2 and y=-1, since (-2)/(-1) > 1.
Since in the first case x and y are both positive and in the second case x and y are not both positive, insufficient.

Statements 1 and 2 combined:
Statement 1: x = y + 1/2.
Statement 2: x/y > 1.
Substituting for x in the inequality:
(y + 1/2)/y > 1.
1 + 1/(2y) > 1.
1/(2y) > 0
1/y > 0
Thus, y>0.
Since y>0 and x = y + 1/2, we know that x>1/2.
Sufficient.

The correct answer is C.

First take-away:
The approach above combined two techniques: algebra and plugging in values.
Many DS questions are best solved using a combination of these two techniques.

Second take-away:
Given an equation with 2 variables (such as x = y + 1/2) and an inequality with the same 2 variables (such as x/y > 1), use the equation to substitute for one of the variables in the inequality.
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by Brent@GMATPrepNow » Mon May 28, 2018 4:56 am
Are x and y both positive?
1) 2x - 2y =1
2) x/y >1
Target question: Are x and y both positive?

Statement 1: 2x - 2y = 1
There are several pairs of numbers that satisfy this condition. Here are two:
Case a: x = 1 and y = 0.5, in which case x and y are both positive
Case b: x = -0.5 and y = -1, in which case x and y are not both positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x/y > 1
This tells us that x/y is positive. This means that either x and y are both positive or x and y are both negative. Here are two possible cases:
Case a: x = 4 and y = 2, in which case x and y are both positive
Case b: x = -4 and y = -2, in which case x and y are not both positive
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2
Statement 1 tells us that 2x - 2y = 1.
Divide both sides by 2 to get: x - y = 1/2
Solve for x to get x = y + 1/2

Now take the statement 2 inequality (x/y > 1) and replace x with y + 1/2 to get:
(y + 1/2)/y > 1
Rewrite as: y/y + (1/2)/y > 1
Simplify: 1 + 1/(2y) > 1
Subtract 1 from both sides: 1/(2y) > 0
If 1/(2y) is positive, then y must be positive.

Statement 2 tells us that either x and y are both positive or x and y are both negative.
Now that we know that y is positive, it must be the case that x and y are both positive
Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by [email protected] » Mon May 28, 2018 10:07 pm
Hi VJesus12,

DS questions can often be solved with a variety of tactics. Beyond the "math" approaches that can be used, you can also TEST Values (as Brent's solution showcased) and often use Number Properties to your advantage.

Here, we're asked if X and Y are BOTH positive? This is a YES/NO question.

Fact 1: 2X - 2Y = 1

I'm going to do a little algebra to rewrite this equation....
2X = 1 + 2Y
X = 1/2 + Y

Now we know that X MUST be bigger than Y and we can find some quick values to TEST.
If.....
Y = 0
X = 1/2
Then the answer to the question is NO

Y = 1
X = 1.5
Then the answer to the question is YES
Fact 1 is INSUFFICIENT

Fact 2: X/Y > 1

Using Number Properties, we know that X and Y are either BOTH POSITIVE (YES answer) or BOTH NEGATIVE (NO answer).
Fact 2 is INSUFFICIENT

Combined, we know....
X = 1/2 + Y
X is GREATER than Y
X/Y > 1
X and Y have the same sign

Since X is greater than Y AND they have the same sign, the ONLY way for X/Y > 1 to be possible is when they're BOTH POSITIVE. If they were both negative, then Y would be greater than X....and we know that that's not possible). The answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich
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