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by Anju@Gurome » Fri Mar 22, 2013 3:48 am
dhairya275 wrote:Is a³ > a²?
(1) 1/a > a
(2) a� > a³
a³ > a² ---> (a³ - a²) > 0 ---> a²(a - 1) > 0
As a² ≥ 0, (a - 1) must be greater than 0.
Hence, a > 1

So, we need to determine whether a > 1 or not.

Statement 1: 1/a > a
If a > 0 ---> a² < 1 ---> 0 < a < 1 ---> NO
If a < 0 ---> a² > 1 ---> a < -1 ---> NO

Sufficient

Statement 2: a� > a³ ---> (a� - a³) > 0 ---> a³(a² - 1) > 0
Hence, either of the following is possible...
  • 1. a³ > 0 and a² > 1 ---> a > 1 ---> YES
    2. a³ < 0 and a² < 1 ---> -1 < a < 0 ---> NO
Not sufficient

The correct answer is A.
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by GMATGuruNY » Fri Mar 22, 2013 6:30 am
dhairya275 wrote:Is a^3 > a^2?
(1) 1/a > a
(2) a^5 > a^3
a³ > a² only if a≠0.
If a≠0, then a² is positive, implying that we can safely divide each side of the question stem by a²:
a³/a² > a²/a²
a>1.

Question rephrased: Is a>1?

Statement 1: 1/a > a
No value greater than 1 will work here:
SUFFICIENT.

Statement 2: a� > a³
a� > a³ only if a≠0.
If a≠0, then a² is positive, implying that we can safely divide each side here by a²:
a�/a² > a³/a²
a³ > a.
It's possible that a=2, since 2³ > 2.
It's possible that a=-1/2, since (-1/2)³ > -1/2.
INSUFFICIENT.

The correct answer is A.
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by rahul.sehgal@btgchampion » Sat Mar 23, 2013 8:00 am
Mitch,
Just to clarify one point -

When we say a^3 > a^2, we also know that this is possible only if a>0 in addition to a#0, right ?

kindly let me know in case i am missing anything.
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by GMATGuruNY » Sat Mar 23, 2013 8:19 am
rahulsehgal wrote:Mitch,
Just to clarify one point -

When we say a^3 > a^2, we also know that this is possible only if a>0 in addition to a#0, right ?

kindly let me know in case i am missing anything.
In this case, a>0.
As my rephrase of the question stem shows, a³ > a² only if a>1.

But the reasoning is not contingent on a>0.
To divide by a², the only requirement is that a≠0.

Another example:
a� > a²
Since a� > a² only if a≠0, we can safely divide each side by a²:
a�/a² > a²/a²
a² > 1.

The resulting inequality implies that a>1 or a<-1.
Thus, to divide by a², it is NOT necessary that a>0.
The only requirement is that a≠0.
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