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Math Expert V
Joined: 02 Sep 2009
Posts: 55273
If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 59% (02:24) correct 41% (02:29) wrong based on 279 sessions

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If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?

(1) |a| + |b| = a + b

(2) a > b

Kudos for a correct solution.

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Manager  Joined: 22 Oct 2014
Posts: 88
Concentration: General Management, Sustainability
GMAT 1: 770 Q50 V45 GPA: 3.8
WE: General Management (Consulting)
Re: If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?  [#permalink]

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3
Statement (1): From the information given, we can conclude that both, a and b are not negative. The two absolute values must be positive (the question stem tells us they are not 0), so a + b must also be positive. As the absolute values have the same sign, a and be must also have the same sign, otherwise their sum would not be equal to the sum of their absolute values.

Picking a few number pairs, we quickly realize that that information is sufficient. This also makes sense theoretically. Going from $$\frac{1}{a}$$ to $$\frac{1}{(a+b)}$$ will always decrease the term for positive a and b, while going from $$\frac{1}{a}$$ to $$\frac{1}{a}+\frac{1}{b}$$ will always increase the term.

So statement 1 is sufficient.

Statement (2): Here we can easily show that this is not sufficient by picking numbers. Choose e.g. 2 and 2, so we get $$\frac{1}{4}<1$$, which is obviously true. But picking 2 and -1 will give $$1<\frac{1}{2}-1$$, which is obviously wrong.

Therefore, answer A is correct.
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Intern  Joined: 06 Jul 2011
Posts: 8
Re: If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?  [#permalink]

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1
Lets Consider stmt1:
From stmt1 we can say that both a and b are positive. Then only |a| + |b| = a + b .
now form the question we have to prove that
1/(a+b) < 1/a + 1/b
or we can rephrase that we have to show that
[ab-(a+b)^2]/ab(a+b) < 0

In this case denominator cannot be less than zero as since a and b are positive, ab has to be positive and so do a + b
so the numerator has to be less than zero
so ab-(a+b)^2 < 0
which comes down to a^2 + b ^2 > -ab
which will always be true.

We are able to derive a conclusion using statement 1 .

If we consider 2. Straight away we can say that it's not sufficient.

Hence Option A is the answer.
Director  D
Joined: 05 Mar 2015
Posts: 998
If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?  [#permalink]

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littlewarthog wrote:
Statement (1): From the information given, we can conclude that both, a and b are not negative. The two absolute values must be positive (the question stem tells us they are not 0), so a + b must also be positive. As the absolute values have the same sign, a and be must also have the same sign, otherwise their sum would not be equal to the sum of their absolute values.

Picking a few number pairs, we quickly realize that that information is sufficient. This also makes sense theoretically. Going from $$\frac{1}{a}$$ to $$\frac{1}{(a+b)}$$ will always decrease the term for positive a and b, while going from $$\frac{1}{a}$$ to $$\frac{1}{a}+\frac{1}{b}$$ will always increase the term.

So statement 1 is sufficient.

Statement (2): Here we can easily show that this is not sufficient by picking numbers. Choose e.g. 2 and 2,so we get $$\frac{1}{4}<1$$, which is obviously true. But picking 2 and -1 will give $$1<\frac{1}{2}-1$$, which is obviously wrong.

Therefore, answer A is correct.

1) a and B has to be positive forIaI+IbI=a+b to be correct .
so any posive value will satisfy the question.
sufficient.
2) a>b

consider both to be positive in 1st case and negative in 2nd case ,we get different answers.
a=2 & b=1 .then yes.
a=-2 & b=-3 then no.
Not sufficient.

Ans A.
Manager  B
Joined: 03 Dec 2014
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Location: India
Concentration: General Management, Leadership
GMAT 1: 620 Q48 V27 GPA: 1.9
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Re: If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?  [#permalink]

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Bunuel wrote:
Bunuel wrote:
If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?

(1) |a| + |b| = a + b

(2) a > b

Kudos for a correct solution.

The correct answer is A.

when I rearrange the question stem, I get:
ab<(a+b)^2

in above situation, both stems seem sufficient. Is my approach correct? Please help.
Math Expert V
Joined: 02 Aug 2009
Posts: 7685
Re: If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?  [#permalink]

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2
robu wrote:
Bunuel wrote:
Bunuel wrote:
If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?

(1) |a| + |b| = a + b

(2) a > b

Kudos for a correct solution.

The correct answer is A.

when I rearrange the question stem, I get:
ab<(a+b)^2

in above situation, both stems seem sufficient. Is my approach correct? Please help.

Hi robu,
You cannot cancel out term on both sides by cross multiplication..
you get the two terms on the same side and then simplify..
1/(a+b)< 1/a + 1/b..
1/(a+b)< (a+b)/ab..
1/(a+b)- (a+b)/ab <0..
{ab-(a+b)^2}/ab(a+b)<0...
now you see, you have two situations..
where {ab-(a+b)^2}<0 and ab(a+b)>0...
and where {ab-(a+b)^2}>0 and ab(a+b)<0...

you can furhter simplify, but your observation may not be correct, when you are dealing with inequality and variables..
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Director  D
Joined: 05 Mar 2015
Posts: 998
Re: If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?  [#permalink]

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when I rearrange the question stem, I get:
ab<(a+b)^2

in above situation, both stems seem sufficient. Is my approach correct? Please help.[/quote]

it means we can not cross multiply in inequalities question? please clarify.[/quote]

robu

one can cross multiply if one is sure about signs of both sides.
but in my opinion it is better not to cross multiply....
Intern  Joined: 14 Jun 2016
Posts: 17
Location: United States
GMAT 1: 700 Q48 V37 WE: Engineering (Pharmaceuticals and Biotech)
If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?  [#permalink]

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1
A.

(1) |a| + |b| = a + b tells us that both values a and b are positive; otherwise, |a| + |b| ≠ a + b if any value is a negative. Therefore, knowing the variables are positive, you can manipulate the equation. 1/(a+b) < (a+b)/ab. Cross multiply and since numerator values are positive, direction of sign is maintained: ab<(a+b)(a+b). Here you can see that the right side will become a^2 +2ab+b^2, definitely greater than ab.

(2) a>b, too many scenarios if a is positive b is negative, or both positive, and one positive one negative.
Manager  B
Joined: 22 Feb 2016
Posts: 87
Location: India
Concentration: Economics, Healthcare
GMAT 1: 690 Q42 V47 GMAT 2: 710 Q47 V39 GPA: 3.57
Re: If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?  [#permalink]

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1
VERY SIMPLE AND LUCID ANSWER

what is given?
AB= 0
hence both A and B are non zero but they can be both positive and negative that we don't know

and a and b also do not have the same value, They are distinct numbers.

Look at statement 1 |A|+|B|=A+B ie, the distance of A and B is positive. Hence A and B are positive numbers. Pick any positive number you will get a strict YES or NO answer.
SUFFICIENT

Statement 2: Does not give us any idea about the sign of the variable it just shows a comparision. NS

A is the answer:)

Experts Do let me know if I am correct. I opt simple ways to save time.
Intern  B
Joined: 05 Jan 2019
Posts: 22
Location: Spain
GMAT 1: 740 Q49 V42 Re: If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?  [#permalink]

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Hi Bunuel

Could you enlighten us with your wisdom in this question?

I cannot elucidate a reliable strategy to get the answer in less than 2:30 mins.

Thanks! Re: If ab ≠ 0 and a + b ≠ 0, is 1/(a+b)< 1/a + 1/b ?   [#permalink] 18 Mar 2019, 03:37
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