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If 0 < x < y, then which of the following MUST be true

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by Brent@GMATPrepNow » Fri Apr 21, 2017 5:29 am
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

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by DavidG@VeritasPrep » Fri Apr 21, 2017 7:13 am
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
Brent original!

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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by prada » Sun Apr 23, 2017 2:27 pm
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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by Matt@VeritasPrep » Fri Apr 28, 2017 12:37 am
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! :D
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by Matt@VeritasPrep » Fri Apr 28, 2017 12:38 am
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
:)
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.)
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by Brent@GMATPrepNow » Fri May 25, 2018 9:50 am
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
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