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

Krk wrote:Analyzing the options available:
1. a/b >0
This means (a/b) can be any positive value above 0.
This ensures that a and b are positive.
This must be true.[/spoiler]

Be careful Krk - if we know that a/b>0, then we know that they are both EITHER positive OR negative. We are not ensured that they are each positive. But, thankfully we do not need them to both be positive. If we rearrange the original (see my earlier post), we only know that the product (ab) is between 0 and 1 - or positive. Knowing that a and b have the same sign means that their quotient will also be positive - so (I) must be true.