vinni.k wrote:*source:- MGMAT
If 1 > 1 - ab > 0, which of the following must be true ?
I. a/b > 0
II. a/b < 1
III. ab < 1
(A). I only
(B). II only
(C). III only
(D). I and II only
(E). I and III only
Answer is E
Your inputs will be helpful.
Thanks & Regards
Vinni
Hi Vinni!
Step 1 is to rearrange the stem inequality into something a bit easier to look at (it would be easier to understand the qualities of ab if we have it isolated)
First let's subtract a 1 from all sides (remember - there are 3 sides here!):
1 > 1 - ab > 0
0 > -ab > -1
Now, we can multiply through by -1 to get a positive value for the product ab (don't forget to flip the signs!):
0 < ab < 1
So now we know that the product ab is (a) positive (so must have the same sign - both positive or both negative) and (b) a proper fraction (between 0 and 1).
Now we can test the choices, starting with the "easiest" based on our rephrase of the question:
III. ab < 1
- We know explicitly that ab is less than 1, so this must be TRUE. Eliminate choices A, B, and D.
**Because our choices are now Either I & III or III only, we don't even need to test the 2nd bullet point (I will later for completion).
I. a/b > 0
- If the signs on a and b are the same, then BOTH the product AND the quotient will always be positive (positive divided by positive = positive, and negative divided by negative = positive). TRUE.
Therefore the answer is
BOTH I and III, so E.
But for completeness, let's check bullet point II.
II. a/b < 1
- There are 2 cases because we do not know the signs on a and b.
(A) If a and b are both positive, we can multiply both sides by b and see that this bullet is saying a<b. We only know that the product of the 2 is a fraction, we cannot know the relative sizes of a and b so we cannot know if this is true.
(B) If a and b are both negative, we can multiply both sides by b AND flip the sign (because b is negative). We now see that this would be asking if a>b? Again, we only know that the product of the 2 is a fraction but we have no way of knowing the relative sizes of a and b - we cannot know if this will be true.
Therefore bullet II may not be true.
Hope this clears it up!!
Whit
