If x, y, and z are positive numbers, is x > y > z ? : GMAT Data Sufficiency (DS)
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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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30 Jan 2014, 22:46
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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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30 Jan 2014, 22:46
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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.

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

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30 Jan 2014, 23:56
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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:
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!

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Kudos (+1) if you find this post helpful.

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

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31 Jan 2014, 01:05
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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31 Jan 2014, 12:53
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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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01 Feb 2014, 09:26
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.

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

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05 Feb 2014, 22:54
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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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05 Feb 2014, 23:11
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Expert's post
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 >).

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 $$5>1$$ --> $$3-5<4-1$$.

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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05 Feb 2014, 23:26
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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07 Oct 2015, 02:52
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Re: If x, y, and z are positive numbers, is x > y > z ?   [#permalink] 07 Oct 2015, 02:52
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