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# Which of the following numbers is closest to 100*(11-sqrt(119)) ?

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GMATH Teacher
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Joined: 12 Oct 2010
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Which of the following numbers is closest to 100*(11-sqrt(119)) ?  [#permalink]

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14 Feb 2019, 13:56
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Difficulty:

95% (hard)

Question Stats:

9% (01:39) correct 91% (01:57) wrong based on 53 sessions

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GMATH practice exercises (Quant Class 12)

Which of the following numbers is closest to $$\,100\left( {11 - \sqrt {119} } \right)\,$$ ?

(A) 8.5
(B) 9.2
(C) 9.9
(D) 10.6
(E) 11.3

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Re: Which of the following numbers is closest to 100*(11-sqrt(119)) ?  [#permalink]

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14 Feb 2019, 14:31
fskilnik wrote:
GMATH practice exercises (Quant Class 12)

Which of the following numbers is closest to $$\,100\left( {11 - \sqrt {119} } \right)\,$$ ?

(A) 8.5
(B) 9.2
(C) 9.9
(D) 10.6
(E) 11.3

I'm sure there is a smarter way to do this, but it's clear that $$\sqrt{119}$$ is something slightly less than $$11$$

However, the answer choices are not spread far enough apart to be done at this point.

So then I thought about solving $$(10+x)^2 = 119$$ where I know $$x<1$$
Trying $$x=0.9$$ gives us $$10^2+2*10*0.9+0.9^2$$
Bingo, this is $$100+18+0.81$$ which is very close to $$119$$ but a little less, so our x was a little too low

Now to solve $$100*(11-10.9) = 100* 0.1 = 10$$

Well shoot we know it's a little less but is it $$9.9$$ or $$9.2$$?

To have gotten $$9.9$$, we would have needed $$x = .901$$. That would not have been enough to get all the way to $$119$$ in $$10^2+2*10*0.901+0.901^2$$

Therefore, best guess is that B is the right answer
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Re: Which of the following numbers is closest to 100*(11-sqrt(119)) ?  [#permalink]

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14 Feb 2019, 15:51
3
fskilnik wrote:
GMATH practice exercises (Quant Class 12)

Which of the following numbers is closest to $$\,100\left( {11 - \sqrt {119} } \right)\,$$ ?

(A) 8.5
(B) 9.2
(C) 9.9
(D) 10.6
(E) 11.3

100(11 - √119)(11 + √119) = 100(11² - √119²) = 100(121-119) = 100*2 = 200.

When the correct answer is multiplied by 11+√119, the result must be as close as possible to 200.
11+√119 = 11 + (a bit less than 11) = a bit less than 22.
Since 9*22 = 198, multiplying 9.2 by a bit less than 22 will yield a product close to 200.

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Re: Which of the following numbers is closest to 100*(11-sqrt(119)) ?  [#permalink]

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14 Feb 2019, 18:11
2
1
fskilnik wrote:
GMATH practice exercises (Quant Class 12)

Which of the following numbers is closest to $$\,100\left( {11 - \sqrt {119} } \right)\,$$ ?

(A) 8.5
(B) 9.2
(C) 9.9
(D) 10.6
(E) 11.3

$$?\,\,\,:\,\,\,\,100\left( {11 - \sqrt {119} } \right)\,\,\,{\rm{approx}}{\rm{.}}$$

$$\left( {11 - \sqrt {119} } \right)\left( {11 + \sqrt {119} } \right) = {11^2} - 119 = 2\,\,\,\,\, \Rightarrow \,\,\,\,\,11 - \sqrt {119} = {2 \over {11 + \sqrt {119} }}$$

$$100 < 119 < 121\,\,\,\, \Rightarrow \,\,\,\,10 < \sqrt {119} < 11\,\,\,\,\mathop \Rightarrow \limits^{ + 11} \,\,\,\,21 < 11 + \sqrt {119} < 22\,\,\,\, \Rightarrow \,\,\,\,{1 \over {22}} < {1 \over {11 + \sqrt {119} }} < {1 \over {21}}$$

$${2 \over {22}} < {2 \over {11 + \sqrt {119} }} < {2 \over {21}}\,\,\,\,\, \Rightarrow \,\,\,\,100 \cdot {1 \over {11}} < \underbrace {100\left( {11 - \sqrt {119} } \right)}_{{\rm{focus}}\,{\rm{!}}} < 100 \cdot {2 \over {21}}$$

$$\left. \matrix{ {{100} \over {11}} = {{99 + 1} \over {11}} = 9{1 \over {11}}\,\, \cong \,\,9.1 \hfill \cr {{200} \over {21}} = {{210 - 10} \over {21}} = 10 - {{10} \over {21}} = 9{{11} \over {21}}\,\, \cong \,\,9.5 \hfill \cr} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\left( {\rm{B}} \right)$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: Which of the following numbers is closest to 100*(11-sqrt(119)) ?  [#permalink]

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17 Feb 2019, 16:03
200/(11+root119)
=200/(11+10.9)
=9.13
Re: Which of the following numbers is closest to 100*(11-sqrt(119)) ?   [#permalink] 17 Feb 2019, 16:03
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