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505-555 Level|   Inequalities|   Roots|                                       
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 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
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 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.

Answer: A.
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 23 Mar 2017, 23:44
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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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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 27 Jun 2014, 02:53
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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.

Answer: A.

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!

Please help me to clarify, thank you so much!

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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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 27 Jun 2014, 02:36
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Bunuel
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.

Answer: A.

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!

Please help me to clarify, thank you so much!
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 19 Aug 2014, 07:01
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Bunuel
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.

Answer: A.


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 :).
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 19 Aug 2014, 08:06
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tobiasfr
Bunuel
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.

Answer: A.


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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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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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 19 Jul 2017, 19:58
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Hi Bunuel,

I couldn't understand the 2nd part.

The objective is the check if k>3n (after simplification).
From option 2 we find K>2n , so it's also sufficient to answer that K is not greater than 3n.

Please explain.
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 19 Jul 2017, 21:59
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IWilWin
Hi Bunuel,

I couldn't understand the 2nd part.

The objective is the check if k>3n (after simplification).
From option 2 we find K>2n , so it's also sufficient to answer that K is not greater than 3n.

Please explain.

Say a question asks: is x > 3? We got that x > 2. Is this sufficient to answer the question? No, because if x= 2.5, then the answer is NO but if x = 4, then the answer is YES.
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 11 Oct 2017, 11:46
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Hi Bunuel,

I couldn't understand the 2nd part.

The objective is the check if k>3n (after simplification).
From option 2 we find K>2n , so it's also sufficient to answer that K is not greater than 3n.

Please explain.

Say a question asks: is x > 3? We got that x > 2. Is this sufficient to answer the question? No, because if x= 2.5, then the answer is NO but if x = 4, then the answer is YES.

Bunuel : I always have a confusion in questions such as these...
We know that on simplifying the question stem we get k>3n
The second premise is reduced to k>2n
Now my understanding is that the second premise is sufficient for us to say that k is not greater than 3n, hence answer choice D. When it itself deduces to k>2n isn't it enough for us to answer the original question stem whether k>3n? In this case No k is not greater than 3n
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 11 Oct 2017, 21:03
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IWilWin
Hi Bunuel,

I couldn't understand the 2nd part.

The objective is the check if k>3n (after simplification).
From option 2 we find K>2n , so it's also sufficient to answer that K is not greater than 3n.

Please explain.

Say a question asks: is x > 3? We got that x > 2. Is this sufficient to answer the question? No, because if x= 2.5, then the answer is NO but if x = 4, then the answer is YES.

Bunuel : I always have a confusion in questions such as these...
We know that on simplifying the question stem we get k>3n
The second premise is reduced to k>2n
Now my understanding is that the second premise is sufficient for us to say that k is not greater than 3n, hence answer choice D. When it itself deduces to k>2n isn't it enough for us to answer the original question stem whether k>3n? In this case No k is not greater than 3n

No. Say a question asks is x > 3?

(1) x > 2. Is this sufficient to answer the question whether x is greater than 3? No. If x = 100, then we'd have an YES answer to the question but if x = 2.5 (so if x is any number from 2 to 3), then we'd have a NO answer to the question.

In a Yes/No Data Sufficiency questions, statement(s) is sufficient if the answer is “always yes” or “always no” while a statement(s) is insufficient if the answer is "sometimes yes" and "sometimes no".
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 26 Nov 2017, 00:17
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We know that on simplifying the question stem we get k>3n
The second premise is reduced to k>2n
Now my understanding is that the second premise is sufficient for us to say that k is not greater than 3n, hence answer choice D. When it itself deduces to k>2n isn't it enough for us to answer the original question stem whether k>3n? In this case No k is not greater than 3n[/quote]

No. Say a question asks is x > 3?

(1) x > 2. Is this sufficient to answer the question whether x is greater than 3? No. If x = 100, then we'd have an YES answer to the question but if x = 2.5 (so if x is any number from 2 to 3), then we'd have a NO answer to the question.

In a Yes/No Data Sufficiency questions, statement(s) is sufficient if the answer is “always yes” or “always no” while a statement(s) is insufficient if the answer is "sometimes yes" and "sometimes no".[/quote]

This explanation..... this is going to save lots of lives on the GMAT. Thank you Bunuel!
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 31 Jul 2019, 14:56
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Is √(n+k)>2√n , yes/no?

Condition n>0 (I) k>0 (I) [n and k are positive integers]

Simplify:
(√(n+k) / √n)^2 > 2
n+k/n>4

n+k>4n

Therefore is k>3n... yes/no?
or
Is k>3 where the least possible value of n is 1
:. Possible values of k = 4,5,6....

St 1: k>3n ...Sufficient

Possible values of k= 4, 5, 6...

St 2: K>2n

possible values of k = 3,4,5....

Because K can be equal to 3 or greater than 3 it is not sufficient.
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 01 Jun 2020, 20:55
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Bunuel
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.

Answer: A.


How can you square both sides? I squared in some questions, and it turned out to be wrong. Could you please help me know in which questions I can square both the sides and which questions I cant.
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 02 Jun 2020, 00:46
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Bunuel
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.

Answer: A.


How can you square both sides? I squared in some questions, and it turned out to be wrong. Could you please help me know in which questions I can square both the sides and which questions I cant.

We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality). Here both sides are under the square root, so non-negative, so we can square.

Adding, subtracting, squaring etc.: Manipulating Inequalities.
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 14 Apr 2021, 05:00
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[b]Solution:[/b]

In any inequality, if the signs are same, we can square, cancel or cross multiply.
Here, both parts of the inequality are positive, we can square it and rephrase the question stem as
Is n + k > 4n ?
St (1):- k > 3n
Adding n on both sides n + k >3n (Sufficient)

St (2)
n + k > 3n
=> k>2n
This doesn’t tell us if n + k >4n (Insufficient)

Option (a)

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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 23 Sep 2022, 09:01
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Hi Bunuel I read through this read and I'm still having some trouble understanding one thing -- why does it matter that this question specifies "n and k are positive integers"? A value under a square root would always need to be positive anyway, and either way n must have the same sign on both sides, so why does it matter?
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If n and k are positive integers, is (n + k)^(1/2) > 2n^(1/2)? 21 Oct 2023, 13:49
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