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

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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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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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Kudos [?]: 80118 [0], given: 10027

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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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
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Kudos [?]: 80118 [1] , given: 10027

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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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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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: 34527
Followers: 6313

Kudos [?]: 80118 [0], given: 10027

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] 07 Aug 2016, 11:37
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