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please help with this difficult problem

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please help with this difficult problem

by abcdefg » Sun Jul 12, 2009 1:49 pm
I've attached a pic of the problem at the solution. Sorry for not being able to black out the solution part. The suggested solution by manhattan gmat certainly makes sense but it involves a lot of manipulation that I couldn't really think of on the spot.

When I was doing the problem, I just crossed multiplied and started canceling things out and ended up with the equation [1/(x-y) = 2].

How do I manipulate x/y from [1/(x-y) = 2]? Thanks.
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by shibal » Sun Jul 12, 2009 2:04 pm
way to complicated, try to get the answers and plug in back... (IMO)
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by truplayer256 » Sun Jul 12, 2009 2:12 pm
Well, I got x/y=9 by doing the following:

We have:

sqrt(x)+sqrt(y)/x-y=2sqrt(x)+2sqrt(y)/x+2sqrt(xy)+y

We can rewrite the above equation as:

sqrt(x)+sqrt(y)/x-y=2(sqrt(x)=sqrt(y))/(sqrt(x)+sqrt(y))^(2)

Now cross multiply:

(sqrt(x)+sqrt(y))^(3)=2(sqrt(x)+sqrt(y))(x-y)

One of the sqrt(x)+sqrt(y)'s cancel and you're left with

(sqrt(x)+sqrt(y))^(2)=2x-2y

x+2sqrt(xy)+y=2x-2y

2sqrt(xy)+3y=x

x-2sqrt(xy)-3y=0

Now factor:

(sqrt(x)-3sqrt(y))(sqrt(x)+sqrt(y))=0

sqrt(x)-3sqrt(y)=0

sqrt(x)=3sqrt(y)

Square both sides:

x=9y

x/y=9

This problem was very time consuming. I really don't think a problem like this would be on the GMAT.
How do I manipulate x/y from [1/(x-y) = 2]? Thanks.
There's no way you can get x/y from 1/(x-y)=2

1=2x-2y. That's as far as you can get.
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by abcdefg » Sun Jul 12, 2009 2:20 pm
But I got [1/(x-y) = 2] just by canceling out so I didn't make any illegal operations. How can it be a dead end?
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by rah_pandey » Mon Jul 13, 2009 2:57 am
after cancellation you should get (√x+√y)^2=2(x-y). Please recheck
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