Is (A/B)^3 < (AB)^3 ? (1) A > 0 (2) AB > 0

is (A/B)^3 < (AB)^3 ?

STATEMENT (1)- A > 0
but we don't have any information about B
if A = 4 B =\(\frac{1}{4}\) then (A/B)^3 > (AB)^3---is (A/B)^3 < (AB)^3 ? --NO

if A = 4 B = \(\frac{-1}{4}\) then (A/B)^3 < (AB)^3---is (A/B)^3 < (AB)^3 ? --YES

from here we can't get a definite answer
so, INSUFFICIENT

STATEMENT (2)- AB > 0
from here we know A and B are either positive or negative
if A = 4 B = \(\frac{1}{4}\) then (A/B)^3 > (AB)^3---is (A/B)^3 < (AB)^3 ? --NO

if A = 4 B = 2 then (A/B)^3 < (AB)^3---is (A/B)^3 < (AB)^3 ? --YES

from here we can't get a definite answer
so, INSUFFICIENT

combining both statements together
we know AB>0 and A>0 this tells us B>0

if A = 4 B = \(\frac{1}{4}\) then (A/B)^3 > (AB)^3---is (A/B)^3 < (AB)^3 ? --NO

if A = 4 B = 2 then (A/B)^3 < (AB)^3---is (A/B)^3 < (AB)^3 ? --YES

from here we can't get a definite answer
so, INSUFFICIENT

E is the correct answer

(A/B)^3 < (AB)^3 ?


(1) A>0
Let's A = 1, B = -2; insert A,B value, we've got -1/8 < -8. NO
But A = 1, B = 2; 1/8 < 8. Yes

So, A is sufficient

(2) AB>0
Let's A = 1, B = 2; insert A,B value, we've got 1/8 < 8. Yes
But A = 2, B = 1; 8 < 8 No.
So, 2 alone is not suffcient


(1) + (2); A>0 and AB >0, thus both A,B > 0
If A = 1, B =2; 1/8 < 8 Yes.
If A =2 , B =1 ; 8 < 8 No.

Not sufficient. Therefore E is the answer

Is \((\frac{A}{B})^3 < (AB)^3\) ?

No relation between A and B is given so many possibilities exist.

(1) \(A > 0\)
Here either \(B > A\) for both integer value and non-integer value of positive B.
Example: Let A = 2 & B = 3 then
\((\frac{2}{3})^3 < (2*3)^3\) YES

OR

\(A > B\) for either positive value of B or negative value of B.
Example: Let A = 2 & B = 1 then
\((\frac{2}{1})^3 < (2*1)^3\) NO

INSUFFICIENT.

(2) \(AB > 0\)

Two cases are possible here:

(a) Both A and B are +ve
Take A = 2 & B = 3 then
\((\frac{2}{3})^3 < (2*3)^3\) YES

(b) Both A and B are -ve
Take \(A = \frac{-1}{2}\) & \(B = \frac{-1}{3}\) then
\(((\frac{-1}{2})/(\frac{-1}{3}))^3 < ((\frac{-1}{2})*(\frac{-1}{3}))^3\) NO

INSUFFICIENT.

Together 1) and 2)

We have \(A > 0\) and \(B > 0\)
Again Take A = a2 & B = 3 then
\((\frac{2}{3})^3 < (2*3)^3\) YES

And

Take \(A = \frac{1}{2}\) & \(B = \frac{1}{3}\) then
\(((\frac{1}{2}/\frac{1}{3})^3 < ((\frac{1}{2})*(\frac{1}{3}))^3\) NO

INSUFFICIENT.

Answer (E).

Is \((\frac{A}{B})^{3}<(AB)^{3}\)?

\(A^{3}(B^{3}-(\frac{1}{B})^{3}\))>0

\(A^{3}*(B-\frac{1}{B})*(B^{2}+B*\frac{1}{B}+(\frac{1}{B})^{2}\))>0

\((B^{2}+B*\frac{1}{B}+(\frac{1}{B})^{2}\)) is always greater than zero.
--> \(A^{3}*(B-\frac{1}{B})\)>0 ???

(1) A > 0

(2) AB > 0

Statement1:
A > 0
--> (B-\(\frac{1}{B}\))>0 ???

If B=2, then 2-\(\frac{1}{2}\)>0 (Yes)
If B=\(\frac{1}{2}\), then \(\frac{1}{2}\)-2>0 (NO)

Insufficient.

Statement2:
AB > 0
--> Both A and B are Positive or Negative:
\(A^{3}*(B-\frac{1}{B})\)>0

if A=B=2, then --> \(2^{3}(2-\frac{1}{2})\)>0 (yes)
if A=B=\(\frac{1}{2}\), then --> \((\frac{1}{2})^{3}(\frac{1}{2}-2)\)>0 (NO)

Insufficient.

Taken together 1 and 2,
A>0 and AB>0 --> B>0.

\(A^{3}*(B-\frac{1}{B})\)>0 --> (B-\(\frac{1}{B}\))>0 ???

if B=2, then --> (2-\(\frac{1}{2}\))>0 (yes)
if B=\(\frac{1}{2}\), then -->(\(\frac{1}{2}\)-2)>0 (NO)

Insufficient

The answer is E.

#1
A>0 so A is +ve but no relation of B with A insufficient
#2
AB>0
so either both AB are +ve or -ve also they can be same value /integer/ fraction no given info
insufficient
from 1 &2
we can say that AB are +ve but whether A=B ; A>B or B<A cannot be determined
IMO E


Is(AB)3<(AB)3 ?


(1) A>0

(2) AB>0

IMO it's E.
Because even after combining if we take A=100 and B=1 then NO is the answer.
And if A=100 and B=2 then YES is the answer.

rephrase: \((A/B)^3<(AB)^3…A/B<AB…A/B<AB\); Is \(A/B<AB\)?

case 1: \(A,B=(1,2)…A/B<AB…1/2<2:true\)
case 2: \(A,B=(10,1)…A/B<AB…10<10:false\)

(1) A>0: case 1 and 2, insufi.
(2) AB>0: case 1 and 2, insufi.
(1&2) A,B>0: case 1 and 2, insufi.

Answer (E)

Is (A/B)^3 <(AB)^3 ?

(1) A>0
if A>0, then
\((A/B)^3<(AB)^3\)
\(A^3/B^3<A^3*B^3\)
\(1/B^3<B^3\)
if B = 1, \(1/1^3<1^3\) is not possible
if B = 2, \(1/2^3<2^3\) is possible
so, no unique answer

(2) AB>0
A>0 & B>0 (I) or A<0 & B<0 (II)
condition (I) is same as (1), so no unique answer
we don't have to check condition (II)

(1) & (2)
A>0 (from 1), so B>0
again no unique solution if we take B = 1 or 2
So, answer is E