GMAT Club World Cup 2022 (DAY 3): If |a| > |b| > |c|, is a*b^3*c^3 > a

given that
\(|a| > |b| > |c|\)
target is \(a*b^3*c^3 > a*b^4*c^2\)

#1\(a > b > c\)
possible when a,b,c are all +ve integers or fractions
3,2,1 or 1/2,1/3,1/4
we get yes to target fractions and no when integers
insufficient
#2
\(a + b > 0\)
c is not known
insufficient
from 1 &2
again we will have two cases as in #1
insufficient
option E is correct

If |a|>|b|>|c|, is a∗b3∗c3>a∗b4∗c2 ?

(1) a>b>c
(2) a+b>0

Solution :

My understanding is that given |a|>|b|>|c||a|>|b|>|c| along with statement 2, we can deduce that a is positive. however we don't know anything about b and c, but from statement 1 we know that if b is negative then c is also negative - in most scenarios 'a∗b3∗c3>a∗b4∗c2' will be false

\(a*b^3*c^3 > a*b^4*c^2\)

Statement 1 Only -
\(a > b > c\)

If a = -ve, b and c are also -ve.
\(|a| * |b|^3 * |c|^3 < |a| * |b|^4 * |c|^2\)
\(|c| < |b|\) - YES

If a = +ve, b = +ve, c = -ve
\(-(|a| * |b|^3 * |c|^3) > |a| * |b|^4 * |c|^2\)
\(-|c| > |b|\) - NO

2 different answers. Hence, INSUFFICIENT

Statement 2 Only -
\(a + b > 0\)

If a = -ve, b is +ve, c is -ve
\(|a| * |b|^3 * |c|^3 > -(|a| * |b|^4 * |c|^2)\)
\(|c| > - |b|\) - YES

If a = +ve, b = +ve, c = -ve
\(-(|a| * |b|^3 * |c|^3) > |a| * |b|^4 * |c|^2\)
\(-|c| > |b|\) - NO

2 different answers. Hence, INSUFFICIENT

1 & 2 Together -
a & b are +ve

If a = +ve, b = +ve, c = -ve
\(-(|a| * |b|^3 * |c|^3) > |a| * |b|^4 * |c|^2\)
\(-|c| > |b|\) - NO

If a = +ve, b = +ve, c = +ve
\(|a| * |b|^3 * |c|^3 > |a| * |b|^4 * |c|^2\)
\(|c| > |b|\) - NO

Same answers. Hence, SUFFICIENT

Hence, answer is C.

If |a|>|b|>|c|, is a∗b3∗c3>a∗b4∗c2?

(1) a>b>c
let's say a=3,b=2,c=1
3.8.1<3.16.1
since |a|>|b|>|c|
putting negative will result result in opposite if given
sufficient

(2) a+b>0
C can be anything
suppose a=4,b= 2 or -2 both are possible c can b 1 or 0 or -1
insufficient
IMO A

St 1 implies a,b,c are positive
a>b>c
Therefore
ab^3c^3 > ab^4c^2
= c > b not true
Therefore definitely No
Hence sufficient

St2. a+b >0
No info about c ie positive or negative also b can be pisitive or negative
Not sufficient

Answer: A

Posted from my mobile device

If you solve the given question, it basically asks us if c > b?

Using statement 1, we can clearly arrive at the answer easily since it's stated.

Using statement 2, we just know that a > -b. We don't really know anything about c here.

So, option A is the best bet imo

Correct answer is choice D

what we are trying to find is if ab3c2(c-b) > 0
from equation 1 and the equation given in the question stem, we can confirm that the numbers are positive and a > b > c
hence equation 1 is good enough

for equation 2 and the question stem , we can tell a b c are positive
hence equation 2 is good enough.

Therefore the correct answer is choice D

\(a*b^3*c^3 > a*b^4*c^2\)
\(a*b*(c-b) > 0\)

(1) \(a > b > c\)
\(|a| > |b| > |c|\)
\(a*b*(c-b) < 0\) --- Sufficient

(2) \(a + b > 0\)
Nothing is known about relationship of b and c. ---- InSufficient

Ans A

Constraint: It is given than absolute values of a > b > c.

Statement 1 states a > b > c
This means that a and b have to be positive, keeping in mind the constraint. C can either be -ve/0/+ve.
After plugging in different values of a, b and c- The answer to the given question will always be No. Hence, this statement is sufficient.
Eg. a=3, b=2, c=-1/0/1

Statement 2 states a + b > 0
This means that a will always be positive whereas b can either be +ve or -ve, keeping in mind the constraint. b cannot be 0 because it should be greater than c in absolute terms. Here, c can be any value.
After plugging in different values of a, b and c- The answer to the given question will always be No. Hence, this statement is sufficient.
Eg. a=3, b=-2, c=-1/0/1

Hence, answer should be D.

Given

|a| > |b| > |c|

Interpretation

(The distance of a from 0) > (The distance of b from 0) > (The distance of c from 0)

Question

\(a*b^{3}*c^{3} > a * b ^ {4} * c ^{2}\)

\(a*b^{3}*c^{3} - a * b ^ {4} * c ^{2} > 0\)

\(a*b^{3}*c^{2}*(c-b) > 0\)

Statement 1

All we know from this statement is a lies to the right of b and b lies to the right of c as shown below on the number line

-----------c-----------b-----------a-----------

We do not know in which position does 0 lies, however based on the above constraint we know that zero lies in either of the positions -

-----------c--0--------b-----------a-----------

-------0---c-----------b-----------a-----------

In both case -
a = +ve
b = +ve
c - b = -ve
\(c^2\) = +ve


\(a*b^{3}*c^{2}*(c-b) > 0\)

+ve * +ve * +ve * -ve = -ve

We have a definite answer "No"

Statement 2

a + b > 0

We know that a is farther away from b with respect to 0, hence a needs to be +ve for this to hold true.

Now, with reference to the given constraints we know that (The distance of b from 0) > (The distance of c from 0), however nothing is given on their positing.

-------0---c-----------b-----------a---------
-----------c--0--------b-----------a---------
----------b-------0--c-----------a-----------
----------b--c----0--------------a-----------

In any of the four case, we will either get \(b^3\) as -ve or (c-b) as -ve

Hence, this statement is also sufficient to answer a definite No.

IMO D

Please refer to the attachment for solution of this problem.

[quote="Bunuel"]If \(|a| > |b| > |c|\), is \(a*b^3*c^3 > a*b^4*c^2\) ?

(1) \(a > b > c\)
(2) \(a + b > 0\)



1)a>b>c Given |a|>|b|>|c|. Combining these 2 inequalities the minimum value of c can be -1
Example 1 - a=3,b=2,c=1
LHS =a*b^3*c^3 =24 RHS= a*b^4*c^2=48
LHS<RHS
Example 2 a=3,b=2,c=-1
LHS=-24 RHS=48
LHS<RHS
Now if we consider c=0 ,in that case LHS = RHS
For fractions also same pattern will follow.
Thus LHS<RHS or LHS=RHS.
Thus LHS is never greater than RHS .1 is sufficient.B,C,E out
2)a+b>0 Given |a|>|b|>|c|.Mininum value of c=-1
Example 1 - a=4 b=-3,c=1 ,LHS=a*b^3*c^3 =-98 RHS= a*b^4*c^2=324
LHS<RHS and if we consider c=0 instead of 1 then LHS=RHS
Thus LHS is not greater than RHS
2 is sufficient.
D is the correct choice

Asked: If \(|a| > |b| > |c|\), is \(a*b^3*c^3 > a*b^4*c^2\) ?
Is abˆ3cˆ3 - abˆ4cˆ2 > 0?
Is abˆ3cˆ2(c-b) > 0?

(1) \(a > b > c\)
a>b>c>0 since \(|a| > |b| > |c|\)
c-b<0
abˆ3cˆ2(c-b)<0
SUFFICIENT

(2) \(a + b > 0\)
a & b both can not be negative.
Case 1: a > b > 0
b > c or c-b<0 in all cases including c<0
abˆ3cˆ2(c-b) < 0
Case 2: a > 0 > b
If c > 0; a> c>0> b; ; a>0; bˆ3<0; c-b> 0; abˆ3cˆ2(c-b) < 0
If c < 0; a > 0 > c > b; a> 0; bˆ3 < 0; c-b>0; abˆ3cˆ2(c-b) < 0
In all cases abˆ3cˆ2(c-b) < 0
SUFFICIENT

IMO D

Set inequality to one side and factor for:
ab^3c^2(c-b)>0 ??
ab* b^2c^2 * (c-b)>0 ?? left side becomes 0 if any of a,b,c, or c-b = 0

Statement (1) tells you that a and b both (+), c is closer to 0 than B is. also --> (c-b) is negative
ab (+) * (bc)^2 * (c-b) (-) cannot be (+)
Sufficient

Statement (2) tells you a>b and a is (+) Insufficient

Choice A is the answer.

We know mods are in the given inequality form however we do not know their signs.
We can also figure that a and b can't be 0 since their mods are both more than |c| and no number is less than 0
1) a>b>c
If a, b and c are all positive, then RHS>LHS because a higher exponent is associated with the higher variable
If c= 0, then a and b can only take positive values. Here RHS=LHS. NS

2) a+b>0
a is positive and |b|<|a| thus b can be positive or negative. We do not know anything about c. If c is 0, they will equal otherwsie they won't. Same two scenarios as above can play out here. NS

1 and 2:
We still do not know value of c and the same 2 scenarios can take place again. NS

Ans E

We have \(|a| > |b| > |c|\)

Statement 1:
\(a > b > c\) So, a,b and c have to be positive.
If a,b, and c are positive then we can reduce the inequality in the question by canceling out the same variables from both sides.
\(a*b^3*c^3 > a*b^4*c^2\) = \(c>b\).
Is \(c>b\)?

from statement 1, \(b>c\); so it is not true.
Sufficient.

Statement 2:
\(a + b > 0\)
case 1: \(a>o\) & \(b>0\), in this case we will always get \(b>c\)
case 2: \(a>o\) & \(b<0\), since \(|a| > |b|\), it will suffice \(a + b > 0\) even if \(b<0\).
But in this case \(a*b^3*c^3 > a*b^4*c^2\) can be reduced to \(bc > b^2\). We need to check if \(bc > b^2\)?
For any values of b and c, we will always get \(b^2 > bc\)
Sufficient.

Answer D