Is abc < 0?

Is abc < 0?
1) ab>0
2) b/c<0

i) ab have same signs => a = -ve, b=-ve; a=+ve, b=+ve

c can be either positive or negative and thus abc can be either => Not Sufficient

ii) b/c < 0 => b and c have different signs

a can be either positive or negative => Not Sifficient

Combining i) and ii) a and b have same signs and b and c have different signs

I) a = +ve b = +ve c= -ve => abc = -ve
II) a = -ve, b=-ve, c = +ve => abc = +ve

Not Sufficient

Answer - E

1 ab>0
if a is + then b is +, or if a is - the b is -. we can not know if c is - or +. therefore we cannot say abc is - or +
2 b/c<0
if b is + then c is -, or if b is - then c is +. we can not know if a is - or +. therefore we cannot say abc is - or +

let's check both ab>0 and b/c<0
if a is + then b is + and c is -, so abc is -
or if a is - the b is - and c is +, so abc is +. since we cannot have an one answer, E is correct.

Is abc < 0?
1) ab>0
2) b/c<0

option 1) ab> 0 but we do not know about c
c could be >0 or < 0 and hence abc could be <0 or >0

Option 2) b/c<0 which means either one of the two b or c is >0 while other is<0
but still we do not know about a

Combining both ,
b/c<0
let b >0 and c<0
ab>0 hence abc<0

or b<0 and c>0
abc>0

hence we do not get the unique answer

E is the answer

Is abc < 0?
Is abc -ve?
For abc to be negative, any one or all three has to be -ve.

1) ab>0
a and b, both can be +ve or a and b, both can be -ve.
If a and b, both are positive, then abc can or can not be -ve based on sign of c and both are positive, then also abc can or can not be -ve based on sign of c. So, insufficient.

2) b/c<0
Either of b or c is -ve but both can't be -ve at a time.
If b=-ve, then abc can or can not be -ve based on sign of a and c.
If c=-ve, then abc can or can not be -ve based on sign of a and b. So, insufficient.

1) + 2)
1) states that a and b both have same sign
2) states that b and c have opposite sign.
If a (and b) have +ve sign then c must have -ve sign. So, abc is -ve.
If a (and b) have -ve sign then c must have +ve sign. So, abc is +ve.
So, insufficient.

Answer is E.

Is abc < 0?

1) ab>0
case 1- a = +ve , b = +ve
we still dont know about c
if c = -ve then abc < 0
if c = +ve then abc >0
no definite answer

case 2- a = -ve , b = -ve
we still dont know about c
if c = -ve then abc < 0
if c = +ve then abc >0
no definite answer

so this statement is INSUFFICIENT

2) b/c<0
case 1- b = +ve , c = -ve
we still dont know about a
if a = -ve then abc > 0
if a = +ve then abc < 0
no definite answer

case 2- b = -ve , c = +ve
we still dont know about a
if a = -ve then abc > 0
if a = +ve then abc < 0
no definite answer

so this statement is INSUFFICIENT

COMBINING both statements together

case1 - b = +ve , c = -ve , a = +ve
then abc < 0

case 2 - b = -ve , c = +ve , a = -ve
then abc > 0
no definite answer
so, INSUFFICIENT

E is the answer

1) ab>0
that means either a<0 and b<0 or a>0 and b>0
not sufficient

2) b/c<0
thus b and c are of opposite signs
not sufficient

combine both
a and b are of same sign and b and c are of opposite signs
lets c be + thus b and a negative
thus abc>0
not c be - thus b and a positive
thus abc <0

hence both are possible
No sufficient
E

Solution:

Question 9: Is \(abc < 0\)?

1) \(ab>0\)
2) \( \frac{b}{c}< 0\)

In this question, we need to find out whether abc > 0.

First statement: \(ab > 0 \), which implies that a and b carry the same signs, either both are positive or both are negative. However, on the basis on this information alone we cannot find whether \(abc > 0 \) as nothing about c is known. For instance, if c < 0 and it is known that ab > 0 then, abc < 0 and if c > 0 then abc > 0. Thus, statement 1 alone is not sufficient.

Second statement: \( \frac{b}{c}< 0\). In this case, we do not know about a. We only know that b and c have opposite signs. Thus, this information alone is not sufficient to answer the question.

Combining both the statements, we know:

\(ab > 0\) in which a and b are either both positive or both negative.
and \(\frac{b}{c} < 0\)

If a and b are positive then, it implies that c is negative as \(\frac{b}{c }< 0\). Thus, \(abc < 0\) in this case.
However, if a and b are negative then, c is positive. In this case,\( abc > 0 \).

Thus, even two statements together are not sufficient to know whether \(abc < 0\). Hence, Option E is the correct answer.

Statement 1 tells us that (ab) is a positive number
but what about c, c could either be a positive or a negative number.
so (1) is not enough. We are left with options B,C and E.

Statement 2 tells us that b/c < 0
this means that b could be a negative number or c could be a negative number.
both cannot be negative at the same time.
And nothing is mentioned about a,
so this statement is insufficient.
We are left with options C and E.

Combining both statements,
if a=1, b=2 , c=-1
ab>0 b/c<0 , abc <0
a=-2, b=-2, c=1, abc>0

So both the statements are insufficient.
So we are left with E.

Solution:

Statement 1. ab>0, implying a&b are positive, but nothing is given about c which could be > or < 0. Insufficient.
Statement 2. b/c<0 => either b<0 or c<0, so nothing can be deduced about c so Insufficient.


Combining both we get: ab is > 0 => b>0 and c<0 , so yes abc<0.
Both statements are together sufficient.

Answer C.