Que - if x and y are positive which of the following must b greater than 1/root(x+y)
a)root((x+y)/2x)
b)(root x + root y)/(x+y)
c)(root x- root y)/(x +y )

a)1/root(x+y) > root ((x+y)/2x)
squaring, 1/(x+y) >(x+y)/2x or (x+y)^2<2x. if x=1,y=3; 16<2 false
b) (root x + root y) / (x+y ) >1/root(x+y)
root x + root y > root (x+y)
squaring both sides,
x + y + 2root(xy) > x+y
root(xy)>0 true always

c)(root x - root y) / (x+y ) >1/root(x+y)
root x - root y > root (x+y)
squaring both sides,
x + y - 2root(xy) > x+y
root(xy)<0 false

Thus only b) is correct.

Why can't we just pick numbers for x and y in the question stem expression and see which Roman Numeral choices are greater w/ the same x and y values?

I understand that the answer will depend on which values you pick for x and y, but I'm under the impression that since it says it MUST be true, then any x and y value work and the expression will ALWAYS be greater or ALWAYS be less than.

Can someone please clarify? Thx

Anurag@Gurome wrote:Refer to this post : https://www.beatthegmat.com/those-radica ... tml#320556

Hi,

Could you please help us with an easier solution!

TIA,
GK

At gmatkiss,
Very simple solution to this. You don't have to get into inequalities at all. I'm sure u know the method of comparing two fraction is cross multiplication eg 11/7 and 8/6, by cross multiplying u get
66 56. The product on the left corresponds to the fraction on the left side and similarly with the right side fraction. 66>56 so 11/7>8/6. This rule only applies to +ve fractions. If both fractions are negative i.e. -11/7 and -8/6 After cross multiplying u get -66 and -56. Right side greater than left side so -8/6> -11/7. If one is negative and one is +ve then no need to cross multiply as the -ve fraction is obviously smaller.

Coming to the problem
if x and y are positive which of the following must b greater than 1/√(x+y)
a)√(x+y)/2x
b)(√x + √y)/(x+y)
c)(√x- √y)/(x +y )

Compare each one with 1/√(x+y)
A)√(x+y)/2x and 1/√(x+y)
Cross multiplying u get,
x+y and 2x. U can plug in numbers for this by urself and u'd conflicting answers. So this statement is not always true

B)(√x + √y)/(x+y) and 1/√(x+y)
(√x + √y)√(x+y) and (x+y)

Dividing both sides by √(x+y) we get
(√x + √y) and (x+y)/√(x+y) So we have

(√x + √y) and √(x+y)
Squaring both sides we get
x+2√xy+y and x+y
Left hand fraction greater than right side fraction so B is greater than 1/√(x+y)

C)(√x- √y)/(x +y ) and 1/√(x+y)
Cross multiplying we get
(√x- √y)√(x+y) and x+y
Dividing both sides by √x+y
(√x- √y) and x+y
Squaring both sides we have
x-2√xy+y and x+y. Since x and y are both +ve the left side is obviously smaller than the right side so C is smaller than 1/√(x+y). Answer is B Only