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M11-26

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Bunuel
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M11-26 [#permalink] New post  16 Sep 2014, 00:45
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35% (01:26) correct 65% (01:07) wrong based on 318 sessions
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Is \(a^7*b^2*c^3 \gt 0\)?



(1) \(bc \lt 0\)

(2) \(ac \gt 0\)
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Bunuel
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Re M11-26 [#permalink] New post  16 Sep 2014, 00:45
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Is \(a^7*b^2*c^3 \gt 0\)?

For the inequality \(a^7*b^2*c^3 > 0\) to hold true, the following conditions must be satisfied:

(i) \(a\) and \(c\) should be either both positive or both negative: in this case \(a^7*c^3\) will be positive;

(ii) \(b\) should not be zero: if \(b = 0\), then the entire expression (\(a^7*b^2*c^3\)) will be 0, not greater than 0, irrespective of the values of \(a\) and \(c\).

(1) \(bc \lt 0\).

From this, we know that \(b \ne 0\). However, we do not have information about \(a\) and \(c\). Therefore, this is not sufficient.

(2) \(ac \gt 0\).

This implies that \(a\) and \(c\) are either both positive or both negative. But, we don't know about \(b\): if \(b=0\), then the expression will be equal to 0. Thus, this is also not sufficient.

(1)+(2) We have both necessary conditions met: \(b \ne 0\) and \(a\) and \(c\) have the same sign. Sufficient.


Answer: C
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Senthil7
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Re: M11-26 [#permalink] New post  20 Aug 2016, 04:48
I think this is a high-quality question and I agree with explanation.
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asricharan
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Re: M11-26 [#permalink] New post  16 Jan 2017, 11:37
I think this is a high-quality question and I agree with explanation.
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Bunuel
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Re: M11-26 [#permalink] New post  20 May 2023, 10:51
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Re: M11-26 [#permalink] New post  20 May 2023, 11:49
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Bunuel wrote:
Is \(a^7*b^2*c^3 \gt 0\)?



(1) \(bc \lt 0\)

(2) \(ac \gt 0\)



The conditions that should be satisfied
a) None should be 0
b) ac >0

Thus, we are looking for:
a) If any of the three are 0, then answer is NO.
b) If none are 0, then we look for whether ac >0.

(1) \(bc \lt 0\)
Just tells us that b and c are of opposite sign and are not 0.
Nothing about a.
Insufficient

(2) \(ac \gt 0\)
This tells us that ac >0, but we do not know whether b is 0 or otherwise.
Insufficient


Combined
Both conditions met and answer is YES.


C
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Re: M11-26 [#permalink] New post  26 May 2023, 09:02
To solve this Data Sufficiency question we need to show that 1) a, b and c are all different from 0, 2) a and c have opposite signs

Let's evaluate each statement separately:
Statement (1) is insufficient to answer but tells us that b is not equal to zero.
Statement (2) is also insufficient alone, but enables us to determine that a and c have opposite signs

You need the combination of these 2 statements to answer this question. The right answer is (C)
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Re: M11-26 [#permalink] New post  13 Jul 2023, 02:00
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a7 ∗ b2 ∗ c3 = (ac)3 ∗ a4 * b2
Since ac > 0
and a4 * b2 will be > 0 (product of square of a2 and b)
Statement B is sufficient by itself.
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Bunuel
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Re: M11-26 [#permalink] New post  13 Jul 2023, 02:12
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sarakuma wrote:
a7 ∗ b2 ∗ c3 = (ac)3 ∗ a4 * b2
Since ac > 0
and a4 * b2 will be > 0 (product of square of a2 and b)
Statement B is sufficient by itself.


You should read solutions more carefully:
    (2) \(ac \gt 0\).

    This implies that \(a\) and \(c\) are either both positive or both negative. But, we don't know about \(b\): if \(b=0\), then the expression will be equal to 0. Thus, this is also not sufficient.

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Re: M11-26 [#permalink]
13 Jul 2023, 02:12
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