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# If n and k are positive integers, is (n+k)^1/2>2n^1/2?

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Manager
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If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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26 Dec 2012, 05:31
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If n and k are positive integers, is $$\sqrt{n+k}>2\sqrt{n}$$

(1) k > 3n
(2) n + k > 3n
[Reveal] Spoiler: OA
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Joined: 02 Sep 2009
Posts: 39640
Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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26 Dec 2012, 05:36
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If n and k are positive integers, is $$\sqrt{n+k}>2\sqrt{n}$$?

Both parts of the inequality are positive, thus we can square it, to get "is $$n+k>4n$$?" --> is $$k>3n$$?

(1) k > 3n. Sufficient.

(2) n + k > 3n --> $$k>2n$$. Not sufficient.

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If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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27 Jun 2014, 02:36
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Bunuel wrote:
If n and k are positive integers, is $$\sqrt{n+k}>2\sqrt{n}$$?

Both parts of the inequality are positive, thus we can square it, to get "is $$n+k>4n$$?" --> is $$k>3n$$?

(1) k > 3n. Sufficient.

(2) n + k > 3n --> $$k>2n$$. Not sufficient.

Hi Bunuel,

I don't understand why n & k are positive then two parts of the inequility $$\sqrt{n+k}>2\sqrt{n}$$ are positive.

What I understand is that: n, k>0 => n+k>0 => $$\sqrt{n+k}$$ might be positive or negative. e.g: x=9>0 --> $$\sqrt{x}$$ = 3 or -3
Same thought or n!

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Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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27 Jun 2014, 02:53
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LucyDang wrote:
Bunuel wrote:
If n and k are positive integers, is $$\sqrt{n+k}>2\sqrt{n}$$?

Both parts of the inequality are positive, thus we can square it, to get "is $$n+k>4n$$?" --> is $$k>3n$$?

(1) k > 3n. Sufficient.

(2) n + k > 3n --> $$k>2n$$. Not sufficient.

Hi Bunuel,

I don't understand why n & k are positive then two parts of the inequility $$\sqrt{n+k}>2\sqrt{n}$$ are positive.

What I understand is that: n, k>0 => n+k>0 => $$\sqrt{n+k}$$ might be positive or negative. e.g: x=9>0 --> $$\sqrt{x}$$ = 3 or -3
Same thought or n!

When the GMAT provides the square root sign for an even root, such as $$\sqrt{x}$$ or $$\sqrt[4]{x}$$, then the only accepted answer is the positive root.

That is, $$\sqrt{9}=3$$, NOT +3 or -3. In contrast, the equation $$x^2=9$$ has TWO solutions, +3 and -3. Even roots have only non-negative value on the GMAT.

Odd roots will have the same sign as the base of the root. For example, $$\sqrt[3]{125} =5$$ and $$\sqrt[3]{-64} =-4$$.

Hope it helps.
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Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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27 Jun 2014, 05:06
Bunuel wrote:
LucyDang wrote:
Bunuel wrote:
If n and k are positive integers, is $$\sqrt{n+k}>2\sqrt{n}$$?

Both parts of the inequality are positive, thus we can square it, to get "is $$n+k>4n$$?" --> is $$k>3n$$?

(1) k > 3n. Sufficient.

(2) n + k > 3n --> $$k>2n$$. Not sufficient.

Hi Bunuel,

I don't understand why n & k are positive then two parts of the inequility $$\sqrt{n+k}>2\sqrt{n}$$ are positive.

What I understand is that: n, k>0 => n+k>0 => $$\sqrt{n+k}$$ might be positive or negative. e.g: x=9>0 --> $$\sqrt{x}$$ = 3 or -3
Same thought or n!

When the GMAT provides the square root sign for an even root, such as $$\sqrt{x}$$ or $$\sqrt[4]{x}$$, then the only accepted answer is the positive root.

That is, $$\sqrt{9}=3$$, NOT +3 or -3. In contrast, the equation $$x^2=9$$ has TWO solutions, +3 and -3. Even roots have only non-negative value on the GMAT.

Odd roots will have the same sign as the base of the root. For example, $$\sqrt[3]{125} =5$$ and $$\sqrt[3]{-64} =-4$$.

Hope it helps.

I got it, thank you!!
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Joined: 07 Aug 2014
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Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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19 Aug 2014, 07:01
Bunuel wrote:
If n and k are positive integers, is $$\sqrt{n+k}>2\sqrt{n}$$?

Both parts of the inequality are positive, thus we can square it, to get "is $$n+k>4n$$?" --> is $$k>3n$$?

(1) k > 3n. Sufficient.

(2) n + k > 3n --> $$k>2n$$. Not sufficient.

How do you get to the conclusion that $$\sqrt{n+k}>2\sqrt{n}$$? Even if k > 3n it can be either n+k > 4n or n+k < 4n, since we don't know more about n.

Thanks .
Math Expert
Joined: 02 Sep 2009
Posts: 39640
Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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19 Aug 2014, 08:06
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Expert's post
tobiasfr wrote:
Bunuel wrote:
If n and k are positive integers, is $$\sqrt{n+k}>2\sqrt{n}$$?

Both parts of the inequality are positive, thus we can square it, to get "is $$n+k>4n$$?" --> is $$k>3n$$?

(1) k > 3n. Sufficient.

(2) n + k > 3n --> $$k>2n$$. Not sufficient.

How do you get to the conclusion that $$\sqrt{n+k}>2\sqrt{n}$$? Even if k > 3n it can be either n+k > 4n or n+k < 4n, since we don't know more about n.

Thanks .

Not sure how you get the above...

Anyway, the question asks whether $$\sqrt{n+k}>2\sqrt{n}$$? After algebraic manipulations shown in my solution the question becomes: is $$k>3n$$? The first statement answers this question, which makes it sufficient.
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Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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07 Oct 2015, 03:14
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Manhattan GMAT Explanation Appears Wrong [#permalink]

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07 Aug 2016, 10:16
Problem:
If n and k are positive integers, is $$\sqrt{n+k}$$ > 2 $$\sqrt{n}$$?

(1) k > 3n
(2) n + k > 3n

------
By squaring both sides of the equation, we are left with:

n + k > 4n

In its simplest form the equation is:

k > 3n
----

According to Manhattan GMAT 12th edition, the second statement is insufficient.
Perhaps I am overlooking some mathematical principle, but if one compares the following two equations one should be able to conclude if the statement is true or not.

statement (2) from problem: n + k > 3n
original equation squared: n + k > 4n

Any help or guidance would be much appreciated.
Math Expert
Joined: 02 Sep 2009
Posts: 39640
Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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07 Aug 2016, 11:37
ebliss wrote:
Problem:
If n and k are positive integers, is $$\sqrt{n+k}$$ > 2 $$\sqrt{n}$$?

(1) k > 3n
(2) n + k > 3n

------
By squaring both sides of the equation, we are left with:

n + k > 4n

In its simplest form the equation is:

k > 3n
----

According to Manhattan GMAT 12th edition, the second statement is insufficient.
Perhaps I am overlooking some mathematical principle, but if one compares the following two equations one should be able to conclude if the statement is true or not.

statement (2) from problem: n + k > 3n
original equation squared: n + k > 4n

Any help or guidance would be much appreciated.

Merging topics. Please refer to the discussion above.

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Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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09 Dec 2016, 14:39
Hi,

If it were not given that n and k are positive integers, then taking the underroot of this expression (Underoort n+k > 2 underroot n) would have resulted in absolute value form as l n + k l > 4 l n l. Am I right?

Just wanted to make sure that I get the concept right.

Regards,
H
Intern
Joined: 05 Dec 2016
Posts: 5
If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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12 Mar 2017, 06:54
HarveyKlaus wrote:
Hi,

If it were not given that n and k are positive integers, then taking the underroot of this expression (Underoort n+k > 2 underroot n) would have resulted in absolute value form as l n + k l > 4 l n l. Am I right?

Just wanted to make sure that I get the concept right.

Regards,
H

Nope. It still would have given the positive value, however, we won't be sure of the sign of the n and k if the statement "they are +ve integers" isn't mentioned.

\sqrt{100-36}
\sqrt{64} = 8
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Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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23 Mar 2017, 12:53
Careless mistake!
Can someone share a link on how to reduce careless mistake please?
Frustrating to miss this kind of easy questions
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Manager
Joined: 01 Dec 2016
Posts: 112
Location: Cote d'Ivoire
Concentration: Finance, Entrepreneurship
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Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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23 Mar 2017, 12:54
Careless mistake!
Can someone share a link on how to reduce careless mistake please?
Frustrating to miss this kind of easy questions
_________________

What was previously thought to be impossible is now obvious reality.
In the past, people used to open doors with their hands. Today, doors open "by magic" when people approach them

Math Expert
Joined: 02 Sep 2009
Posts: 39640
Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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23 Mar 2017, 23:44
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guialain wrote:
Careless mistake!
Can someone share a link on how to reduce careless mistake please?
Frustrating to miss this kind of easy questions

Check below topics:

Careless Mistakes on GMAT Math BY MIKE MCGARRY, MAGOOSH;
3 Deadly Mistakes you must avoid in LCM-GCD Questions BY EGMAT;
Do you make these 3 mistakes in GMAT Even-Odd Questions? BY EGMAT;
Common Quant Mistakes That You Must Avoid by VERITAS PREP.

Hope it helps.
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Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2? [#permalink]

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23 Mar 2017, 23:57
Thanks a lot Bunnuel.
Very useful
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Re: If n and k are positive integers, is (n+k)^1/2>2n^1/2?   [#permalink] 23 Mar 2017, 23:57
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