Brent@GMATPrepNow wrote:If 0 < x < y, then which of the following MUST be true?
I) (x + 2)/(y + 2) > x/y
II) (x - y)/x < 0
III) 2x/(x + y) < 1
A) I only
B) I and II only
C) I and III only
D) II and III only
E) I, II and III
Answer:
E
Source:
www.gmatprepnow.com
Difficulty level: 650
Let's examine each statement individually:
A) (x + 2)/(y + 2) > x/y
Since y is POSITIVE, we can safely take the given inequality and multiply both sides by y to get: (y)(x+2)/(y+2) > x
Also, if y is POSITIVE, then (y+2) is POSITIVE, which means we can safely multiply both sides by (y+2) to get: (y)(x+2) > x(y+2)
Expand: xy + 2y > xy + 2x
Subtract xy from both sides: 2y > 2x
Divide both sides by 2 to get: y > 2
Perfect! This checks out with the given information that says 0 < x < y
So, statement A is TRUE
B) (x - y)/x < 0
Let's use number sense here.
If x < y, then x - y must be NEGATIVE
We also know that x is POSITIVE
So, (x - y)/x = NEGATIVE/POSITIVE = NEGATIVE
In other words, it's TRUE that (x - y)/x < 0
Statement B is TRUE
C) 2x/(x + y) < 1
More number sense...
If x is positive, then 2x is POSITIVE
If x and y are positive, then x + y is POSITIVE
If x < y, then we know that x + x < x + y
In other words, we know that 2x < x + y
If 2x < x + y, then the FRACTION 2x/(x + y) must be less than 1
Statement C is TRUE
Answer: E
Cheers,
Brent
