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# root{463} is between

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root{463} is between [#permalink]

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12 Aug 2012, 08:31
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Difficulty:

5% (low)

Question Stats:

91% (01:27) correct 9% (00:45) wrong based on 204 sessions

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$$\sqrt{463}$$ is between

(A) 21 and 22
(B) 22 and 23
(C) 23 and 24
(D) 24 and 25
(E) 25 and 26

What is the fastest way to solve this problem?
[Reveal] Spoiler: OA
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Re: [square_root]463[/square_root] is between [#permalink]

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12 Aug 2012, 08:35
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The way I solved it was by realizing

20^2= 400

so 21^2 = 20^2 +20 + 1*21 = 441

and

22^2 = 20^2 +20*2 + 2*21 = 440 + 40 + 42 > 500

so \sqrt{463} is >21 but <22

still wasn't that fast, but it made me think there might be some formula or something.
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Re: [square_root]463[/square_root] is between [#permalink]

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12 Aug 2012, 08:58
JustinDragna wrote:
The way I solved it was by realizing

20^2= 400

so 21^2 = 20^2 +20 + 1*21 = 441

and

22^2 = 20^2 +20*2 + 2*21 = 440 + 40 + 42 > 500

so \sqrt{463} is >21 but <22

still wasn't that fast, but it made me think there might be some formula or something.

Your reasoning is perfect. And there is no magic formula to help here.
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Re: [square_root]463[/square_root] is between [#permalink]

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12 Aug 2012, 18:27
JustinDragna wrote:
The way I solved it was by realizing

20^2= 400

so 21^2 = 20^2 +20 + 1*21 = 441

and

22^2 = 20^2 +20*2 + 2*21 = 440 + 40 + 42 > 500

so \sqrt{463} is >21 but <22

still wasn't that fast, but it made me think there might be some formula or something.

The other way is to find the square root of 463 first and then compare it, but even this method will require atleast 3-4 iterations and is pretty long.
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Re: root{463} is between [#permalink]

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06 Sep 2014, 15:59
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Re: root{463} is between [#permalink]

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08 Sep 2014, 22:52
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$$21^2 = 441$$

$$22^2 = 484$$

Answer = A
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Re: root{463} is between [#permalink]

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09 Dec 2015, 05:49
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EvaJager wrote:
JustinDragna wrote:
The way I solved it was by realizing

20^2= 400

so 21^2 = 20^2 +20 + 1*21 = 441

and

22^2 = 20^2 +20*2 + 2*21 = 440 + 40 + 42 > 500

so \sqrt{463} is >21 but <22

still wasn't that fast, but it made me think there might be some formula or something.

Your reasoning is perfect. And there is no magic formula to help here.

We can actually avoid the second part. We saw that moving from 20 to 21, the increase was 441-400 = 41. So, moving from 21-22, the increase would definitely be greater than 41. 441+41 > 463. Thus it must lie between 21 and 22.

+Kudos, if this helped!
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Re: root{463} is between   [#permalink] 09 Dec 2015, 05:49
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# root{463} is between

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