pushpitkc wrote:

If a<b and c<b, is abc<a?

1. ab < 0

2. ac > 0

Source:

Experts GlobalIMO C.

I will approach this question with cases approach!Q: Is abc < a ?

based on sign of a we can simplify the question further... which will help us in the cases.Case 1) a > 0 : then the Q : bc < 1 ? ( cancelling a )

Case 2) a < 0 : then the Q : bc > -1 ? ( cancelling negative a )

1. ab < 0

Case 1)

not possible here as b > a : so if a > 0 ... b also > 0 and ab < 0 not possible.

Case 2) a < 0. If ab < 0 then b > 0.

Our Q for case 2 is : Is bc > -1 ? we do not know if c > 0 or not.

Hence 1. is Insufficient!2. ac > 0

Case 1) a > 0 . If ac > 0 then c > 0. Since we know b > c... hence b > 0.

Q : is bc < 1 ? could be either yes or no depending on magnitude of b & c.

Case 2) a < 0. If ac > 0 then c < 0. Since we know b > c ... b could be positive or negative.

Our Q: is bc > -1? could be yes or no depending on if b is positive or negative.

Hence 2. is insufficient.(1) + (2) together...

Case 1) a > 0 is not possible as ab < 0. and b > a.

Case 2) a must be less than zero.

Our question is : Is bc > -1? ac > 0 hence c > 0 & b > c hence b >0

hence bc > 0.

Hence the answer to the question in this case is "YES" always. This means both 1. and 2. together are sufficient to answer.

Hence Option (C) is our answer.

Best,

Gladi