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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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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, 02:38.
Last edited by NickHalden on 24 Jun 2015, 10:15, 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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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]
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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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

Originally posted by GMATinsight on 25 Jun 2015, 02:59.
Last edited by GMATinsight on 25 Jun 2015, 03:32, 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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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]
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]
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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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]
answer is E.. ab has to be positive, greater than 0 but less than 1
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Re: If 1 > 1 - ab > 0, which of the following must be true? [#permalink]
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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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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]
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]
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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Re: If 1 > 1 - ab > 0, which of the following must be true? [#permalink]
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]
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]
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
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Re: If 1 > 1 - ab > 0, which of the following must be true? [#permalink]
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.
Re: If 1 > 1 - ab > 0, which of the following must be true? [#permalink]
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