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# Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1

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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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05 Aug 2015, 06:37
Bunuel wrote:
arijitb1980 wrote:
Bunuel wrote:
Are x and y both positive?
(1) 2x-2y=1
(2) x/y>1

(1) 2x-2y=1. Well this one is clearly insufficient. You can do it with number plugging OR consider the following: x and y both positive means that point (x,y) is in the I quadrant. 2x-2y=1 --> y=x-1/2, we know it's an equation of a line and basically question asks whether this line (all (x,y) points of this line) is only in I quadrant. It's just not possible. Not sufficient.

(2) x/y>1 --> x and y have the same sign. But we don't know whether they are both positive or both negative. Not sufficient.

(1)+(2) Again it can be done with different approaches. You should just find the one which is the less time-consuming and comfortable for you personally.

One of the approaches:
$$2x-2y=1$$ --> $$x=y+\frac{1}{2}$$
$$\frac{x}{y}>1$$ --> $$\frac{x-y}{y}>0$$ --> substitute x --> $$\frac{1}{y}>0$$ --> $$y$$ is positive, and as $$x=y+\frac{1}{2}$$, $$x$$ is positive too. Sufficient.

Hello Bunuel,

I am preparing for GMAT and you are a master in quantitatives,I salute you. However this problem I did not understand why it is C.I got E by plugging in numbers.
Considerin Stmt (1)
2x-2y=1
Means x-y=1/2
Now i used 2 numbers x=2,y=1.5 and x= -1.5 and y= -2
So (1) is insufficient as both x and y can be both +ve or -ve.
Stmt(2)
x/y > 1
Means x >y
If i use the above numbers x=2,y=1.5 and x= -1.5 and y= -2
then this holds true but cannot determine if x and y are both positive or negetive.
Combining (1) and (2) I still cannot determine if x and y are both positive or negetive .
So for me answer is E .
Please let me know what am I missing here.

Thanks

As stated above, x= -1.5 and y= -2 does not satisfy the second statement, hence you cannot use these values when plugging numbers.

Hi,

While I see that translating (x/y)>1 into x>y must be wrong, as clearly it lets me use a different range of numbers (and is the reason I chose E instead of C), I don't understand what is wrong with my algebra.

Why can't I multiply (x/y) * y > 1 * y and get x>y?

Thank you.
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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05 Aug 2015, 07:07
1
Matthias15 wrote:

Hi,

While I see that translating (x/y)>1 into x>y must be wrong, as clearly it lets me use a different range of numbers (and is the reason I chose E instead of C), I don't understand what is wrong with my algebra.

Why can't I multiply (x/y) * y > 1 * y and get x>y?

Thank you.

Hi Matthias15

If (x/y) > 1

Case-1: If y >0 then x > y

Case-2: If y <0 then x < y

Please note: Multiplying a Negative Number on both sides require the Inequation sign to flip [as mentioned in Case-2]

That's where you are wrong.

I hope it helps!
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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24 Sep 2015, 16:00
What happens when y = o?

X = 0.5 and y = 0, satisfies i and x/y is indeed > 0 since anything divided by 0 is infinity. But, despite this, y is not positive as it '0'.

Even in this case -

"One of the approaches:
2x−2y=1 --> x=y+12
xy>1 --> x−yy>0 --> substitute x --> 1y>0 --> y is positive, and as x=y+12, x is positive too. Sufficient."

The moment you assume y = 0, you can easily see that 1/y is infinity that is greater than 0 and hence, satisfies the equation.

Am i missing something. My answer would be 'E'
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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24 Sep 2015, 16:16
3
1
Manbehindthecurtain wrote:
Are x and y both positive?

(1) 2x-2y = 1
(2) x/y > 1

Target question: Are x and y both positive?

Statement 1: 2x - 2y = 1
There are several pairs of numbers that satisfy this condition. Here are two:
Case a: x = 1 and y = 0.5, in which case x and y are both positive
Case b: x = -0.5 and y = -1, in which case x and y are not both positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x/y > 1
This tells us that x/y is positive. This means that either x and y are both positive or x and y are both negative. Here are two possible cases:
Case a: x = 4 and y = 2, in which case x and y are both positive
Case b: x = -4 and y = -2, in which case x and y are not both positive
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2
Statement 1 tells us that 2x - 2y = 1.
Divide both sides by 2 to get: x - y = 1/2
Solve for x to get x = y + 1/2

Now take the statement 2 inequality (x/y > 1) and replace x with y + 1/2 to get:
(y + 1/2)/y > 1
Rewrite as: y/y + (1/2)/y > 1
Simplify: 1 + 1/(2y) > 1
Subtract 1 from both sides: 1/(2y) > 0
If 1/(2y) is positive, then y must be positive.

Statement 2 tells us that either x and y are both positive or x and y are both negative.
Now that we know that y is positive, it must be the case that x and y are both positive
Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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24 Sep 2015, 16:21
2
jitendra31 wrote:
What happens when y = o?
X = 0.5 and y = 0, satisfies i and x/y is indeed > 0 since anything divided by 0 is infinity. But, despite this, y is not positive as it '0'.
'

It may seem that 0.5/0 = infinity, but this is not the case.
If we approach 0 from the positive side, then it looks like 0.5/0 is a REALLY BIG POSITIVE NUMBER
0.5/0.1 = 5
0.5/0.01 = 50
0.5/0.001 = 500
0.5/0.0001 = 5000
0.5/0.00001 = 50000
etc.

But what if we approach 0 from the NEGATIVE side:
0.5/(-0.1) = -5
0.5/(-0.01) = -50
0.5/(-0.001) = -500
0.5/(-0.0001) = -5000
0.5/(-0.00001) = -50000
Here it looks like 0.5/0 will be a REALLY BIG NEGATIVE NUMBER

This is why we say that x/0 is undefined.

Cheers,
Brent
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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06 May 2016, 06:42
1
LM wrote:
Are x and y both positive?

(1) 2x - 2y = 1
(2) x/y > 1

Solution:

We need to determine whether x and y are both positive.

Statement One Alone:

2x – 2y = 1

Simplifying statement one we have:

2(x – y) = 1

x – y = ½

The information in statement one is not sufficient to determine whether x and y are both positive. For instance if x = 1 and y = ½, x and y are both positive; however if x = -1/2 and y = -1, x and y are not both positive. We can eliminate answer choices A and D.

Statement Two Alone:

x/y > 1

Using the information in statement two, we see that x and y can both be positive or both be negative. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using the information in statements one and two we know that x – y = ½ and that x/y > 1. Isolating x in the equation we have: x = ½ + y. We can now substitute ½ + y for x in the inequality x/y > 1 and we have:

(1/2 + y)/y > 1

(1/2)/y + y/y > 1

1/2y + 1 > 1

1/2y > 0

Thus, y must be greater than zero. Also, since x/y is greater than one, x also must be greater than zero.

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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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22 Jun 2016, 21:17
There are 2 variables (x and y) in the original condition. In order to match the number of variables and the number of equations, we need 2 equations. Since the condition 1) and 2) each has 1 equation, there is high chance that the correct answer is C. Using 1) and 2), from 2(x-y)=1>0 we get 2(x-y)>0, x>y. Using 2), if we multiply both sides by y^2, we get xy>y^2, xy-y^2>0, y(x-y)>0. Since x-y>0, we get y>0. From x>y>0, x>0. The answer is always yes and the condition is sufficient. Hence, the correct answer is C.

For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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22 Jun 2016, 22:03
Manbehindthecurtain wrote:
Are x and y both positive?

(1) 2x-2y = 1
(2) x/y > 1

Statement 1
2x-2y = 1

x - y = (1/2)

x = 3/2; y = 1 ; both x and y positive

x = 1/4, y = -1/4; x is positive, y is negative

so not sufficient

Statement 2

(x/y) > 1

x = 3/2; y = 1 ; (x/y) is more than 1. both x and y positive

x = -2; y = -1 ; (x/y) is more than 1. both x and y negative

so not sufficient

Combining Statement 1 and 2

x - y = 1/2

x = y+(1/2)

x/y > 1

{y+(1/2)} / y is more than 1

1+(1/2y) > 1

(1/2y) > 0

y has to be positive. because x = y+1/2, so x also has to be positive.

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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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22 Jan 2017, 12:47
What about the example of y=0? Then x/y is surely greater than 1, hence I chose answer E.
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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22 Jan 2017, 13:50
Top Contributor
1
vgmatv wrote:
What about the example of y=0? Then x/y is surely greater than 1, hence I chose answer E.

x/0 can be really big OR really small...

From my earlier post:

If we approach 0 from the positive side, then it looks like 0.5/0 is a REALLY BIG POSITIVE NUMBER
0.5/0.1 = 5
0.5/0.01 = 50
0.5/0.001 = 500
0.5/0.0001 = 5000
0.5/0.00001 = 50000
etc.

But what if we approach 0 from the NEGATIVE side:
0.5/(-0.1) = -5
0.5/(-0.01) = -50
0.5/(-0.001) = -500
0.5/(-0.0001) = -5000
0.5/(-0.00001) = -50000
Here it looks like 0.5/0 will be a REALLY BIG NEGATIVE NUMBER

This is why we say that x/0 is undefined.

Cheers,
Brent
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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20 Jun 2017, 22:27
TaN1213 wrote:
Bunuel wrote:
Are x and y both positive?
(1) 2x-2y=1
(2) x/y>1

(1) 2x-2y=1. Well this one is clearly insufficient. You can do it with number plugging OR consider the following: x and y both positive means that point (x,y) is in the I quadrant. 2x-2y=1 --> y=x-1/2, we know it's an equation of a line and basically question asks whether this line (all (x,y) points of this line) is only in I quadrant. It's just not possible. Not sufficient.

(2) x/y>1 --> x and y have the same sign. But we don't know whether they are both positive or both negative. Not sufficient.

(1)+(2) Again it can be done with different approaches. You should just find the one which is the less time-consuming and comfortable for you personally.

One of the approaches:
$$2x-2y=1$$ --> $$x=y+\frac{1}{2}$$
$$\frac{x}{y}>1$$ --> $$\frac{x-y}{y}>0$$ --> substitute x --> $$\frac{1}{y}>0$$ --> $$y$$ is positive, and as $$x=y+\frac{1}{2}$$, $$x$$ is positive too. Sufficient.

Discussed here: http://gmatclub.com/forum/ds1-93964.htm ... approaches and also here along with other hard inequality problems: http://gmatclub.com/forum/inequality-an ... 86939.html

Hope it helps.

Hi Buñuel,

If we put x=1/2 & y=0 , would you please explain how are both statements together sufficient? Both statements hold true for these values of x and y, yet 0 is not positive.

Regards.
chris558 wrote:
Are x and y both positive?

1) 2x-2y=1
2(x-y)=1
x-y=1/2
-->3/4-1/4=1/2....YES
-->-1/4-(-3/4)=1/2...NO
INSUFFICIENT

2) x/y>1
This just means that x and y have the same sign. They're either both positive or both negative.
INSUFFICIENT

1&2)
x=1/2+y

(1/2+y)/y>1
y/2 + 1 > 1
y/2 > 0 which means that Y is greater than 0. And since both x and y have the same sign, both x and y are Positive. YES.

Sent from my Redmi Note 4 using GMAT Club Forum mobile app

y cannot be 0 because in this case x/y will not be defined (division by 0 is not allowed) and not greater than 1 as given in the second statement.
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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21 Jun 2017, 13:36
Are x and y both positive?

(1) 2x-2y = 1

2(x-y) = 1

x-y = 1/2 = 05

Lets substitute some values and check:

x = 0.1 & y = -0.4

0.1 - (-0.4) = 0.5 ==============> Answer to the question is NO

x = 1 & y = 0.5

1 - 05 = 0.5 =================> Answer to the questions is YES

As we are getting multiple answers, Statement (1) is Not Sufficient.

(2) x/y > 1

Which means both x & y have same sing, they can be either positive or both negative.

As we are getting multiple answers, statement (2) us Not Sufficient.

Now, lets combine both (1) and (2)

we get:

x = y + 1/2

also, we know x/y>1

which means: (y+1/2)/y>1

which means y is > 0

as y>0, x has to be > 0 as well. So we can come to the conclusion that both x & y are positive.

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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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15 Jul 2017, 22:46
First of all, everyone should really read Bunel's earlier posts before just tagging him. He must have written the same explanation five times!

Let me try:

Statement 1:
2x - 2y =1
2*(x-y) = 1
x - y =0.5

Ok cool, but you can probably tell intuitively that a difference of 0.5 doesn't really help. X and Y could be 1 and 0.5 or 0.25 and -0.25.

So this is not really that helpful, however it was important to do this algebra because, often on DS questions, you need to see how the statements work together and simpler expressions are much easier to work with. So, always at least try and simplify stuff wherever possible as the GMAT really seems to reward that reflex.

Statement 2:
x/y > 1

This should be an immediate reaction: same sign. If you don't immediately think that, practice until you do.

Now, as Bunel mentioned, there are two approaches. Number picking and algebra. Whenever possible, I try to use algebra. It's really personal preference on a question like this, but I have a panic that goes off in my head where I am scared I am missing a potential extreme value. So I try to use algebra to avoid that feeling.

Whenever you have an equation and an inequality, you always want to substitute the equation into the inequality (the other way around is not really permitted).

Notice that it actually doesn't matter which value you isolate. Because you know x and y have the same sign, you are just trying to figure out the sign of either one of them.

equation 1: x = y + 0.5
equation 2: x/y > 1

Now you can clean up the second equation as Bunel did, or you can go straight for it.

(y + 0.5)/y > 1
y/y + 0.5/y >1
1 + 0.5/y > 1
0.5y > 0

So a positive number (0.5) multiplied by y is greater that 0. Ok so y has to be positive. And if y is positive......so is x (because statement 2 told us that).

(C)

Keep practicing everyone!
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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05 Apr 2018, 17:11
1
1
Are x and y both positive?

GRAPHIC APPROACH.

Notice that the question is basically asks whether the point (x, y) is in the first quadrant.

(1) $$2x - 2y = 1$$. Draw line $$y=x-\frac{1}{2}$$:

Not sufficient.

(2) $$\frac{x}{y} > 1$$. Draw line $$\frac{x}{y}=1$$. The solutions is the green region:

Not sufficient.

(1)+(2) Intersection is the portion of the blue line which lies in the first quadrant. Sufficient.

Attachment:

graph.png [ 6.13 KiB | Viewed 1777 times ]

Attachment:

graph %282%29.png [ 5.95 KiB | Viewed 1782 times ]

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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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30 Apr 2018, 07:42
Hi everyone,

Sorry but i dont understand the condition that 1/2y > 0 so y must be positive. But i think y can also be negative because when y is negative so 2y is still positive and 1/2y > 0 anyway. Where am i wrong? Please help. Thank you so much
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 [#permalink]

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30 Apr 2018, 07:54
Hungluu92vn wrote:
Hi everyone,

Sorry but i dont understand the condition that 1/2y > 0 so y must be positive. But i think y can also be negative because when y is negative so 2y is still positive and 1/2y > 0 anyway. Where am i wrong? Please help. Thank you so much

For $$\frac{1}{2y}>0$$ to be true y must be positive. It cannot be negative because in this case 2y = 2*negative = negative, so $$\frac{1}{2y}=\frac{1}{negative}=negative<0$$.
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1   [#permalink] 30 Apr 2018, 07:54

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