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Are X and Y both positive
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800_or_bust
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(1) Not sufficient. x = 1, y = 1/2 OR x = 1/2, y = 0, etc...shubh425 wrote:Please explain.
(2) Not sufficient. This implies x and y have the same sign, and x has a larger absolute value than y. But they could both be positive, or both be negative.
Combined (1) and (2) are sufficient. If x and y were both negative and x has a larger absolute value than y, then 2x - 2y has to evaluate to be negative. You would be adding a positive term to a negative term with a larger absolute value, and the result would always be negative. Thus, x and y must both be positive. The answer is C.
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Target question: Are x and y both positive?Are x and y both positive?
1) 2x - 2y =1
2) x/y >1
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
Answer = C
Cheers,
Brent
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Dear Brent,Brent@GMATPrepNow wrote:Target question: Are x y both positive?Are x y both positive?
1) 2x - 2y =1
2) x/y >1
Statement 2: x/y > 1
This tells us that x/y is positive. This means that either x y are both positive or x y are both negative. Here are two possible cases:
Case a: x = 4 y = 2, in which case x y are both positive
Case b: x = -4 y = -2, in which case x y are not both positive
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Brent
Could you please clear this doubt of mine.
IS x/y>1 != x>y
I solved this problem in this process
Since x/y>1 it should also follow x>y
if x=3 , y=2
3/2>1 and 3>2.
whereas if x=-4 y=-2
x/y >1
but x>y is false
so B is sufficient , both x y should be positive.
Where am I thinking wrong?
Regards
Teja
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Brent@GMATPrepNow
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The part above (in green) is incorrect.evs.teja wrote:
Dear Brent,
Could you please clear this doubt of mine.
IS x/y>1 != x>y
I solved this problem in this process
Since x/y>1 it should also follow x>y
if x=3 , y=2
3/2>1 and 3>2.
whereas if x=-4 y=-2
x/y >1
but x>y is false
so B is sufficient , both x y should be positive.
Where am I thinking wrong?
Regards
Teja
You have taken x/y > 1 and multiplied both sides by y to get x > y
We can only do this if we are CERTAIN that y is positive.
For example, we know that 3/2 > 1
If we multiply both sides by 2, we get 3 > 2. Great!
However, what about this case: (-3)/(-2) > 1
If we multiply both sides by -2, we get -3 > -2. Oops!
Since can cannot be certain that y is positive in the inequality x/y > 1, we cannot multiply both sides by y to get x > y
For more on this, see the following video: https://www.gmatprepnow.com/module/gmat ... /video/979
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Danny@GMATAcademy
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I think most points are lost on this question when test takers try to multiply both sides of the inequality in statement 2 by y to get x > y.
The reason we cannot do this is that we don't know whether y is positive or negative. Recall that for inequalities, if you multiply or divide by a negative, you have to flip the direction of the inequality sign ( 3 > 2 ----> -3 < -2). But if we don't know the sign of the thing your multiplying both sides by, we don't know if the inequality sign stays the same or flips around.
Often, I find it very helpful to keep track of both options. For example, here:
x/y > 1
case 1: if y pos: x > y
case 2: if y neg: x < y
For this question, doing this might help make the subsequent deductions a little easier. Statement 1 can be written as x = y + 1/2. In other words, x is 1/2 greater than y. And if x is greater, we can't be in case 2 above. We must be in case 1. y must be positive. And if y is positive, and x is greater than y, x muust be positive too.
The reason we cannot do this is that we don't know whether y is positive or negative. Recall that for inequalities, if you multiply or divide by a negative, you have to flip the direction of the inequality sign ( 3 > 2 ----> -3 < -2). But if we don't know the sign of the thing your multiplying both sides by, we don't know if the inequality sign stays the same or flips around.
Often, I find it very helpful to keep track of both options. For example, here:
x/y > 1
case 1: if y pos: x > y
case 2: if y neg: x < y
For this question, doing this might help make the subsequent deductions a little easier. Statement 1 can be written as x = y + 1/2. In other words, x is 1/2 greater than y. And if x is greater, we can't be in case 2 above. We must be in case 1. y must be positive. And if y is positive, and x is greater than y, x muust be positive too.
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800_or_bust
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Remember when you multiply and divide across an equality, the direction of the sign changes if the value being multiplied or divided is negative. Thus, we cannot say that if x/y > 1 that x > y unless we know for certain that y is positive.evs.teja wrote:Dear Brent,Brent@GMATPrepNow wrote:Target question: Are x y both positive?Are x y both positive?
1) 2x - 2y =1
2) x/y >1
Statement 2: x/y > 1
This tells us that x/y is positive. This means that either x y are both positive or x y are both negative. Here are two possible cases:
Case a: x = 4 y = 2, in which case x y are both positive
Case b: x = -4 y = -2, in which case x y are not both positive
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Brent
Could you please clear this doubt of mine.
IS x/y>1 != x>y
I solved this problem in this process
Since x/y>1 it should also follow x>y
if x=3 , y=2
3/2>1 and 3>2.
whereas if x=-4 y=-2
x/y >1
but x>y is false
so B is sufficient , both x y should be positive.
Where am I thinking wrong?
Regards
Teja
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Matt@VeritasPrep
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We could also say
x - y = 1/2
and
x/y > 1
x/y - 1 > 0
(x - y)/y > 0
Combining the two, we have
(1/2)/y > 0
So y must be positive. Since y > 0, we know x = 1/2 + (something greater than 0), so x is also positive.
This keeps us away from the sign-flipping hazard, and is easier for some students to see (and believe!)
x - y = 1/2
and
x/y > 1
x/y - 1 > 0
(x - y)/y > 0
Combining the two, we have
(1/2)/y > 0
So y must be positive. Since y > 0, we know x = 1/2 + (something greater than 0), so x is also positive.
This keeps us away from the sign-flipping hazard, and is easier for some students to see (and believe!)
