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# If a and b are positive, is (a^(-1) + b^(-1))^(-1) less than (a^(-1)*b

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If a and b are positive, is (a^(-1) + b^(-1))^(-1) less than (a^(-1)*b [#permalink]

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18 Dec 2010, 14:01
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If a and b are positive, is is $$(a^{-1}+b^{-1})^{-1}$$ less than $$(a^{-1}*b^{-1})^{-1}$$?

(1) a = 2b
(2) a + b > 1
[Reveal] Spoiler: OA

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Ajit

Last edited by Bunuel on 08 Oct 2017, 08:46, edited 1 time in total.
Renamed the topic and edited the question.

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Re: If a and b are positive, is (a^(-1) + b^(-1))^(-1) less than (a^(-1)*b [#permalink]

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18 Dec 2010, 14:16
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ajit257 wrote:
If a and b are positive, is (a-1 + b-1)-1 less than (a-1b-1)-1?
(1) a = 2b
(2) a + b > 1

Question: is $$(a^{-1}+b^{-1})^{-1}<(a^{-1}*b^{-1})^{-1}$$? --> $$(\frac{1}{a}+\frac{1}{b})^{-1}<(\frac{1}{ab})^{-1}$$ --> $$\frac{ab}{a+b}<ab$$, as $$a$$ and $$b$$ are positive we can reduce by $$ab$$ and finally question becomes: is $$a+b>1$$?

(1) a = 2b --> is $$3b>1$$ --> is $$b>\frac{1}{3}$$, we don't know that, hence this statement is not sufficient.
(2) a + b > 1, directly gives an answer. Sufficient.

P.S. ajit257 you should type the question so that it's clear which is an exponent, which is subtraction, and so on.
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Re: If a and b are positive, is (a^(-1) + b^(-1))^(-1) less than (a^(-1)*b [#permalink]

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12 Dec 2013, 05:10
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Bunuel wrote:
ajit257 wrote:
If a and b are positive, is (a-1 + b-1)-1 less than (a-1b-1)-1?
(1) a = 2b
(2) a + b > 1

Question: is $$(a^{-1}+b^{-1})^{-1}<(a^{-1}*b^{-1})^{-1}$$? --> $$(\frac{1}{a}+\frac{1}{b})^{-1}<(\frac{1}{ab})^{-1}$$ --> $$\frac{ab}{a+b}<ab$$, as $$a$$ and $$b$$ are positive we can reduce by $$ab$$ and finally question becomes: is $$a+b>1$$?

(1) a = 2b --> is $$3b>1$$ --> is $$b>\frac{1}{3}$$, we don't know that, hence this statement is not sufficient.
(2) a + b > 1, directly gives an answer. Sufficient.

P.S. ajit257 you should type the question so that it's clear which is an exponent, which is subtraction, and so on.

Hey Bunuel,

once again, this is a little bit fast for me.

I follow your first and third step to reduce the question, but I don't get the second.

I'd explained myself that $$(\frac{1}{ab})^{-1}$$ = $$1*(\frac{ab}{1})$$ so we have ab on the right side.

But I don't follow what you did do reduce the left side. Could you explain in detail?

Thank you!

Kudos [?]: 46 [1], given: 19

Math Expert
Joined: 02 Sep 2009
Posts: 42356

Kudos [?]: 133204 [0], given: 12439

Re: If a and b are positive, is (a^(-1) + b^(-1))^(-1) less than (a^(-1)*b [#permalink]

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12 Dec 2013, 05:14
unceldolan wrote:
Bunuel wrote:
ajit257 wrote:
If a and b are positive, is (a-1 + b-1)-1 less than (a-1b-1)-1?
(1) a = 2b
(2) a + b > 1

Question: is $$(a^{-1}+b^{-1})^{-1}<(a^{-1}*b^{-1})^{-1}$$? --> $$(\frac{1}{a}+\frac{1}{b})^{-1}<(\frac{1}{ab})^{-1}$$ --> $$\frac{ab}{a+b}<ab$$, as $$a$$ and $$b$$ are positive we can reduce by $$ab$$ and finally question becomes: is $$a+b>1$$?

(1) a = 2b --> is $$3b>1$$ --> is $$b>\frac{1}{3}$$, we don't know that, hence this statement is not sufficient.
(2) a + b > 1, directly gives an answer. Sufficient.

P.S. ajit257 you should type the question so that it's clear which is an exponent, which is subtraction, and so on.

Hey Bunuel,

once again, this is a little bit fast for me.

I follow your first and third step to reduce the question, but I don't get the second.

I'd explained myself that $$(\frac{1}{ab})^{-1}$$ = $$1*(\frac{ab}{1})$$ so we have ab on the right side.

But I don't follow what you did do reduce the left side. Could you explain in detail?

Thank you!

Sure.

$$(\frac{1}{a}+\frac{1}{b})^{-1}$$;

$$(\frac{b+a}{ab})^{-1}$$;

$$\frac{ab}{b+a}$$.

Does this make sense?
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Re: If a and b are positive, is (a^(-1) + b^(-1))^(-1) less than (a^(-1)*b [#permalink]

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12 Dec 2013, 06:29
Bunuel wrote:

Sure.

$$(\frac{1}{a}+\frac{1}{b})^{-1}$$;

$$(\frac{b+a}{ab})^{-1}$$;

$$\frac{ab}{b+a}$$.

Does this make sense?

Yeah, now I see it. Guess my head was just overloaded with math --> it's really clear now! Thanks!

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Re: If a and b are positive, is (a^(-1) + b^(-1))^(-1) less than (a^(-1)*b [#permalink]

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08 Oct 2017, 05:41
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Re: If a and b are positive, is (a^(-1) + b^(-1))^(-1) less than (a^(-1)*b   [#permalink] 08 Oct 2017, 05:41
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