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GMATPrep : A simple yet confusing DS question on numbers

by anirban_lax » Thu Sep 09, 2010 7:55 pm
Hi

I encountered this problem in course of a practice test and was very confident of my answer till I saw the official answer. Here it is:

Are x and y both positive?

a) 2x - 2y =1
b) x/y > 1

I'm curious to know if someone gets the same answer as mine and hence I'll share the OA later.

Thanks
Anirban
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by Rahul@gurome » Thu Sep 09, 2010 8:19 pm
Solution:
Consider first statement (1) alone.
2x - 2y = 1.
Let x = 1 and y = 1/2.
So 2x - 2y = 1 and x and y are both positive.
Next let x = 1/4 and y = -1/4.
Again 2x - 2y = 1 and x is positive and y is negative.
So we cannot say definitely from statement (1) alone whether x and y are both positive or not.

Next consider statement (2) alone.
It says x/y > 1 or (x-y)/y > 0.
So either case (1) x > y and y > 0
Or case (2) x < y and y < 0.
For example let x = 4 and y = 2, then x/y = 4/2 = 2 > 1.
Here both x and y are positive.
Next let x = -4 and y = -2, then x/y = (-4)/(-2) = 2 >1.
Here both x and y are both negative.
So again from statement (2) alone, we cannot say definitely whether x and y are both positive or not.

Next combine both the statements together and check.
On combining we have that x = y + 1/2 .
Since (x-y)/y > 0, we have that (y+1/2 - y)/y > 0.
Or 1/2y > 0.
Or 2y > 0.
Or y > 0.
Also if y > 0, as seen from case (1), x > y and so x > 0.
Or both x and y are positive.

The correct answer is hence (C).
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by Stuart@KaplanGMAT » Thu Sep 09, 2010 8:27 pm
anirban_lax wrote:Hi

I encountered this problem in course of a practice test and was very confident of my answer till I saw the official answer. Here it is:

Are x and y both positive?

a) 2x - 2y =1
b) x/y > 1

I'm curious to know if someone gets the same answer as mine and hence I'll share the OA later.

Thanks
Anirban
Let's approach with more critical thinking and less algebra.

1) 2x - 2y = 1

dividing both sides by 2:

x - y = 1/2

So, x is .5 greater than y.

Knowing the distance between two numbers tells us nothing about their signs: insufficient.

2) x/y > 1

Since x/y is positive, we know that x and y must have the same sign. So, we could pick two positives OR two negatives: insufficient.

Since x/y > 1 we know that the absolute value of x must be greater than the absolute value of y.

Combined:

From (2), we know that x and y have the same sign.

Since x is .5 greater than y, if we pick two positives we'll also follow x/y > 1.

However, if we pick two negatives, the absolute value of y will be greater than the absolute value of x, violating (2).

Accordingly, only two positives will satisfy both statements: sufficient, choose (C).
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by anirban_lax » Thu Sep 09, 2010 8:42 pm
Ahhhh! I was so close and yet so far.

Thanks a lot for your help! C is indeed the OA.
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by uwhusky » Thu Sep 09, 2010 8:44 pm
Third approach from third expert please =). Preferably even simpler, somehow.
Yep.
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by Ian Stewart » Fri Sep 10, 2010 7:40 am
anirban_lax wrote: Are x and y both positive?

a) 2x - 2y =1
b) x/y > 1
uwhusky wrote:Third approach from third expert please =). Preferably even simpler, somehow.
I can offer one other perspective:

Neither statement is sufficient alone; from Statement 1 we only know that x = y + 1/2 (so x could be -5 and y could be -5.5, say), and from Statement 2 we only know that x/y > 1 (so x could be -3 and y could be -2, say). Clearly x and y could also both be positive using either Statement alone, so we can get a 'yes' answer and a 'no' answer to the question.

Together, from Statement 1 we know that x = y + 1/2, so we know that x > y. If y is negative, we can divide by y on both sides of this inequality (reversing the inequality since we're dividing by a negative) to find that x/y < 1. That can't be true, from Statement 2, so y must be positive. Since x > y, x is also positive, so the two Statements together are sufficient.
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by uwhusky » Fri Sep 10, 2010 8:31 am
Hey Ian,

I have always thought about utilizing the approach of positive number/negative number and flipping the inequality sign, but this may be the first time I am reading an expert doing so. Is there any perceived "red flags" of utilizing this approach, such as confusing yourself or certain numbers do not work?

Thanks again!
Yep.
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by Adi_Pat » Sat Sep 11, 2010 4:40 pm
Ian Stewart wrote:
anirban_lax wrote: Are x and y both positive?

a) 2x - 2y =1
b) x/y > 1
uwhusky wrote:Third approach from third expert please =). Preferably even simpler, somehow.
I can offer one other perspective:

Neither statement is sufficient alone; from Statement 1 we only know that x = y + 1/2 (so x could be -5 and y could be -5.5, say), and from Statement 2 we only know that x/y > 1 (so x could be -3 and y could be -2, say). Clearly x and y could also both be positive using either Statement alone, so we can get a 'yes' answer and a 'no' answer to the question.

Together, from Statement 1 we know that x = y + 1/2, so we know that x > y. If y is negative, we can divide by y on both sides of this inequality (reversing the inequality since we're dividing by a negative) to find that x/y < 1. That can't be true, from Statement 2, so y must be positive. Since x > y, x is also positive, so the two Statements together are sufficient.
I dont get one point here. you've tried to exaplin it above - but im still not clear...

if we use both statements together, we get

x = y + 1/2 and x/y>1
if i read x/y>1 as x>y then i have
x= y+1/2 and x>y...which would give x =1/4 and y = -1/4
but if I read the statement as x/y > 1, then it would imply x,y are both either positive or negative.
I've seen this a few times now, where if I simply the inequality it changes its meaning.

Any advice on this would be appreciated !!!
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by Ian Stewart » Sat Sep 11, 2010 5:30 pm
Adi_Pat wrote: I dont get one point here. you've tried to exaplin it above - but im still not clear...

if we use both statements together, we get

x = y + 1/2 and x/y>1
if i read x/y>1 as x>y then i have
If you see the inequality x/y > 1, and you have no other information, you cannot simply rewrite this as x > y. When you do that, you're multiplying both sides of the inequality by y, and you need to know whether y is positive or negative; if you multiply on both sides of an inequality by a negative, you need to reverse the inequality. So, when you see the inequality x/y > 1, you can conclude that x > y if y is positive, and x < y if y is negative. In other words, one of the following two things must be true:

x > y > 0

or

x < y < 0

but without more information, we don't know which of the two inequalities above is true.

That's the reason some of the solutions to problems similar to the one above might appear a bit convoluted; we need to account for two cases, the case in which y is positive, and the case in which y is negative.
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by Ian Stewart » Sat Sep 11, 2010 5:36 pm
uwhusky wrote:Hey Ian,

I have always thought about utilizing the approach of positive number/negative number and flipping the inequality sign, but this may be the first time I am reading an expert doing so. Is there any perceived "red flags" of utilizing this approach, such as confusing yourself or certain numbers do not work?

Thanks again!
It's always possible, when you see an inequality like x/y > 1, to consider two cases (the case where y is positive and the case where y is negative), to see what must be true, as I did in the post above. In general, any approach that requires you to consider different cases is likely to be more confusing than one that does not require cases, so I suppose that's the only 'red flag' here. I don't often find myself doing this on GMAT inequality questions, because there's usually a 'cleaner' approach available, but it certainly works perfectly well.
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by Adi_Pat » Sun Sep 12, 2010 1:08 pm
Ian Stewart wrote:
Adi_Pat wrote: I dont get one point here. you've tried to exaplin it above - but im still not clear...

if we use both statements together, we get

x = y + 1/2 and x/y>1
if i read x/y>1 as x>y then i have
If you see the inequality x/y > 1, and you have no other information, you cannot simply rewrite this as x > y. When you do that, you're multiplying both sides of the inequality by y, and you need to know whether y is positive or negative; if you multiply on both sides of an inequality by a negative, you need to reverse the inequality. So, when you see the inequality x/y > 1, you can conclude that x > y if y is positive, and x < y if y is negative. In other words, one of the following two things must be true:

x > y > 0

or

x < y < 0

but without more information, we don't know which of the two inequalities above is true.

That's the reason some of the solutions to problems similar to the one above might appear a bit convoluted; we need to account for two cases, the case in which y is positive, and the case in which y is negative.

Makes sense..Thanks !!!
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