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

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

Given |a|>|b|>|c|

is a∗b^3∗c^3 > a∗b^4∗c^ 2?

Statement 1 : a > b > c (given!)


Also, |a|>|b|>|c|

Here we can say that a, b and c are surely positive.

Therefore, question a∗b^3∗c^3 > a∗b^4∗c^ 2 [in this case] can be rewritten as:

Is c>b ?? to which the answer is a 100% NO

Statement A (Sufficient)


Statement 2 : a +b>0

[also it is given |a|>|b|>|c|]. Here we can confidently say that a is positive.

In this case the question can be rewritten as : is b^3∗c^3 > b^4∗c^ 2?


We know that b^4∗c^ 2 ie (RHS) will always be positive since the powers of both b and c are even. Moreover, the expression b^3∗c^3 will always be less than b^4∗c^ 2 since |b|>|c| and power of b is less in LHS than in RHS.

Hence, statement B is sufficient too.


IMO Option D

Given information: \(|a| > |b| > |c|\)
To determine if: \(a*b^3*c^3 > a*b^4*c^2\)?

(1) \(a > b > c\)

So, a>b>c as well as |a|>|b|>|c|. Based on these 2 conditions, we can infer the following:
1) a cannot be negative (because if a=-1, and a>b>c, then b=-2 and c=-3 for example but then their absolute value inequality condition will not satisfy 1<2<3, not the other way around)
2) b cannot be negative (same reason as above)
3) c can be negative
4) a,b,c can all be positive fractions (negative fractions will not satisfy the original inequality condition of the question stem)

Now that we have established certain possibilities regarding values, we can test values:

a b c \(a*b^3*c^3\) \(a*b^4*c^2\) \(a*b^3*c^3 > a*b^4*c^2\)
3 2 1 24 48 NO
1/2 1/3 1/4 1/3456 1/2592 NO

We can test other similar patterns but result will be the same and we have tested both integers as well as fractions
Negative a,b are not possible and having c as negative will only make \(a*b^3*c^3\) as NEGATIVE and the other one as POSITIVE so again it will be a NO as to if greater or not, hence no point checking that with values
Negative fractions will not satisfy original inequality condition

SUFFICIENT

2) \(a + b > 0\)

So, a+b>0 as well as |a|>|b|>|c|. Based on these 2 conditions, we can infer the following:

1) a cannot be negative (because if a is negative, then b will have to be greater than a for a+b>0 to hold true and that will go against the original inequality condition of the question)
2) b and c can be negative as long as they satisfy original inequality condition

Let us test using some values:

a b c \(a*b^3*c^3\) \(a*b^4*c^2\) \(a*b^3*c^3 > a*b^4*c^2\)
3 2 1 24 48 NO
1/2 1/3 1/4 1/3456 1/2592 NO
5 -3 -2 1080 1620 NO
1 -1/2 -1/3 1/216 1/144 NO

Only b or c as negative is one case not tested with values because it will lead to \(a*b^3*c^3\) as negative and then again NO to the question

SUFFICIENT

Answer - D