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605-655 (Medium)|   Arithmetic|   Roots|      
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Bunuel
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Which of the points on the number line above best expresses the expres 10 Sep 2017, 05:17
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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


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B
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generis
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Which of the points on the number line above best expresses the expres 10 Sep 2017, 08:18
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Bunuel

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

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image011.gif
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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

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B
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reachskishore
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Which of the points on the number line above best expresses the expres 11 Sep 2017, 08:45
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genxer123
Bunuel

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

Show 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

Answer
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B
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is there any other way to get this ??
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Madhavi1990
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Which of the points on the number line above best expresses the expres 03 Dec 2017, 13:43
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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!
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Which of the points on the number line above best expresses the expres 03 Dec 2017, 15:06
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Madhavi1990
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!
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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))
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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?) :-)
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Which of the points on the number line above best expresses the expres 03 Dec 2017, 15:34
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genxer123
Madhavi1990
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?) :-)
Show more



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!
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Which of the points on the number line above best expresses the expres 16 Dec 2025, 08:25
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