If ab<0 and bc>0, then which of the following expressions must be nega

If ab<0 and bc>0, then which of the following expressions must be negative?

I. abc
II. ab*2
III. a\(b^2\)*c

Bunuel I have a doubt for the II option.
II option says [ab*2 not a\(b^2\)]
Solving option II;
ab*2 = a*b*2 or a x b x 2

From the question, a and b have opposite signs. The expression is negative in both cases.
ab*2
-a*b*2 = -2ab (when a is negative)
a*-b*2 = -2ab (when b is negative)

I got Answer D (Both II and III are negative)...
Pls Let me know if there's any mistake in my approach.

_________________
Kindly press "+1 Kudos" to appreciate

ab<0 means a & b have opposite signs
bc>0 means b & c have same signs

Combining the two above, a is opposite in sign to both b and c.
So two cases: Either a >0 and b<0, c<0
Or a<0, b>0, c>0

abc is positive in first case
ab^2 is also positive in first case
But ab^2c is negative in both cases (because b^2 is always positive and a/c have opposite signs)

Hence answer is B

Edited the question. II should read \(ab^2\). Thank you.

Thank you Bunuel.
Now I get Answer B (only III is negative).

_________________
Kindly press "+1 Kudos" to appreciate

a and b is negative of each other.
c and b is positive of each other.

So c and a is negative of each other.

Answer is B

While this problem could be solved by considering individual scenarios for the signs of each variable, it is much easier if you simplify/”decouple” the three possibilities so that you can leverage the information provided in the question stem more easily.



The expression in statement III is the easiest to assess in its rewritten form. ab is given as negative in the question stem and bc is given as positive in the question stem and negative times positive is always negative. This guarantees that the expression in statement III is always negative and allows you to eliminate answers (A) and (E).

For the expression in statement I you know that ab is negative but you do not know the sign of c it could be positive or negative. Therefore the expression in statement 1 does not have to be negative. In statement II you again know that ab is negative but you do not know the sign of b it could be positive or negative. The expression in statement II therefore does not have to be negative. The correct answer is (B).

Variables a b c ab<0 bc>0 abc ab^2 acb^2
Case1 + - - - + + + -
Case2 - + + - + - - -

Since ab is less than zero and bc is greater than zero, we have:

Case 1: If b is negative, then a is positive and c is negative.

Case 2: If b is positive, then a is negative and c is positive.

Thus, we see that abc does not have to be negative (if the values of a, b and c are from case 1) and ab^2 does not have to be negative (if the values of a and b are from case 1).

However, since the product of a and c is always negative (in either case) and since b^2 is always positive, ab^2 * c is always negative.

Answer: B