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If x and y are positive, which of the following must be

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Director
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If x and y are positive, which of the following must be [#permalink] New post 12 May 2006, 04:03
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A
B
C
D
E

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If x and y are positive, which of the following must be greater than 1/√(x+y)?
I. √(x+y)/2x,
II. (√x+√y)/(x+y)
III. (√x-√y)/(x+y)

Please explain your solution.
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 [#permalink] New post 12 May 2006, 04:59
Hallo M8,
For I when x=y both stem and A) are equal
For III when x=y nominator is 0 so the ratio is 0 and the number is less than stem for sure.
For II used brute force with values >< than 1 it is also less than stem so seems that NONE of the 3 is bigger than stem
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 [#permalink] New post 12 May 2006, 10:01
Plugging in x=2 and y=2

1/√(x+y) = 1/2

I. √(x+y)/2x = 2/4 = 1/2 not greater so we can rule this out.

II. (√x+√y)/(x+y) = (1.414 + 1.414) / 4 = 2.828/4 > 1/2 so this one could be true, needs further testing

III. (√x-√y)/(x+y) = 0, we can rule this out.

Without further calculation, since None is not a choice here, I will guess II only

But if None is an option we probably need to dig further and it gets ugly. Im sure there are easier way to do this, anyone??

MB what is OA?
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 [#permalink] New post 12 May 2006, 10:19
II is the answer.

If the stem from each of the options, II is the only one that would definitely evaluate to a positive real number. Try it!

New here, so unable to figure out how to type in those 'root' symbols, so unable to show you the calculations - apologies.
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 [#permalink] New post 12 May 2006, 13:14
Its II(2).

subtracting 1/√(x+y) from (√x+√y)/(x+y)
it gives: [√x+√y-√(x+y)]/(x+y)
Here, Square root of individual +ve numbers is always more than square root of sum of numbers.
So this value is always +ve.
Similar reasoning can be used to prove for other statements that we can't conclude for surity.
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 [#permalink] New post 12 May 2006, 21:48
paddyboy wrote:
II is the answer.

New here, so unable to figure out how to type in those 'root' symbols, so unable to show you the calculations - apologies.


Just copy/paste it buddy. :wink:
Waiting for your calculations.
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 [#permalink] New post 12 May 2006, 23:29
In an ideal world, people would leave lazy guys like me in peace, but you make me do the calculations :lol:

Here they are:

Subtracting (1) from stem, we get
1/√(x+y) - √(x+y)/2x
= (2x - x - y)/[2x√(x+y)]
= (x - y)/[2x√(x+y)]

We don't know relative values of x and y, hence can't say if (1) is greater than stem, as we can't say whether the expression is +ve or -ve.

Subtracting (2) from stem, we can definitely say that (2) is greater than stem, as mendiratta has explained.

Subtracting (3) from stem, we get
1/√(x+y) - (√x-√y)/(x+y)
= [√(x+y) - √x + √y]/(x + y)
This is always +ve, which means the stem is always larger than (3).

So answer is (2) <phew>!
  [#permalink] 12 May 2006, 23:29
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