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# If 1 > 1 - ab > 0, which of the following must be true?

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If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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23 Jun 2015, 22:42
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If 1 > 1 - ab > 0, which of the following must be true?

I. a/b > 0
I. 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

Kudos for a correct solution.

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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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24 Jun 2015, 01:36
4
I am going to go with E.

1>1-ab>0; If we split this into two separate equations (1-ab<1 & 1-ab>0), we get the following solutions - ab>0 & ab<1.

I says that a/b>0. Since ab>0, it means that either a,b both are positive or both are negative. In either scenario I holds true.

III says that ab<1. We already got that result be splitting and solving the inequality.

II says that a/b<1. Let's take two cases - Case 1 (a=2, b=0.2) & Case 2 (a=0.2, b=2). II holds true for Case 2 but not Case 1, and therefore cannot always be true.

Which means that only I and III are always true.
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If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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Updated on: 24 Jun 2015, 09:15
1
1
Step 1: Subtracting 1 from all,
0>-ab>-1.
Step 2: Multiplying by -1, signs are reversed. Therefore we get :
0<ab<1
Therefore, we can conclude :
1. both ab have the same sign
2. ab lies between 0 to 1

Hence E !!

Originally posted by NickHalden on 24 Jun 2015, 01:38.
Last edited by NickHalden on 24 Jun 2015, 09:15, edited 1 time in total.
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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24 Jun 2015, 06:27
3
If 1 > 1 - ab > 0, which of the following must be true?

I. a/b > 0
I. 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

Solution -
a. For the part 1 > 1 - ab -> Subtract -1 from both sides gives -ab < 0 -> ab > 0. Both a and b are positive or negative.
b. For the part 1 - ab > 0 -> Add ab on both sides gives -> ab<1.

I. a/b > 0. This will give us both a and b are positive or negative. Meets the condition a above. Sufficient.
II. a/b < 1. This is opposite of condition a above. In Sufficient.
III. ab < 1. This inequality satisfy the condition b above. Sufficient.

Thanks
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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24 Jun 2015, 09:09
Rearranging the 1>1-ab>0, equation we get
1>1-ab or ab>0 ---1
1-ab>0 or ab<1 -----2
A and b both has to be same sign from equation 1, so a/b will always be >0 buy in some cases it will be >2 or <1 .Hence choose 1
From equation 2 it’s clear that ab<1, hence choose 3.
[This can also be solved by taking example where we have d>a>l , and cross verify ]

Thanks,
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If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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Updated on: 25 Jun 2015, 02:32
Bunuel wrote:
If 1 > 1 - ab > 0, which of the following must be true?

I. a/b > 0
I. 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

Kudos for a correct solution.

1 > 1 - ab > 0 has two parts

1 > 1 - ab and 1 - ab > 0
i.e. ab > 0 and 1 > ab

i.e. 0 < ab < 1

I. a/b > 0 will always be true as a and b must have same sign for ab to be between 0 and 1

II. a/b < 1 will not always be true @a=1/3 and b=1/2

III. ab < 1 will always be true as inferred from the given range of ab

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Originally posted by GMATinsight on 25 Jun 2015, 01:59.
Last edited by GMATinsight on 25 Jun 2015, 02:32, edited 1 time in total.
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Posts: 8257
Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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25 Jun 2015, 02:22
1
GMATinsight wrote:
Bunuel wrote:
If 1 > 1 - ab > 0, which of the following must be true?

I. a/b > 0
I. 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

Kudos for a correct solution.

1 > 1 - ab > 0 has two parts

1 > 1 - ab and 1 - ab > 0
i.e. ab > 0 and 1 > ab

i.e. 0 < ab < 1

I. a/b > 0 will always be true as a and b must have same sign for ab to be between 0 and 1

II. a/b < 1 will not always be true @a=1/3 and b=1/2

III. ab < 1 will not always be true as inferred from the given range of ab

ab<1 for all values and has to be true..
a and b have to have same sign so a/b>0..
ans E
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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25 Jun 2015, 02:35
chetan2u wrote:

ab<1 for all values and has to be true..
a and b have to have same sign so a/b>0..
ans E

Yes. A typo error due to copy paste. But thank you! you deserve a Kudos .
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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26 Jun 2015, 06:53
Bunuel wrote:
If 1 > 1 - ab > 0, which of the following must be true?

I. a/b > 0
I. 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

Kudos for a correct solution.

1 > 1 - ab > 0
Hence 0 > -ab > -1
Hence 0< ab < 1

Hence ab is positive and ab is less than 1
if ab is positive, then a and b both are of same sign and a/b is positive.

Hence statement 1 and statement 3 are correct.

Hence option E is correct.
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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27 Jun 2015, 09:25
1
E
1<1-ab<0 or 0<ab<1
I. a/b > 0 - true as ab>0 then a/b>0
I. a/b < 1 - could be
III. ab < 1 - true
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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27 Jun 2015, 09:46
answer is E.. ab has to be positive, greater than 0 but less than 1
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Posts: 61508
Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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29 Jun 2015, 05:22
Bunuel wrote:
If 1 > 1 - ab > 0, which of the following must be true?

I. a/b > 0
I. 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

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

You can manipulate the original compound inequality as follows, making sure to perform each manipulation on every term:

1 > 1 - ab > 0

0 > -a b > -1 Subtract 1 from all three terms.

0 < ab < 1 Multiply all three terms by -1 and flip the inequality signs.

Therefore you know that 0 < ab < 1. This tells you that ab is positive, so a/b must be positive (a and b have the same sign). Therefore, I must be true. However, you do not know whether a/b < 1, so II is not necessarily true. But you do know that ab must be less than 1, so III must be true.

Therefore, the correct answer is (E).
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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30 Jun 2015, 07:15
1
If 1 > 1 - ab > 0, which of the following must be true?

I. a/b > 0
I. a/b < 1
III. ab < 1

If you first deal with the right half of the inequality, you can add ab to both sides to get ab<1. Since III is true, you can eliminate A, B, and D.

Now test I. Looking at the left side of the inequality, add ab to both sides and subtract 1 from both sides to yield ab>0. For ab to be positive, a and b must have the same signs. This will also be true if we divide a by b. Statement I is also true.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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30 Jun 2015, 08:46
0 < 1-ab < 1, since (1-ab) is both positive and less than 1, 0<ab<1, so III is true. For (ab) to be less than 1 and greater than 0 the fraction a/b must be greater than 0 (must be positive), so I is also true. Ans is E?
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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15 Jul 2015, 08:19
Hello Bunuel ,
While solving this problem , i reached up till this inequality ( 0<ab<1) .
However , statement III above says that ab<1 .
This covers a lot many more numbers, which wont be satisfied by the inequality
provided in the stem ( e.g since ab<1 , it will also mean that ab =-3).
Should this not be reason good enough to eliminate statement III (as
this has provided us with values that are unable to satisfy the stem) ?

PS: If the question had asked , "which of the following might be true ?",
then,yes, we could still include statement III.
I am unable to understand this.
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Posts: 61508
Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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15 Jul 2015, 08:27
fetchnitin wrote:
Hello Bunuel ,
While solving this problem , i reached up till this inequality ( 0<ab<1) .
However , statement III above says that ab<1 .
This covers a lot many more numbers, which wont be satisfied by the inequality
provided in the stem ( e.g since ab<1 , it will also mean that ab =-3).
Should this not be reason good enough to eliminate statement III (as
this has provided us with values that are unable to satisfy the stem) ?

PS: If the question had asked , "which of the following might be true ?",
then,yes, we could still include statement III.
I am unable to understand this.

We have that 0 < ab < 1. Now, let me asks you is ab < 1 true?
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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15 Jul 2015, 08:56
Hello,
yes,this is true .ab<1 .
But my confusion is that it also contains values
that are outside the range of the inequality in the
question stem .
Is it that, while solving these type of questions all
the values of the question stem should be tried and checked
with the options and not the other way round ( i.e all values of the
options should fit the range of stem ). If so ,then could u explain to me
the logic behind this, as i am unable to visualize this .

Thanks
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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15 Jul 2015, 09:00
Hello Bunuel,
yes,this is true .ab<1 .
But my confusion is that it also contains values
that are outside the range of the inequality in the
question stem .
Is it that, while solving these type of questions all
the values of the question stem should be tried and checked
with the options and not the other way round ( i.e all values of the
options should fit the range of stem ). If so ,then could u explain to me
the logic behind this, as i am unable to visualize this .

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 61508
Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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15 Jul 2015, 09:03
fetchnitin wrote:
Hello Bunuel,
yes,this is true .ab<1 .
But my confusion is that it also contains values
that are outside the range of the inequality in the
question stem .
Is it that, while solving these type of questions all
the values of the question stem should be tried and checked
with the options and not the other way round ( i.e all values of the
options should fit the range of stem ). If so ,then could u explain to me
the logic behind this, as i am unable to visualize this .

Thanks

I think that you should practice Must or Could be true questions more: search.php?search_id=tag&tag_id=193

Hope it helps.
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Re: If 1 > 1 - ab > 0, which of the following must be true?  [#permalink]

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24 Feb 2018, 07:25
Top Contributor
Bunuel wrote:
If 1 > 1 - ab > 0, which of the following must be true?

I. a/b > 0
I. 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

Kudos for a correct solution.

GIVEN: 1 > 1 - ab > 0
Multiply all 3 sides by -1 to get: -1 < -1 + ab < 0 [since I multiplied by a NEGATIVE number, I had to REVERSE the inequality symbols]
Add 1 to all 3 sides to get: 0 < ab < 1

First, if 0 < ab, then a and b are the SAME SIGN, which means a/b > 0
So, statement I is TRUE
ELIMINATE B and C

Second, our new inequality clearly tells us that ab < 1
So, statement III is TRUE
ELIMINATE A and D

So, we need not even check whether statement II is true.

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Re: If 1 > 1 - ab > 0, which of the following must be true?   [#permalink] 24 Feb 2018, 07:25

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