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Re: If x and y are positive, which of the following must be [#permalink]
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11 Sep 2013, 02:32
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study wrote: If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?
1. \(\frac{\sqrt{x+y}}{2x}\)
2. \(\frac{\sqrt{x}+\sqrt{y}}{x+y}\)
3. \(\frac{\sqrt{x}\sqrt{y}}{x+y}\)
(A) None (B) 1 only (C) 2 only (D) 1 and 3 only (E) 2 and 3 only 1 way to look at this problem is : I) Cross Multiplying LHS : x+y RHS 2x (So y can be anything greater than or less than x), hence ruled out II) LHS : [square_root]x + [square_root]y = [square_root]x+y Now if x were a^2, y were b^2 then LHS : a + b and RHS [square_root]a^2 + b^2 which is nothing but a right triangle with RHS as diagonal and a and b as sides of it. And that the sum of the 2 sides of the triangle is always greater than the third side. Hence this will be true. III) Similarly as with 2, this will become LHS : a  b and RHS : [square_root]a^2 + b^2. Any side will always be greater than the difference between the 2 sides. Hence this will be false. Thus Answer will be C



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Re: If x and y are positive, which of the following must be [#permalink]
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12 Sep 2013, 00:11
study wrote: If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?
1. \(\frac{\sqrt{x+y}}{2x}\)
2. \(\frac{\sqrt{x}+\sqrt{y}}{x+y}\)
3. \(\frac{\sqrt{x}\sqrt{y}}{x+y}\)
(A) None (B) 1 only (C) 2 only (D) 1 and 3 only (E) 2 and 3 only I am not sure why everyone is deriving complex formulas for this problem. I answered this question in 30 seconds. Put x=y=1. Only 2 survives. You can test by a couple of more values. It will always hold good. Hence C.



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Re: If x and y are positive, which of the following must be [#permalink]
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05 Feb 2014, 09:49
I did the following, nothing too complicated
First statement clearly not enough since after cross multiplying we get is y>x? We don't know
For the second statement we have is (sqrt (x) + sqrt (y)(sqrt (x+y)> x+y
x + y = sqrt (x+y) ^2 so what we really need to know is if (sqrt (x) + sqrt (y) > sqrt (x+y) and yes, since sqrt (n) where n is any number is always more than 1 so >1 + >1 is always more than >1. Therefore this statement is true
For the last statement we get something a bit different but still We have sqrt (x+y)(sqrt (x)  sqrt (y)) > x+y
Again x+y = sqrt (x+y)^2
Therefore question is ' Is sqrt (x)  sqrt (y) > sqrt (x+y)
This will never be the case as >1  >1 will never be >1
Therefore our only correct answer choice is the second statement
Hope its clear Cheers J



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Re: If x and y are positive, which of the following must be [#permalink]
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30 Jan 2015, 21:27
Hi all,
If I assume x = 3 and y = 6, then:
Original expression = 1/3
1 evaluates to 1/2 2 evaluates to (1/9)*something 3 evaluates to (1/9)*something
Thus, I get only 1 as an answer.
Any comments?
TO



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Re: If x and y are positive, which of the following must be [#permalink]
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31 Jan 2015, 05:25



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Re: If x and y are positive, which of the following must be [#permalink]
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15 Jun 2015, 04:17
study wrote: If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?
1. \(\frac{\sqrt{x+y}}{2x}\)
2. \(\frac{\sqrt{x}+\sqrt{y}}{x+y}\)
3. \(\frac{\sqrt{x}\sqrt{y}}{x+y}\)
(A) None (B) 1 only (C) 2 only (D) 1 and 3 only (E) 2 and 3 only Interesting ..... It's not a good question What if i take x = y = 2 ....Then c option would work out to be 0 /4 . Nothing in the prompt prevents me from considering x and y to be equal and doing that Option C is not possible as well I think the question should have made it very clear that x is not equal to y



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Re: If x and y are positive, which of the following must be [#permalink]
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15 Jun 2015, 04:19
amlan009 wrote: study wrote: If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?
1. \(\frac{\sqrt{x+y}}{2x}\)
2. \(\frac{\sqrt{x}+\sqrt{y}}{x+y}\)
3. \(\frac{\sqrt{x}\sqrt{y}}{x+y}\)
(A) None (B) 1 only (C) 2 only (D) 1 and 3 only (E) 2 and 3 only Interesting ..... It's not a good question What if i take x = y = 2 ....Then c option would work out to be 0 /4 . Nothing in the prompt prevents me from considering x and y to be equal and doing that Option C is not possible as well I think the question should have made it very clear that x is not equal to y The question asks "which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)", not "which of the following could be greater than \(\frac{1}{\sqrt{x+y}}\)".
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Re: If x and y are positive, which of the following must be [#permalink]
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18 Jun 2015, 00:28
Bunuel wrote: amlan009 wrote: study wrote: If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?
1. \(\frac{\sqrt{x+y}}{2x}\)
2. \(\frac{\sqrt{x}+\sqrt{y}}{x+y}\)
3. \(\frac{\sqrt{x}\sqrt{y}}{x+y}\)
(A) None (B) 1 only (C) 2 only (D) 1 and 3 only (E) 2 and 3 only Interesting ..... It's not a good question What if i take x = y = 2 ....Then c option would work out to be 0 /4 . Nothing in the prompt prevents me from considering x and y to be equal and doing that Option C is not possible as well I think the question should have made it very clear that x is not equal to y The question asks "which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)", not "which of the following could be greater than \(\frac{1}{\sqrt{x+y}}\)". Firstly , i am a huge fan and love the way you help people ...it's amazing Thanks bunuel , and now for the topic I think i have not explained my stance properly ..i am not very happy with the question because must implies true in all cases whereas in the specific case when x and Y are equal positive integers , i don't see the third option to be valid for the question stem anymore Let me explain , Since x and Y are positive integers , i take x = y= 2 ( which i am surely allowed to ) Now , question translates to what is greater than 1 / \sqrt{4} = 1 /2 = 0.5 When the expression c with this numbers translates to 0/(2+2) =0/4 i.e 0 Surely , 0.5 is not greater than 0 . Now, therefore when x and y are same , it is clearly not true . Am i assuming wrong thing's somewhere ?



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Re: If x and y are positive, which of the following must be [#permalink]
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18 Jun 2015, 01:39
amlan009 wrote: Firstly , i am a huge fan and love the way you help people ...it's amazing Thanks bunuel , and now for the topic I think i have not explained my stance properly ..i am not very happy with the question because must implies true in all cases whereas in the specific case when x and Y are equal positive integers , i don't see the third option to be valid for the question stem anymore
Let me explain , Since x and Y are positive integers , i take x = y= 2 ( which i am surely allowed to ) Now , question translates to what is greater than 1 / \sqrt{4} = 1 /2 = 0.5
When the expression c with this numbers translates to 0/(2+2) =0/4 i.e 0 Surely , 0.5 is not greater than 0 .
Now, therefore when x and y are same , it is clearly not true .
Am i assuming wrong thing's somewhere ? The correct answer to the question is C, which says that only II option must be greater than \(\frac{1}{\sqrt{x}+\sqrt{y}}\). Why are talking about the third option?
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Re: If x and y are positive, which of the following must be [#permalink]
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20 Jun 2016, 05:25
Hi Bunuel, Can you please give me the link to a set of these type of problems to practice? Thanks in Advance.
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Re: If x and y are positive, which of the following must be [#permalink]
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Re: If x and y are positive, which of the following must be [#permalink]
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29 Aug 2017, 20:24
Hi Bunuel, For option 2, do we need to consider the negative values of of sqrt(x) and sqrt(y) as well? If x is positive sqrt(x) can be positive or negativebut if we take the negative values then option 2 is not working. For ex, I took x=16 & y = 9 & calculated putting the negative values but it didn't work. Please guide me. Thanks.
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Re: If x and y are positive, which of the following must be [#permalink]
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29 Aug 2017, 22:02
buan15 wrote: Hi Bunuel,
For option 2, do we need to consider the negative values of of sqrt(x) and sqrt(y) as well? If x is positive sqrt(x) can be positive or negativebut if we take the negative values then option 2 is not working. For ex, I took x=16 & y = 9 & calculated putting the negative values but it didn't work.
Please guide me. Thanks. When the GMAT provides the square root sign for an even root, such as \(\sqrt{x}\) or \(\sqrt[4]{x}\), then the only accepted answer is the positive root. That is, \(\sqrt{16}=4\), NOT +4 or 4. Even roots have only a positive value on the GMAT.In contrast, the equation \(x^2=16\) has TWO solutions, +4 and 4. Odd roots have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{64} =4\).
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Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
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Re: If x and y are positive, which of the following must be [#permalink]
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29 Aug 2017, 22:43
study wrote: If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?
1. \(\frac{\sqrt{x+y}}{2x}\)
2. \(\frac{\sqrt{x}+\sqrt{y}}{x+y}\)
3. \(\frac{\sqrt{x}\sqrt{y}}{x+y}\)
(A) None (B) 1 only (C) 2 only (D) 1 and 3 only (E) 2 and 3 only Pick easy substitutions x=2 and y=2 and you will get the answer straight to C
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Re: If x and y are positive, which of the following must be [#permalink]
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29 Aug 2017, 23:23



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Re: If x and y are positive, which of the following must be [#permalink]
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22 Oct 2017, 08:38
I used a simple approach of number picking. pick a small number say 2. it will take about 1 minute to eliminate the wrong answers.



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If x and y are positive, which of the following must be [#permalink]
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25 Oct 2017, 11:33
Most Simple Solution:
Keep x = y amd solve the equation




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