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
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If 0 < x < y, then which of the following MUST be true
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Brent@GMATPrepNow
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DavidG@VeritasPrep
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Brent original!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
I) We could pick numbers here. Say x = 1 and y = 2.
(x + 2)/(y + 2) = 3/4
x/y = 1/2
Indeed 3/4 > 1/2
The inequality will be true no matter what we select.
Algebraically, you could cross multiply (x + 2)/(y + 2) > x/y , to get (xy + 2y)/(xy + 2x) > 1. In this case, we know that the denominator is smaller because x < y and thus 2x < 2y. If the denominator is smaller than the numerator, and we're dealing with positive values, the fraction will be greater than 1. So we know this is true.
II) (x - y)/x < 0
Because we know x is positive, we can multiply both sides by x without worrying about what happens to the inequality sign. So we get x - y < 0. Add y to both sides to get x < y. Well, yes, that's true.
III) 2x/(x + y) < 1
Again, we can multiply both sides by x +y without worrying about the inequality sign flipping because x and y are both positive. So we get 2x < x + y; subtract x and we get x < y. Yep.
The answer is E
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Matt@VeritasPrep
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Let's take them one at a time.
I:
(x + 2)/(y + 2) > x/y
Since x and y are both positive, we cross multiply:
y * (x + 2) > x * (y + 2)
xy + 2y > xy + 2x
2y > 2x
y > x
We were told this already, so this checks out!
II)
(x - y)/x < 0
Since x is positive, we can cross multiply again:
(x - y) < 0*x
(x - y) < 0
x < y
Déjà vu!
III:
2x/(x + y) < 1
You know the drill by now ...
2x < x + y
x < y
Tréjà vu!
I:
(x + 2)/(y + 2) > x/y
Since x and y are both positive, we cross multiply:
y * (x + 2) > x * (y + 2)
xy + 2y > xy + 2x
2y > 2x
y > x
We were told this already, so this checks out!
II)
(x - y)/x < 0
Since x is positive, we can cross multiply again:
(x - y) < 0*x
(x - y) < 0
x < y
Déjà vu!
III:
2x/(x + y) < 1
You know the drill by now ...
2x < x + y
x < y
Tréjà vu!
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Matt@VeritasPrep
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This would be great! Unfortunately, we'd often have to guess, and I'm not sure how accurate the guesses would be. (You'd hope they'd be better than nothing, but they could be harmful.)prada wrote:I like the fact that the level of the question was posted. I wish an expert could tag all questions with approx levels. It gives us an idea
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Brent@GMATPrepNow
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Let's examine each statement individually: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
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
Brent Hanneson - Creator of GMATPrepNow.com
