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If x, y, and z are positive numbers, is x > y > z ?

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If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x, y, and z are positive numbers, is x > y > z ?

(1) xz > yz
(2) yx > yz

Data Sufficiency
Question: 69
Category: Algebra Inequalities
Page: 158
Difficulty: 600


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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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SOLUTION

If x, y, and z are positive numbers, is x > y > z ?

(1) xz > yz. Since z is a positive number we can safely reduce by it: x > y. Not sufficient.

(2) yx > yz. Since y is a positive number we can safely reduce by it: x > z. Not sufficient.

(1)+(2) We know that x is greater than both y and z but we don't know whether y > z. Not sufficient.

Answer: E.
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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We are given x , y, z > 0, all the variables are positive, we also know that multiplying/dividing each side of an inequality with a positive integer does not change the direction of the inequality. So let's analyze the statements.

Statement (1)


xz > yz z can cancel each other without affecting the direction of inequality
we can deduce x > y but don't know anything about z in relation to if it is smaller or greater than the other variables. Not Sufficient


Statement (2)


Similarly we can deduce x > z but we can't place y so Statement 2 alone is not sufficient.. (Be careful NOT to carry over statement 1 on to this one yet!!!)


Now look at:

Statement (1) & (2)


x > y and x > z so now we know is is the greatest but still we don't know whether y > z or y < z

So either of the statements together NOT sufficient. Answer E

Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x, y, and z are positive numbers, is x > y > z ?

(1) xz > yz
(2) yx > yz

Data Sufficiency
Question: 69
Category: Algebra Inequalities
Page: 158
Difficulty: 600


GMAT Club is introducing a new project: The Official Guide For GMAT® Quantitative Review, 2ND Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide For GMAT® Quantitative Review, 2ND Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project:
1. Please provide your solutions to the questions;
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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New post 31 Jan 2014, 02:05
From S1:if xz>yz
and x,y,z are +ve,we can cancel out z to conclude x>y.But nothing is given about z.So insufficient.

From S2:x>z.Nothing is given about y.Insufficient.

Combining the two we get,x>y and x>z.
But whether y>/<z is not known.So insufficient.

Ans.E
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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New post 31 Jan 2014, 13:53
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Ans E

We know that all are positive integers so we can divide by the variable without changing the sign.

From1:
xz>yz --> x>y Insufficient

From2:
yx>yz --> x>z Insufficient

Combining both:
x>z and x>y but we don't know anything about how y compares to z . Insufficient.
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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SOLUTION

If x, y, and z are positive numbers, is x > y > z ?

(1) xz > yz. Since z is a positive number we can safely reduce by it: x > y. Not sufficient.

(2) yx > yz. Since y is a positive number we can safely reduce by it: x > z. Not sufficient.

(1)+(2) We know that x is greater than both y and z but we don't know whether y > z. Not sufficient.

Answer: E.
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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Although the correct answer is E, can we apply the following approach? (if not, then why can't we use it?)
At the beginning, we conclude that (1) and (2) (if used separately) are not sufficient. Then, we combine these 2 inequalities, in other words, we form a system of 2 inequalities. Since x,y,z are variables, and we don’t know their signs, we cannot multiply. But we can subtract (2) from (1). As a result, this operation yields: xz-yx>yz-yz => xz-yx>0 => xz>yx => z>y => x>z>y => C
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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kronint wrote:
Although the correct answer is E, can we apply the following approach? (if not, then why can't we use it?)
At the beginning, we conclude that (1) and (2) (if used separately) are not sufficient. Then, we combine these 2 inequalities, in other words, we form a system of 2 inequalities. Since x,y,z are variables, and we don’t know their signs, we cannot multiply. But we can subtract (2) from (1). As a result, this operation yields: xz-yx>yz-yz => xz-yx>0 => xz>yx => z>y => x>z>y => C


No, that's not correct. You cannot subtract yx > yz from xz > yz because their signs are in the same direction (> and >).

ADDING/SUBTRACTING INEQUALITIES:


You can only add inequalities when their signs are in the same direction:

If \(a>b\) and \(c>d\) (signs in same direction: \(>\) and \(>\)) --> \(a+c>b+d\).
Example: \(3<4\) and \(2<5\) --> \(3+2<4+5\).

You can only apply subtraction when their signs are in the opposite directions:

If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).

Adding/subtracting/multiplying/dividing inequalities: help-with-add-subtract-mult-divid-multiple-inequalities-155290.html

Hope it helps.
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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New post 06 Feb 2014, 00:26
Thank you. You are right!
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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New post 08 Jan 2017, 11:44
When I plug in numbers for

X > Y > Z such as 3 > 2 > 1 or 20 > 10 > 2

the equations from (1) and (2) are always right. Any idea why the answer is still E?

Thanks a lot in advance!
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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New post 08 Jan 2017, 11:54
TomSplashy wrote:
When I plug in numbers for

X > Y > Z such as 3 > 2 > 1 or 20 > 10 > 2

the equations from (1) and (2) are always right. Any idea why the answer is still E?

Thanks a lot in advance!


You plug the numbers that answer YES to the question and then check the equations given in (1) and (2). That#s not correct. When trying to solve an YES/NO DS question with plug-in method you should try to find two sets of numbers (which satisfy given information in the stem and the statements), that give different answer to the question (one YES and another NO). For example, you can check x=10, y=5 and z=1 for an YES answer and x=10, y=1 and z=5 for a NO answer.

Hope it's clear.
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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New post 08 Jan 2017, 15:51
@Bunel: Thanks for the explanation!
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Re: If x, y, and z are positive numbers, is x > y > z ? [#permalink]

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New post 24 Jan 2017, 15:56
1, xz>yz ; x > y Insufficient
2, yx > yz ; x > z Insufficient
3, No data about y and z Insufficient
E
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Re: If x, y, and z are positive numbers, is x > y > z ?   [#permalink] 24 Jan 2017, 15:56
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