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# If a, b, x, and y are positive integers, is a^(-x) > b(-y)?

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Intern
Joined: 27 Oct 2013
Posts: 2
If a, b, x, and y are positive integers, is a^(-x) > b(-y)?  [#permalink]

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Updated on: 07 May 2018, 20:53
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Question Stats:

84% (01:19) correct 16% (01:31) wrong based on 388 sessions

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If a, b, x, and y are positive integers, is $$a^{(-x)} > b^{(-y)}$$?

(1) a < b
(2) x < y

Spoiler: :: OFFICIAL SOLUTION
Attachment:

GmatPrepDS.jpg [ 161.95 KiB | Viewed 4261 times ]

I chose E because if a=1/4, b=1/2, x=1, y=2, wouldn't a^(-x)=b^(-y)?

Originally posted by zbvl on 30 Oct 2013, 18:58.
Last edited by Bunuel on 07 May 2018, 20:53, edited 3 times in total.
Edited the question.
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Re: If a, b, x, and y are positive integers, is a^(-x) > b(-y)?  [#permalink]

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30 Oct 2013, 21:33
zbvl wrote:
If a, b, x, and y are positive integers, is a^(-x)>b(-y)?

(1) a<b

(2) x<y

I chose E because if a=1/4, b=1/2, x=1, y=2, wouldn't a^(-x)=b^(-y)?

The question stem mentions specifically that a,b,x and y are positive INTEGERS. Your choice of a,b doesn't subscribe to that.

From F.S 1, we know that b>a . Thus, for a=2,b=3 and x=y=1, we have $$b^y>a^x$$and thus a YES for the question stem.Again, for a=2,b=3 and x=10,y=1, we have $$b^y<a^x$$, and a NO.Insufficient.

Similarly for F.S 2, Insufficient.

Taking both together, we know that b>a and y>x.Thus:
1.$$b^y>a^y$$
2.$$a^y>a^x$$

Thus, $$b^y>a^x.$$ Sufficient.

C.
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Re: If a, b, x, and y are positive integers, is a^(-x) > b(-y)?  [#permalink]

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10 Dec 2013, 12:15
1
Question: a^-x > b^-y? --> 1/a^x > 1/b^y?

We need to find a definit YES or NO.

(1) a < b If a=2 and b = 8 and x = 1 and y = 1 then YES. But if a = 2 and b = 8 and x = 100 and y = 1 then NO. IS
(2) x < y If a = 2 and b = 8 and x = 1 and y = 2 then YES. But if a = 100 and b = 1 then NO. IS

TOGETHER a < b and x < y ==> plug in numbers Suff. Hence C.
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Re: If a, b, x, and y are positive integers, is a^(-x) > b(-y)?  [#permalink]

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08 Mar 2016, 21:24
Here is my approach
W need to prove that x1/a^x >1/b^x
now we can flip the inequality if they are of same sign while doing the reciprocal..
hence we need to prove => a^x<b^y
statement 1 => no clue of x and y => not sufficient
statement 2 => no clue of a and b => not sufficient
combing them we can say that the bae an exponent of rhs are always greater hence rhs would be greater
thus C
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Re: If a, b, x, and y are positive integers, is a^(-x) > b(-y)?  [#permalink]

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17 Mar 2017, 22:04
We can redefine the question as (1/a)^x > (1/b)^y

S1 a < b We know nothing about x and y here. So if a=3 and b = 4 and x = 1 and y = 1 then YES. But if a = 3 and b = 4 and x = 10 and y = 1 then NO. This statement is insufficient.

S2 x < y We know nothing about a and b here.So if a = 3 and b = 4 and x = 1 and y = 2 then YES. But if a = 10 and b = 1 and x=1 and y=2 then NO. This statement is insufficient.

Taking S1 and S2 together is sufficient to get the answer.

Hence C.
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Re: If a, b, x, and y are positive integers, is a^(-x) > b(-y)?  [#permalink]

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08 May 2018, 06:00
Top Contributor
1
zbvl wrote:
If a, b, x, and y are positive integers, is $$a^{(-x)} > b^{(-y)}$$?

(1) a < b
(2) x < y

Target question: Is a^(-x) > b^(-y)?
This is a great candidate for rephrasing the target question.
Aside: At the bottom of this post, you can find a video with tips on rephrasing the target question

First recognize the following: a^(-x) = 1/(a^x) and b^(-y) = 1/(b^y)
So, we can ask Is 1/(a^x) > 1/(b^y)?
Also, since a and b are POSITIVE, we can be certain that (a^x) is POSITIVE and (b^y) is POSITIVE
So, we can safely take the inequality 1/(a^x) > 1/(b^y) and multiply both sides by (a^x) to get: 1 > (a^x)/(b^y)
Next, we can multiply both sides by (b^y) to get: (b^y) > (a^x)
REPHRASED target question: Is (b^y) > (a^x)?

Statement 1: a < b
No information about x or y.
So, statement 1 is NOT SUFFICIENT

Statement 2: x < y
No information about a or b.
So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
The key here is that all 4 variables are positive.
If b is greater than a AND y is greater than x, we can be certain that (b^y) > (a^x)
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

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Re: If a, b, x, and y are positive integers, is a^(-x) > b(-y)?  [#permalink]

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10 May 2018, 09:45
zbvl wrote:
If a, b, x, and y are positive integers, is $$a^{(-x)} > b^{(-y)}$$?

(1) a < b
(2) x < y

[spoiler=OFFICIAL SOLUTION]
Attachment:
GmatPrepDS.jpg

Rephrasing the question we have:

Is 1/a^x > 1/b^y ?

Is b^y > a^x ?

Statement One Alone:

a < b

Since we do now know anything about x and y, statement one alone is not sufficient to answer the question.

Statement Two Alone:

x < y

Since we do now know anything about a and b, statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Since b is greater than a, and y is greater than x, we know that b^y is greater than a^x.

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Re: If a, b, x, and y are positive integers, is a^(-x) > b(-y)? &nbs [#permalink] 10 May 2018, 09:45
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