If n is positive, is root(n) > 100 ? : GMAT Data Sufficiency (DS)
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If n is positive, is root(n) > 100 ?

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If n is positive, is root(n) > 100 ? [#permalink]

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If n is positive, is \(\sqrt {n} > 100\)?

(1) \(\sqrt {n-1} > 99\)

(2) \(\sqrt {n+1} > 101\)

Can someone write out the algebra on this one, I just want to double check work. Thanks.
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Re: Quant Rev DS #51 [#permalink]

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If n is positive, is \(\sqrt {n} > 100\)?

Is \(\sqrt {n} > 100\)? --> is \(n>100^2\)?

(1) \(\sqrt {n-1} > 99\) --> \(n-1>99^2\) --> \(n>99^2+1\): \(99^2+1\) is less than \(100^2\) (as \(100^2=(99+1)^2=99^2+2*99+1\)), so \(n\) may or may not be more than this value. Not sufficient.

(2) \(\sqrt {n+1} > 101\) --> \(n+1>101^2\) --> \(n>101^2-1=(101-1)(101+1)=100*102\), so \(n>100*102>100^2\). Sufficient.

Answer: B.


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Re: Quant Rev DS #51 [#permalink]

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tonebeeze wrote:
If n is positive, is \(\sqrt {n} > 100\)?

1. \(\sqrt {n-1} > 99\)

2. \(\sqrt {n+1} > 101\)

Can someone write out the algebra on this one, I just want to double check work. Thanks.


Even though you asked for algebra, let me point out that you don't really need algebra to solve this.

When you are considering big numbers, square roots of consecutive numbers differ by very little. e.g. \(\sqrt {10000} = 100\) and \(\sqrt {9999} = 99.995\)... Square roots of even small positive numbers differ by less than 1 e.g. \(\sqrt {1} = 1\) and \(\sqrt {2} = 1.414\) - Difference of just 0.414 \(\sqrt {3} = 1.732\) - Difference of just 0.318. The difference just keeps getting smaller and smaller.

So if \(\sqrt {n-1} > 99\), \(\sqrt {n}\) will be very close to \(\sqrt {n-1}\) and will also be greater than 99. But will it be greater than 100, we cannot say. So not sufficient.

If \(\sqrt {n+1} > 101\), then \(\sqrt {n}\) may not be greater than 101, but it will definitely be greater than 100 since between two consecutive integers, the square root difference will not reach 1 (as shown above).
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Re: If n is positive, is \sqrt {n} > 100 ? 1. \sqrt {n-1} [#permalink]

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New post 09 Apr 2012, 11:07
I didn't understand why statement 1 is not sufficient. Can someone please explain? thanks.
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Re: If n is positive, is \sqrt {n} > 100 ? 1. \sqrt {n-1} [#permalink]

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mymbadreamz wrote:
I didn't understand why statement 1 is not sufficient. Can someone please explain? thanks.


Question asks whether \(n>100^2\).

From (1) we have that \(n>99^2+1\). Now, since \(100^2>99^2+1\), then it's possible that \(n>100^2>99^2+1\), which would mean that the answer is YES, as well as that \(100^2>n>99^2+1\), which would mean that the answer is NO. Two different answers, hence not sufficient.

Hope it's clear.
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Re: If n is positive, is \sqrt {n} > 100 ? 1. \sqrt {n-1} [#permalink]

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mymbadreamz wrote:
I didn't understand why statement 1 is not sufficient. Can someone please explain? thanks.


Stmnt 1 just tells you that \(\sqrt{n-1} > 99\)

Think of 2 diff cases:

\(\sqrt{n-1} = 99.2\)
\(\sqrt{n} = 99.205\)

or

\(\sqrt{n-1} = 112\)
\(\sqrt{n} = 112.015\)

Can you say whether \(\sqrt{n}\) is greater than 100?
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Re: If n is positive, is root(n) > 100 ? [#permalink]

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Re: If n is positive, is root(n) > 100 ? [#permalink]

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New post 24 Dec 2015, 03:34
Important consideration: n is not an integer, not stated explicitly; assuming the same would given an incorrect conclusions

Statement 1 gives many values which could be greater than 99 but not greater than 100, or greater than 100. Insufficient

Statement 2 gives a sure conclusion that it is greater. Hence, sufficient
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Re: If n is positive, is root(n) > 100 ? [#permalink]

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tonebeeze wrote:
If n is positive, is \(\sqrt {n} > 100\)?

(1) \(\sqrt {n-1} > 99\)

(2) \(\sqrt {n+1} > 101\)



Target question: Is √n > 100?

This is a good candidate for REPHRASING the target question.
Take √n > 100 and square both sides to get n > 10,000
So, we get: REPHRASED target question: Is n > 10,000?

Statement 1: √(n - 1) > 99
Square both sides to get n - 1 > 99²
Evaluate: n - 1 > 9801
Add 1 to both sides to get: n > 9802
So, x COULD equal 9803, in which case n < 10,000
Conversely, x COULD equal 10,001, in which case n > 10,000
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: √(n + 1) > 101
Square both sides to get n + 1 > 101²
Evaluate: n + 1 > 10,201
Subtract 1 from both sides to get: n > 10,200
If x is greater than 10,200, then we can be certain that x > 10,000
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer =
[Reveal] Spoiler:
B


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Re: If n is positive, is root(n) > 100 ?   [#permalink] 22 Aug 2016, 14:00
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