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# If a,b,c,d are integers such that both (-2a) * (−3c) * b and

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If a,b,c,d are integers such that both (-2a) * (−3c) * b and [#permalink]
naumyuk wrote:
If A, B, C, and D are integers such that both $$(-2A) * (−3C) * B$$ and $$D * (-2B) * 5A$$ are positive, which of the following expressions is always negative?

1. AxB
2. BxC
3. CxD
4. AxD
5. AxC

Don't pay attention to quotients. They don't have any influence.

To make the expressions easier for understanding rewrite them in the following way:

Expression abc = positive
Expression abd = negative

In order to be positive and negative correspondingly 'abc' and 'abd' components have to have following set of signs:
abc abd
+++ ++-
+-- +-+
-+- -++
--+ ---

So, we can see that for each pair c and d always have different signs. Therefore for all possible options c*d must be negative.

Originally posted by naumyuk on 25 Jan 2016, 09:50.
Last edited by ENGRTOMBA2018 on 25 Jan 2016, 09:53, edited 2 times in total.
Edited the typo and reformatted the solution.
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Re: If a,b,c,d are integers such that both (-2a) * (−3c) * b and [#permalink]
Nice question
Since (-2a)*(-3c)*b> 0 => 6*a*b*c>0 = > a*b*c>0----1
d*(-2b)*(5a)>0 => -10*d*b*a>0- ==> d*b*a<0 ----2

Now let us look at the choices
A and B -- a*b and b*c . Well a*b can be positive in which case c>0 and a*b<0 in which case c<0 . The same thing applies to choice B and hence both are not correct choices
> 0
Let us look at D and E - a*d and a*c respectively. For reasons noted above if b<0 then a*c<0 but if b>0 then a*c >0. Also a*d for the same reason can be +ve and -ve.
Which brings us to C. At this stage we can check this answer and go.. But let us see what is going on

if c<0 then from inequality (1) a*b*c> 0 ==> that a*b <0 . From inequality (2) d*b*a<0 we see that if a*b<0 then d>0 ===> c*d<0
if c>0 then a*b>0 . From (2) d<0 ==> again c*d<0 so answer choice C!
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Re: If a,b,c,d are integers such that both (-2a) * (−3c) * b and [#permalink]
naumyuk wrote:
If a,b,c,d are integers such that both $$(-2a) * (−3c) * b$$ and $$d * (-2b) * 5a$$ are positive, which of the following expressions is always negative?

A) a*b
B) b*c
C) c*d
D) a*d
E) a*c

The given expressions are (ignoring co-efficients):
abc is positive.
-abd is positive.

Note that both have ab common so whatever sign that brings, will be same in both products. If c and d had the same signs, abc and abd would have the same sign so abc and -abd would have opposite signs.
So c and d must have opposite signs so that both abc and -abd have the same sign (positive).

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Re: If a,b,c,d are integers such that both (-2a) * (3c) * b and [#permalink]
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Re: If a,b,c,d are integers such that both (-2a) * (3c) * b and [#permalink]
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