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

Question

If n and k are positive integers, is  \sqrt{n+k}>2\sqrt{n}  ?

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

Bunuel · Accepted Answer

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.

Bunuel · Answer

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.

Bunuel · Answer

When the GMAT provides the square root sign for an even root, such as  \sqrt{x}  or  \sqrt {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 {125} =5  and  \sqrt {-64} =-4 .

Hope it helps.

LucyDang · Answer

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!

tobiasfr · Answer

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

.

Bunuel · Answer

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.

guialain · Answer

Careless mistake!
Can someone share a link on how to reduce careless mistake please?
Frustrating to miss this kind of easy questions

IWilWin · Answer

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.

Bunuel · Answer

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.

avaneeshvyas · Answer

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

Bunuel · Answer

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".

Marshy · Answer

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".

This explanation..... this is going to save lots of lives on the GMAT. Thank you Bunuel!

khan0210 · Answer

Is √(n+k)>2√n , yes/no?

Condition n>0 (I) k>0 (I)

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.

Zaer119 · Answer

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.

Bunuel · Answer

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 .

CrackverbalGMAT · Answer

]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)

Devmitra Sen
GMAT SME

ayakik · Answer

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?

totaltestprepNick · Answer

A https://www.youtube.com/watch?v=Ycb7Ne17k3Q Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience tutoring@totaltestprep.net

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

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