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

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18 Aug 2012, 21:01

Bunuel... For sure, this problem can be easily solved using algebra, just as you have explained before. However, I was trying to use geometry to solve this problem. The equation of the line is x/0.5 + y/-0.5= 1, which means that the x and y intercepts are 0.5 and -0.5 respectively. Hence, the line passes through 1st, 3rd, and 4th quadrants.

In Q1, both x and y are +ve. In Q4, y is negative and hence this is out. Where I am getting confused is Q3: how do I verify whether x/y >1?

Re: Are X and Y both positive? GMAT PREP CAT [#permalink]

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02 Oct 2012, 19:42

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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.

Bunuel i would like tto know how \(\frac{1}{y}>0\) have this : if I have ( y + 1 - y / 2 ) / y > 0 the result should be \(\frac{1}{2y}>0\) and not \(\frac{1}{y}> 0\)

can you please explain ??'

thanks

@edited ............I have seen the explanation in another answer by you ) Ok
_________________

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

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17 Jan 2013, 04:55

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Manbehindthecurtain wrote:

Are x and y both positive?

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

1. x-y = 1/2 This means that the distance between x and y is 1/2 unit and that x is greater than y. But x and y could be positive such as x=5 and y=4.5, OR x and y could be both negative such as x=-4 and y=-4.5

INSUFFICIENT.

2. x/y > 1 This shows that x and y must be positive meaning they are either both (+) or both (-). ex) x/y = 5/2 OR x/y = -5/-2 = 5/2 still > 1

INSUFFICIENT.

Combine. Let x = 5 and y=9/2: 5/(9/2) = 10/9 > 1 - This means when x and y are both positive it could be a solution to x/y > 1 Let x = -4 and y=-9/2: -4/(-9/2) = 8/9 < 1 - This means when x and y are negative it could not be a solution to x/y > 1

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.

Answer is C.

Shouldn't (1/2+y)/y>1 simplify to (1/2y) + 1 > 1 ? Or am I missing something? Still get the right answer following this logic but I believe this step is off.

Re: Are X and Y both positive? GMAT PREP CAT [#permalink]

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11 Apr 2013, 10:06

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.

(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.

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

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14 Sep 2013, 22:23

Hello Bunuel, Request you to please provide your comments on the doubt posted here-

Usually, whenever I see combining an inequality and equation, I substitute the value of one of the variable in the inequality and then analyze the effect. So, going by that approach;

x-y=1/2 ---(1) x/y>1 --(2) Substituting the value of x in equation(2)

(y+1/2)/y>1

Lets assume that y is positive-

(y+1/2) > y

1/2>0 --This means that our assumption is true since 1/2 is greater than Zero. Hence, y > 0

Now, Lets assume that y is negative-

Now, here I'm stuck, I know that multiplying by a negative number changes the sign of the inequality. I'm sure that the sign will be changed but what would be the resulting equation. I mean, do we need to replace y with "-y" in the whole equation. Please clarify. Which of the following would be correct then

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

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14 Sep 2013, 22:39

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imhimanshu wrote:

Hello Bunuel, Request you to please provide your comments on the doubt posted here-

Usually, whenever I see combining an inequality and equation, I substitute the value of one of the variable in the inequality and then analyze the effect. So, going by that approach;

x-y=1/2 ---(1) x/y>1 --(2) Substituting the value of x in equation(2)

(y+1/2)/y>1

Lets assume that y is positive-

(y+1/2) > y

1/2>0 --This means that our assumption is true since 1/2 is greater than Zero. Hence, y > 0

Now, Lets assume that y is negative-

Now, here I'm stuck, I know that multiplying by a negative number changes the sign of the inequality. I'm sure that the sign will be changed but what would be the resulting equation. I mean, do we need to replace y with "-y" in the whole equation. Please clarify. Which of the following would be correct then

a) y+1/2 <y b) y+1/2 < -y c) -y+1/2 < -y

Please help. Thanks

Refer to the highlighted portion : Actually you don't have to take 2 cases at this point: The expression you have is : \(\frac{y+0.5}{y}>1 \to 1+\frac{0.5}{y}>1 \to \frac{1}{y}>0\)--> Hence, y>0.

As for your doubt, if y is negative, we cross-multiply it and get : \(y+0.5<y \to 0>0.5\), which is absurd.

If y is negative, then -y would be positive, and for multiplying a positive quantity, you don't need to flip signs. So , yes expression a is correct.
_________________

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

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19 Nov 2013, 11:29

Hi I get confused in this question. I understand A,B,D are not answer bu confuse in C and E.However,official answer is C.

My approach 1) x=y+(1/2) Not sufficient 2) x/y>1 Not sufficient 1+2) x=y+(1/2) So plugging in a value of y which makes x>y by statement 2 . So If,y=-2.5 which gives x=-2 then No If, y=2 x=2.5 then Yes So answer is E. Please correct me where I am wrong.

Hi I get confused in this question. I understand A,B,D are not answer bu confuse in C and E.However,official answer is C.

My approach 1) x=y+(1/2) Not sufficient 2) x/y>1 Not sufficient 1+2) x=y+(1/2) So plugging in a value of y which makes x>y by statement 2 . So If,y=-2.5 which gives x=-2 then No If, y=2 x=2.5 then Yes So answer is E. Please correct me where I am wrong.

x=-2 and y=-2.5 does not satisfy x/y>1.
_________________

Re: Are X and Y both positive? GMAT PREP CAT [#permalink]

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22 Dec 2013, 00:27

Bunuel wrote:

Are x and y both positive?

\(2x-2y=1\) --> \(x=y+\frac{1}{2}\) \(\frac{x}{y}>1\) --> \(\frac{x-y}{y}>0\) --> substitute x --> \(\frac{1}{y}>0\) --> \(y\) is positive, and as \(x=y+\frac{1}{2}\), \(x\) is positive too. Sufficient.

Hope it helps.

Sorry for the bump but could you elaborate on the last part where you go from x/y>1 to (x-y)/y>0 to 1/y>0 ..?

\(2x-2y=1\) --> \(x=y+\frac{1}{2}\) \(\frac{x}{y}>1\) --> \(\frac{x-y}{y}>0\) --> substitute x --> \(\frac{1}{y}>0\) --> \(y\) is positive, and as \(x=y+\frac{1}{2}\), \(x\) is positive too. Sufficient.

Hope it helps.

Sorry for the bump but could you elaborate on the last part where you go from x/y>1 to (x-y)/y>0 to 1/y>0 ..?

I don't quite follow this algebra

\(\frac{x}{y}>1\) --> \(\frac{x}{y}-1>\) --> \(\frac{x-y}{y}>0\). Now, substitute \(x=y+\frac{1}{2}\) there to get \(\frac{1}{2y}>0\), which further simplifies to \(\frac{1}{y}>0\).

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

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23 Dec 2013, 03:53

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Manbehindthecurtain wrote:

Are x and y both positive?

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

Plug in approach that can be used without thinking much and very likely arrive at the correct answer.

Values to be taken: x positive and negative and find the corresponding values for y based on the statements Note: x and y cannot be of different signs and also x cannot be zero as they will not satisfy (ii)

(i) x=10, we have y =9.5 .Both positive satisfied And now x=-10, we have y=-9.5. Both negative also satisfied .Different results. So (i) alone not sufficient

(ii) x=10, y can be positive. Both positive satisfied . And now x=-10, y can be negative. Both negative also satisfied. So (ii) alone not sufficient

(i) + (ii) x=10, y=9.5 satisfies both the statements . Both positive satisfied . And now x=-10. Value of y is found from (i) and is negative , but we see it does not satisfy (ii). So both cannot be negative .

So we can answer the question using (i) and (ii) together
_________________

Re: Are X and Y both positive? GMAT PREP CAT [#permalink]

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25 Jan 2014, 21:15

Bunuel wrote:

\(\frac{x}{y}>1\) does not mean that \(x>y\). If both \(x\) and \(y\) are positive, then \(x>y\), BUT if both are negative, then \(x<y\). What you are actually doing when writing \(x>y\) from \(\frac{x}{y}>1\) is multiplying both parts of inequality by \(y\): never multiply (or reduce) an inequality by variable (or by an expression with variable) if you don't know the sign of it or are not certain that variable (or expression with variable) doesn't equal to zero.

So from (2) \(\frac{x}{y}>1\), we can only deduce that \(x\) and \(y\) have the same sigh (either both positive or both negative).

See the complete solution of this problem in my previous post.

Hope it helps.

Hi Bunuel,

Please can you elaborate on the below part

\(\frac{x}{y}>1\) does not mean that \(x>y\). If both \(x\) and \(y\) are positive, then \(x>y\), BUT if both are negative, then \(x<y\).

for this case i took ii as X>Y, then taking X = -3 and Y = -3.5 i can satisfy ii in both ways if X and Y are -ve or X and Y are +ve. But have read in this forum that we should first change the inequality to x/y > 1. Why do we need to do that? Any theory around this will be helpful.

Thanks in advance
_________________

“Confidence comes not from always being right but from not fearing to be wrong.”

\(\frac{x}{y}>1\) does not mean that \(x>y\). If both \(x\) and \(y\) are positive, then \(x>y\), BUT if both are negative, then \(x<y\). What you are actually doing when writing \(x>y\) from \(\frac{x}{y}>1\) is multiplying both parts of inequality by \(y\): never multiply (or reduce) an inequality by variable (or by an expression with variable) if you don't know the sign of it or are not certain that variable (or expression with variable) doesn't equal to zero.

So from (2) \(\frac{x}{y}>1\), we can only deduce that \(x\) and \(y\) have the same sigh (either both positive or both negative).

See the complete solution of this problem in my previous post.

Hope it helps.

Hi Bunuel,

Please can you elaborate on the below part

\(\frac{x}{y}>1\) does not mean that \(x>y\). If both \(x\) and \(y\) are positive, then \(x>y\), BUT if both are negative, then \(x<y\).

for this case i took ii as X>Y, then taking X = -3 and Y = -3.5 i can satisfy ii in both ways if X and Y are -ve or X and Y are +ve. But have read in this forum that we should first change the inequality to x/y > 1. Why do we need to do that? Any theory around this will be helpful.

Thanks in advance

Sorry but I don't follow what you mean...
_________________

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

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29 Jan 2014, 22:28

Is it safe to solve this kind of questions based on logic?

I didn't jump into calculations/plug-ins, since statement (1) is clearly insufficient. And statement (2) states that x & y both have the same sign, so combining them together, the result of subtraction is a positive number, and given from (2) that they have the same sign, then they both must be positive.

gmatclubot

Re: Are x and y both positive? (1) 2x-2y = 1 (2) x/y > 1
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29 Jan 2014, 22:28

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