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Math Expert V
Joined: 02 Sep 2009
Posts: 58335
If x>y>0, which of the following must be true:  [#permalink]

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Question Stats: 79% (01:03) correct 21% (01:12) wrong based on 767 sessions

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If x>y>0, which of the following must be true:

I. x^2>y^2
II. x^3 > y^3
III. |x|>y

A. I only
B. II only
C. III only
D. II and III
E. I, II and III

Kudos for a correct solution.

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Re: If x>y>0, which of the following must be true:  [#permalink]

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1
Bunuel wrote:
If x>y>0, which of the following must be true:

I. x^2>y^2
II. x^3 > y^3
III. |x|>y

A. I only
B. II only
C. III only
D. II and III
E. I, II and III

I would go wih E here

I. the issue is to check whether proper fractions can disprove this statement. Pick y=0,1 and pick x=0,101 $$x^2$$ is going to be greater than $$y^2$$ try with x= √0,6 and y= √0,5 $$x^2$$ is greater than $$y^2$$.

II. We don't have to worry about even/odd powers since our values are greater than zero. Thus our considerations on I. hold on II.

III. We don't have to worry about the modulus. |x| is x when x>0 this is our case. x>y thus |x| is greater than y.

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Manager  S
Joined: 24 Jun 2014
Posts: 51
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Re: If x>y>0, which of the following must be true:  [#permalink]

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1
Assuming x and y to be integers
X=3,Y=2
X^2=3^2=9
Y^2=2^2=4
So I is true

Now lets look at option III because if option 3 is true E is the answer or else A
As per question X>Y>0 Hence |X| >Y

So OA=E
Manager  Joined: 14 Jul 2014
Posts: 88
Re: If x>y>0, which of the following must be true:  [#permalink]

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1
Ans - E

Since given both X & Y are +ve,

Stmnt 1 - Quickly plugged in decimals ( x = 0.2, y = 0.1) and (x = 1.2 and y = 1.1) and integers

Stmnt 2 - No need for any computation
Infact , irrespective of the sign , whenever given that X > Y, any odd power will always hold true
i.e. x^3 > y^3, x^5 > y^5, x^7 > y^7,
x^1/3 > y^1/3 (cube root)...etc

Stmnt 3 - No need for any computation - Modulas does not matter as the constraint in the Q is that both X & Y are +ve
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Posts: 40
Re: If x>y>0, which of the following must be true:  [#permalink]

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1
When two positive integers x & y are compared by the given condition x>y, then the only case when y>x will be when the power is -ve. So it satisfies all the options i, ii & iii.

Therefore option e
Math Expert V
Joined: 02 Sep 2009
Posts: 58335
Re: If x>y>0, which of the following must be true:  [#permalink]

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1
Bunuel wrote:
If x>y>0, which of the following must be true:

I. x^2>y^2
II. x^3 > y^3
III. |x|>y

A. I only
B. II only
C. III only
D. II and III
E. I, II and III

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION

Solution: E.

The "trick" here is that you are apt to expect a trick. Clearly all three statements hold for integers (if x = 2 and y = 1, then x^2 = 4 and y^2 = 1, and x^3 = 8 and y^3 = 1). But you may expect for fractions to be different - if you square, say, 1/2, then it gets smaller (1/4). But, still, the larger fraction will remain larger when squared or cubed. Take for example 1/2 and 1/3 each squared. 1/2 --> 1/4, and 1/3 --> 1/9. The smaller fraction becomes even smaller. Statement III is true simply by definition - the absolute value of a positive number is just that number.
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Re: If x>y>0, which of the following must be true:  [#permalink]

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1
Bunuel wrote:
Bunuel wrote:
If x>y>0, which of the following must be true:

I. x^2>y^2
II. x^3 > y^3
III. |x|>y

A. I only
B. II only
C. III only
D. II and III
E. I, II and III

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION

Solution: E.

The "trick" here is that you are apt to expect a trick. Clearly all three statements hold for integers (if x = 2 and y = 1, then x^2 = 4 and y^2 = 1, and x^3 = 8 and y^3 = 1). But you may expect for fractions to be different - if you square, say, 1/2, then it gets smaller (1/4). But, still, the larger fraction will remain larger when squared or cubed. Take for example 1/2 and 1/3 each squared. 1/2 --> 1/4, and 1/3 --> 1/9. The smaller fraction becomes even smaller. Statement III is true simply by definition - the absolute value of a positive number is just that number.

if I consider X= 1/3 and y=1/2 , then, X square is not greater that y Square. and same is true for cube power. In my view only o'ption three holds must be true. please clarify. thanks in advance.
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Re: If x>y>0, which of the following must be true:  [#permalink]

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robu wrote:
if I consider X= 1/3 and y=1/2 , then, X square is not greater that y Square. and same is true for cube power. In my view only o'ption three holds must be true. please clarify. thanks in advance.

You can not consider x=1/3 and y=1/2 as this will make y>x but per the given condition x>y

Hope this helps.
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GMAT 1: 540 Q39 V26 GMAT 2: 680 Q46 V37 Re: If x>y>0, which of the following must be true:  [#permalink]

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Plug values

For any two numbers x and y such that x>y>0 the squares and cubes are also in the same inequality. However, x^2 is smaller than x in case of 0<x<1; so is the cube.

Also |x| = x for x>0 or -x for x <0. Given that x>0, this too is positive and so is the result.

Hence all options are true.
E
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GRE 1: Q169 V154 Re: If x>y>0, which of the following must be true:  [#permalink]

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We just need to remember here that if x and y are both greater than zero => and x>y => x^n>y^n is true
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Re: If x>y>0, which of the following must be true:  [#permalink]

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Bunuel wrote:
Bunuel wrote:
If x>y>0, which of the following must be true:

I. x^2>y^2
II. x^3 > y^3
III. |x|>y

A. I only
B. II only
C. III only
D. II and III
E. I, II and III

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION

Solution: E.

The "trick" here is that you are apt to expect a trick. Clearly all three statements hold for integers (if x = 2 and y = 1, then x^2 = 4 and y^2 = 1, and x^3 = 8 and y^3 = 1). But you may expect for fractions to be different - if you square, say, 1/2, then it gets smaller (1/4). But, still, the larger fraction will remain larger when squared or cubed. Take for example 1/2 and 1/3 each squared. 1/2 --> 1/4, and 1/3 --> 1/9. The smaller fraction becomes even smaller. Statement III is true simply by definition - the absolute value of a positive number is just that number.

plz correct me if i am wrong, it means this is good for integers as well as for fractions too.. (provided positive)
thanks
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Re: If x>y>0, which of the following must be true:  [#permalink]

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_________________ Re: If x>y>0, which of the following must be true:   [#permalink] 01 Jan 2018, 06:05
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