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x, y - inequality - number proprety

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by GMATGuruNY » Sat Aug 29, 2015 2:19 pm
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.
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 » Sat Aug 29, 2015 2:19 pm
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
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by sandipgumtya » Sun Aug 30, 2015 6:16 am
Clearly the statements individually are insufficient.Combined"X=Y+1/2 and X/Y>1 i.e (Y+Y/2)>1 which implies Y is positive.so X also has to be positive here.Ans C
Experts is my approach correct.Or is there any faster method?
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by [email protected] » Sun Aug 30, 2015 8:53 am
Hi sandipgumtya,

It sounds like you're comfortable with the Number Properties that are built into this question, but when dealing with DS questions, you should be careful about simply describing anything as 'clearly sufficient' or 'clearly insufficient' - especially when you do not explain WHY. Tougher DS questions can include subtle options or possibilities that might not be immediately apparent - unless you do the necessary work to prove that those options exist, you might end up making a minor mistake (and end up getting questions wrong without even knowing it).

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by Max@Math Revolution » Mon Aug 31, 2015 6:09 am
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.


Are x and y both positive?

1) 2X - 2y = 1
2) x/Y > 1

==> In the original condition we have 2 variables (x,y) thus we need 2 equations. Since there are 1 each in 1) and 2), C is likely the answer, and it actually is.

Using both 1) & 2) together, 2x-2y=1>0 thus 2x-2y>0, 2(x-y)>0, x-y>0. In case of 2), if we multiply y^2 to both sides (the inequality sign does not change since the i multiplied positive value)) we get xy>y^2 ==> xy-y^2>0, y(x-y)>0 thus x-y>0. therefore from con 1) y>0, and x-y>0 ==> x>y>0. Having x>0 and y>0 makes the answer yes, thus it is sufficient. Therefore the answer is C.


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by sandipgumtya » Mon Aug 31, 2015 10:15 am
Hi Rich,
I have tested value for both of these before arriving at the conclusion.Anyway thanks for ur words of caution.But i am not able to get the context .Pl explain a bit more.

Thanks
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by Max@Math Revolution » Fri Sep 04, 2015 3:57 am
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.


Are x and y both positive?

1) 2X - 2y = 1
2) x/Y > 1
transforming the original condition and the question by variable approach method, we have 2 variables 2개(x,y). we need 2 more equations to match the number of equations and variables, and since there is 1 each in 1) and 2), C is likely the answer.

Using both 1) and 2), 2x-2y=1>0, 2x-2y>0, x-y>0, x>y. from x/y>1, multiplying both sides by y^2(since multiplying by square values maintain the direction of inequality signs) we have xy>y^2, xy-y^2>0, y(x-y)>0, therefore being x-y>0. Since y>0 and also x>y>0, meaning we have x>0, the answer is yes. The conditions are sufficient, therefore the answer is C.



If you know our own innovative logics to find the answer, you don't need to actually solve the problem.
www.mathrevolution.com
- The one-and-only World's First Variable Approach for DS and IVY Approach for PS that allow anyone to easily solve GMAT math questions.

- The easy-to-use solutions. Math skills are totally irrelevant. Forget conventional ways of solving math questions.

- The most effective time management for GMAT math to date allowing you to solve 37 questions with 10 minutes to spare

- Hitting a score of 45 is very easy and points and 49-51 is also doable.

- Unlimited Access to over 120 free video lessons at https://www.mathrevolution.com/gmat/lesson

- Our advertising video at https://www.youtube.com/watch?v=R_Fki3_2vO8
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by Matt@VeritasPrep » Fri Sep 04, 2015 6:11 pm
Let's work backwards here.

S2 gives us two scenarios:

if y > 0, then x > y

or

if y < 0, then y < x

So this is close, but not quite sufficient.

S1 on its own tells us that x = y + 1/2. So x > y, but we don't know if they're both positive, both negative, or one of each.

Together, however, S1 gives us x > y, and S2 tells us that this is only possible if y > 0. So we have x > y and y > 0, forcing both variables to be positive. Sufficient!
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