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square root(x+y)

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by dreamv » Tue Feb 21, 2012 2:35 pm
If x and y are positive, which of the following must be greater than 1/square root(x+y)?

I. Square root(x+y)/2x
II. (Square root x + square root y)/x+y
III. (square root x - square root y)/x+y

A. I only
B. II only
C. III only.
D. I and II
E. I, II and III
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by pemdas » Tue Feb 21, 2012 2:57 pm
IOM b
condition I
sqrt(x+y)/2x > 1/sqrt(x+y)
x+y>2x and y>x Not always true

condition II
(sqrt(x)+sqrt(y))/(x+y) > 1/sqrt(x+y)
sqrt(x+y)((sqrt(x)+sqrt(y)) > x+y
sqrt(x)+sqrt(y) > sqrt(x+y) Always true

condition III
the same as condition true but with difference on LHS
sqrt(x)-sqrt(y) > sqrt(x+y) Not always true
dreamv wrote:If x and y are positive, which of the following must be greater than 1/square root(x+y)?

I. Square root(x+y)/2x
II. (Square root x + square root y)/x+y
III. (square root x - square root y)/x+y

A. I only
B. II only
C. III only.
D. I and II
E. I, II and III
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comparing roots

by GMATGuruNY » Tue Feb 21, 2012 4:25 pm
dreamv wrote:If x and y are positive, which of the following must be greater than 1/square root(x+y)?

I. Square root(x+y)/2x
II. (Square root x + square root y)/x+y
III. (square root x - square root y)/x+y

A. I only
B. II only
C. III only.
D. I and II
E. I, II and III
Let x=1 and y=1.
1/√(x+y) = 1/√(1+1) = 1/√2.
Eliminate any expression not greater than 1/√2.

Scanning options I, II and III, we can quickly see that the numerator of III is equal to 0:
(√x-√y)/(x+y) = (√1-√1)/(1+1) = 0.
Eliminate any answer choice that includes III (C and E).

I: √(x+y)/2x
√(1+1)/(2*1) > 1/√2
√2/2 > 1/√2.
Cross-multiplying, we get:
√2 *√2 > 2*1
2>2.
Doesn't work.
Eliminate any remaining answer choice that includes I (A and D).

The correct answer is B.
Last edited by GMATGuruNY on Mon Jan 15, 2018 5:45 am, edited 1 time in total.
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by imhimanshu » Wed Feb 29, 2012 8:47 pm
Hey Mitch,
Why did you go for x=y = 1? you could have gone for other values as well?? Is there any reasoning behind it...

I also solved this way in my exam.. i.e x= y = 1 and got the answer.. however, i am not sure whether this will click in real exam or not.

Thanks
GMATGuruNY wrote:
dreamv wrote:If x and y are positive, which of the following must be greater than 1/square root(x+y)?

I. Square root(x+y)/2x
II. (Square root x + square root y)/x+y
III. (square root x - square root y)/x+y

A. I only
B. II only
C. III only.
D. I and II
E. I, II and III
Let x=1 and y=1.
1/√(x+y) = 1/√(1+1) = 1/√2.
Eliminate any expression not greater than 1/√2.

Scanning options I, II and III, we can quickly see that the numerator of III is equal to 0:
(√x-√y)/(x+y) = (√1-√1)/(1+1) = 0.
Eliminate any answer choice that includes III (C and E).

I: √(x+y)/2x
√(1+1)/(2*1) > 1/√2
√2/2 > 1/√2.
Cross-muliplying, we get:
√2 *√2 > 2*1
2>2.
Doesn't work.
Eliminate any remaining answer choice that includes I (A and D).

The correct answer is B.
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