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Math Expert V
Joined: 02 Sep 2009
Posts: 58432
If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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Difficulty:   5% (low)

Question Stats: 80% (00:57) correct 20% (01:08) wrong based on 87 sessions

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If $$ab^2 > 0$$ and $$ac < 0$$, then which of the following must be true?

I. $$ab >0$$

II. $$b^2c < 0$$

III. $$a*c^2 > 0$$

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 58432
Re: If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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Bunuel wrote:
If $$ab^2 > 0$$ and $$ac < 0$$, then which of the following must be true?

I. $$ab >0$$

II. $$b^2c < 0$$

III. $$a*c^2 > 0$$

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

Official Solution:

If $$ab^2 > 0$$ and $$ac < 0$$, then which of the following must be true?

I. $$ab >0$$

II. $$b^2c < 0$$

III. $$a*c^2 > 0$$

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

For the product of two numbers to be positive they must have the same sign and since the square of a number is always more than or equal to 0, then $$ab^2 > 0$$ implies that $$a>0$$ and $$b \neq 0$$

Since $$a>0$$, from above, then $$ac < 0$$ implies that $$c < 0$$

So, we have that: $$a > 0$$, $$c < 0$$ and $$b \neq 0$$

Let's check the options:

I. $$ab >0$$. Not true if $$b < 0$$

II. $$b^2c < 0$$. Always true since $$b^2 > 0$$ and $$c < 0$$

III. $$a*c^2 > 0$$. Always true since $$a > 0$$ and $$c^2 > 0$$

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General Discussion
Director  G
Joined: 09 Mar 2018
Posts: 994
Location: India
Re: If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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Bunuel wrote:
If $$ab^2 > 0$$ and $$ac < 0$$, then which of the following must be true?
I. $$ab >0$$

II. $$b^2c < 0$$

III. $$a*c^2 > 0$$

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

IMO D

If $$ab^2 > 0$$ and $$ac < 0$$

From above you can imply the following
a is +ive, b can be +ive or -ive and c is -ive

I. $$ab >0$$, True or False , by using the above

II. $$b^2c < 0$$, Always true

III. $$a*c^2 > 0$$, Always true
_________________
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Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9701
Location: Pune, India
Re: If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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Bunuel wrote:
If $$ab^2 > 0$$ and $$ac < 0$$, then which of the following must be true?

I. $$ab >0$$

II. $$b^2c < 0$$

III. $$a*c^2 > 0$$

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

Given:
$$ab^2 > 0$$ - Since b^2 must be positive only, a would be positive too (to get positive product)

and $$ac < 0$$ - Since a is positive (obtained above), c must be negative to get negative product

So we have 'a' positive, 'c' negative and no idea about 'b'.

I. $$ab >0$$
Since we have no idea about b, we don't know whether ab will be positive or not.

II. $$b^2c < 0$$
b^2 will be positive (note that none of these is 0) and c is negative so product will be negative. This is true.

III. $$a*c^2 > 0$$
a is positive and c^2 will be positive too. So product is positive. This is true too.

_________________
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VP  D
Joined: 31 Oct 2013
Posts: 1465
Concentration: Accounting, Finance
GPA: 3.68
WE: Analyst (Accounting)
Re: If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

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Bunuel wrote:
If $$ab^2 > 0$$ and $$ac < 0$$, then which of the following must be true?

I. $$ab >0$$

II. $$b^2c < 0$$

III. $$a*c^2 > 0$$

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

Given,

$$ab^2>0.$$

a must be positive. a>0. It is not possible to tell whether b is positive or negative as $$b^2>0$$. don't know anything about b.

ac<0. we know that a>0. So, c<0.

I. $$ab >0$$

a>0. don't know anything about b. could be true. MUST not one.

II. $$b^2c < 0$$

b>0 and c<0. Must be true.

III. $$a*c^2 > 0$$

a>0 and $$c^2>0$$. Must be true.

D is the correct answer.
GMAT Club Legend  D
Joined: 18 Aug 2017
Posts: 5031
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: If ab^2 > 0 and ac < 0, then which of the following must be true?  [#permalink]

Show Tags

Bunuel wrote:
If $$ab^2 > 0$$ and $$ac < 0$$, then which of the following must be true?

I. $$ab >0$$

II. $$b^2c < 0$$

III. $$a*c^2 > 0$$

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

for expression ab^2>0 & ac<0 to be valid
a +ve , b -ve or +ve and c -ve

in that case only
case
II. $$b^2c < 0$$

III. $$a*c^2 > 0$$
would be valid
IMO D Re: If ab^2 > 0 and ac < 0, then which of the following must be true?   [#permalink] 23 Jan 2019, 06:40
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If ab^2 > 0 and ac < 0, then which of the following must be true?

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