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# Which of the points on the number line above best expresses the expres

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Math Expert
Joined: 02 Sep 2009
Posts: 44598
Which of the points on the number line above best expresses the expres [#permalink]

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10 Sep 2017, 05:17
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45% (medium)

Question Stats:

55% (00:50) correct 45% (00:46) wrong based on 184 sessions

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Which of the points on the number line above best expresses the expression $$\frac{\sqrt{2}-2}{\sqrt{3}-3}$$?

A. A
B. B
C. C
D. D
E. E

[Reveal] Spoiler:
Attachment:

image011.gif [ 456 Bytes | Viewed 1201 times ]
[Reveal] Spoiler: OA

_________________
SC Moderator
Joined: 22 May 2016
Posts: 1544
Re: Which of the points on the number line above best expresses the expres [#permalink]

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10 Sep 2017, 08:18
1
KUDOS
Bunuel wrote:

Which of the points on the number line above best expresses the expression $$\frac{\sqrt{2}-2}{\sqrt{3}-3}$$?

A. A
B. B
C. C
D. D
E. E

[Reveal] Spoiler:
Attachment:
image011.gif

Use approximations and straight arithmetic to find the value for the point on the line that best expresses:

$$\frac{\sqrt{2}-2}{\sqrt{3}-3}$$

$$\sqrt{2}$$ ≈ 1.4 and $$\sqrt{3}$$ ≈ 1.7

$$\frac{1.4-2}{1.7-3}$$ ≈

$$\frac{-.6}{-1.3}$$ ≈

$$\frac{6}{12}$$ ≈ $$\frac{1}{2}$$ (the answer is a bit larger than that because I rounded down)

On the number line, the point that best expresses that value is

[Reveal] Spoiler:
B

_________________

At the still point, there the dance is. -- T.S. Eliot
Formerly genxer123

Intern
Joined: 21 Nov 2016
Posts: 45
Re: Which of the points on the number line above best expresses the expres [#permalink]

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11 Sep 2017, 08:45
1
KUDOS
genxer123 wrote:
Bunuel wrote:

Which of the points on the number line above best expresses the expression $$\frac{\sqrt{2}-2}{\sqrt{3}-3}$$?

A. A
B. B
C. C
D. D
E. E

[Reveal] Spoiler:
Attachment:
image011.gif

Use approximations and straight arithmetic to find the value for the point on the line that best expresses:

$$\frac{\sqrt{2}-2}{\sqrt{3}-3}$$

$$\sqrt{2}$$ ≈ 1.4 and $$\sqrt{3}$$ ≈ 1.7

$$\frac{1.4-2}{1.7-3}$$ ≈

$$\frac{-.6}{-1.3}$$ ≈

$$\frac{6}{12}$$ ≈ $$\frac{1}{2}$$ (the answer is a bit larger than that because I rounded down)

On the number line, the point that best expresses that value is

[Reveal] Spoiler:
B

is there any other way to get this ??
Senior Manager
Joined: 15 Jan 2017
Posts: 361
Re: Which of the points on the number line above best expresses the expres [#permalink]

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03 Dec 2017, 13:43
3
KUDOS
You could simplify. 1.414(- 0.414) / 1.713(-0.713)
= -ve 0...6 (cause 4*4 =16)/ -ve 0.9 (0.3 *0.3= 0.009)--> 0.6/0.9(for simplicity)
--> 0.2/0.3 (-ve cancels out, direction is positive now)
--> approx 0.66 which is slightly more than 1/2 so closest value is B.

Please let me know if above method was right. Kudos please, if you like this solution!
SC Moderator
Joined: 22 May 2016
Posts: 1544
Which of the points on the number line above best expresses the expres [#permalink]

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03 Dec 2017, 15:06
You could simplify. 1.414(- 0.414) / 1.713(-0.713)
= -ve 0...6 (cause 4*4 =16)/ -ve 0.9 (0.3 *0.3= 0.009)--> 0.6/0.9(for simplicity)
--> 0.2/0.3 (-ve cancels out, direction is positive now)
--> approx 0.66 which is slightly more than 1/2 so closest value is B.

Please let me know if above method was right. Kudos please, if you like this solution!

Madhavi1990 , I like this solution, but I don't understand
1) how you got TO -.414, e.g.

You did not multiply by -1. That would have yielded $$\frac{2-\sqrt{2}}{3-\sqrt{3}}\approx{\frac{.59}{1.27}}$$;

And 2) how you got from $$\frac{-.414}{-.713}$$ to $$\frac{0.6}{0.9}$$.

Maybe I'm slow on the uptake today. I don't understand your shorthand here, for example :
Quote:
1.414(- 0.414) / 1.713(-0.713)
= -ve 0...6 (cause 4*4 =16)/ -ve 0.9 (0.3 *0.3= 0.009)--> 0.6/0.9(for simplicity))

Would you please explain how you derived the first fraction, then how you got from the first fraction to the second? (And in steps with full words?)
_________________

At the still point, there the dance is. -- T.S. Eliot
Formerly genxer123

Senior Manager
Joined: 15 Jan 2017
Posts: 361
Re: Which of the points on the number line above best expresses the expres [#permalink]

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03 Dec 2017, 15:34
1
KUDOS
genxer123 wrote:
You could simplify. 1.414(- 0.414) / 1.713(-0.713)
= -ve 0...6 (cause 4*4 =16)/ -ve 0.9 (0.3 *0.3= 0.009)--> 0.6/0.9(for simplicity)
--> 0.2/0.3 (-ve cancels out, direction is positive now)
--> approx 0.66 which is slightly more than 1/2 so closest value is B.

Please let me know if above method was right. Kudos please, if you like this solution!

Madhavi1990 , I like this solution, but I don't understand
1) how you got TO -.414, e.g.

You did not multiply by -1. That would have yielded $$\frac{2-\sqrt{2}}{3-\sqrt{3}}\approx{\frac{.59}{1.27}}$$;

And 2) how you got from $$\frac{-.414}{-.713}$$ to $$\frac{0.6}{0.9}$$.

Maybe I'm slow on the uptake today. I don't understand your shorthand here, for example :
Quote:
1.414(- 0.414) / 1.713(-0.713)
= -ve 0...6 (cause 4*4 =16)/ -ve 0.9 (0.3 *0.3= 0.009)--> 0.6/0.9(for simplicity))

Would you please explain how you derived the first fraction, then how you got from the first fraction to the second? (And in steps with full words?)

Hi, apologies. I will be clearer from now on. I tried to use approximations based on powers and cyclicity. So 2^1 = ends in 2, 2^2 = 4, 2^3 = 8 and similiarly, for 4: 4^1 = 4, 4^2 = 16, 4^3 = 64. So the ending number matters. I don't know if this is the best solution, but it saved time for me,and seemed fine to apply as a theory

root 2(1-root2) = 1.414(1 - 1.414) = approximately with ending numbers 4*4 = 16 . So the number could be (negative) 0. (some number of zeroes) followed by 6. So I took -0.6 (as a possible approximation)

Root 3(1- root 3) = 1.73(1- 1.73) = 1.73 (-0.73) = approximately ending in 0...(some zeroes) 9 (3*3 = 9)

Then I combined the numerator/ denominator = 0.6/0.9 (it becomes positive now, as -ve cancels out) --> this is approximately 0.2/0.3 which 0.66 just a little more than half. B seemed suitable.

Please let me know if there are any errors conceptually or otherwise. Would love to know it, and hopefully not repeat the same error. Thank you!
Re: Which of the points on the number line above best expresses the expres   [#permalink] 03 Dec 2017, 15:34
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