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# If n is positive, is \sqrt {n} > 100 ? 1. \sqrt {n-1}

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If n is positive, is \sqrt {n} > 100 ? 1. \sqrt {n-1} [#permalink]  11 Jan 2011, 17:57
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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]  11 Jan 2011, 18:08
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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.

 ! Please post PS questions in the PS subforum: gmat-problem-solving-ps-140/Please post DS questions in the DS subforum: gmat-data-sufficiency-ds-141/No posting of PS/DS questions is allowed in the main Math forum.

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Re: Quant Rev DS #51 [#permalink]  11 Jan 2011, 19:30
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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]  09 Apr 2012, 12:07
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]  09 Apr 2012, 12:19
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I didn't understand why statement 1 is not sufficient. Can someone please explain? thanks.

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]  09 Apr 2012, 22:05
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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 \sqrt {n} > 100 ? 1. \sqrt {n-1}   [#permalink] 09 Apr 2012, 22:05
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