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# Are x and y both negative? (1) 3x - 3y = 1 (2) x/y < 1

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Math Expert
Joined: 02 Sep 2009
Posts: 49275
Are x and y both negative? (1) 3x - 3y = 1 (2) x/y < 1  [#permalink]

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26 Nov 2017, 23:27
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Difficulty:

55% (hard)

Question Stats:

61% (01:21) correct 39% (01:21) wrong based on 57 sessions

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Are x and y both negative numbers?

(1) 3x - 3y = 1
(2) x/y < 1

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Math Expert
Joined: 02 Sep 2009
Posts: 49275
Re: Are x and y both negative? (1) 3x - 3y = 1 (2) x/y < 1  [#permalink]

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26 Nov 2017, 23:28
Bunuel wrote:
Are x and y both negative numbers?

(1) 3x - 3y = 1
(2) x/y < 1

Similar question from GMAT Prep: https://gmatclub.com/forum/are-x-and-y- ... 63377.html
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Re: Are x and y both negative? (1) 3x - 3y = 1 (2) x/y < 1  [#permalink]

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26 Nov 2017, 23:43
Bunuel wrote:
Are x and y both negative numbers?

(1) 3x - 3y = 1
(2) x/y < 1

Hi Bunuel

Stmnt 1: x - y= 1/3.. Not Sufficient.
Stmnt 2: y-x/y > 0.. Not Sufficient..

1 +2.. (-1/3)/y > 0.. which means y = -ve number
x = y + 0.33.. Here, x can be both +ve and -ve.

Cab you please let me know if there is anything wrong with the approach.
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Re: Are x and y both negative? (1) 3x - 3y = 1 (2) x/y < 1  [#permalink]

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27 Nov 2017, 01:20
(1) 3x-3y=1
x-y=1/3
case 1, both x and y are -ve:
x-y=-8/3-(-9/3)=1/3
case 2, = x and y are +ve:
x-y=3-8/3=1/3

Insufficient

(2) from this statement we understand, that |x|<|y|
(1) + (2)
Combining, we eliminate case 2 from reasoning in (1) statement taken the condition in (2), so we get that both statements are valid only when x and y take -ve values
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Are x and y both negative? (1) 3x - 3y = 1 (2) x/y < 1  [#permalink]

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Updated on: 29 Nov 2017, 13:30
Bunuel wrote:
Are x and y both negative numbers?

(1) 3x - 3y = 1
(2) x/y < 1

Statement 1 tells us $$x > y$$ by $$\frac{1}{3}$$
Insufficient, as $$x$$ and $$y$$ could be any sign.

Statement 2 tells us $$\frac{x}{y}$$ is less than one. Again just the same info in statement 1 $$(x>y)$$, except not as exact. Insufficient.

Now let's step back and consider both equations together. One or both variables have to be negative given both statements. Since we have 2 different cases with the same result, then neither statement is sufficient. Hence E.

While there are infinite numerical examples, $$-1$$ and $$\frac{-2}{3}$$ or $$\frac{1}{6}and \frac{-1}{6}$$ are prime examples.

CASE 1 $$(\frac{-2}{3}, -1)$$

$$\frac{-2}{3} - (-1)$$ = $$\frac{1}{3}$$... (1)

$$\frac{2}{3}$$ is absolutely less than $$1$$.... (2)

CASE 2 $$(\frac{1}{6}, \frac{-1}{6})$$

$$\frac{1}{6} - (\frac{-1}{6})$$ = $$\frac{1}{3}$$... (1)
$$\frac{x}{y} = -1$$ $$<1$$... (2)

Since both variables could negative or one could, then the answer is E.

Note that I wrongly assumed that the one negative and positive coordinates (case 2) won't make a difference and the answer should be C. However, amanvamagmat below called my attention to this error in his post.

Best,

Originally posted by rulingbear on 28 Nov 2017, 16:54.
Last edited by rulingbear on 29 Nov 2017, 13:30, edited 2 times in total.
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Re: Are x and y both negative? (1) 3x - 3y = 1 (2) x/y < 1  [#permalink]

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28 Nov 2017, 23:24
1
(1) 3x-3y=1 or x-y = 1/3. This just tells us that x>y, we dont know about their signs. Both could be positive, both could be negative, or we can have x positive, y negative. Insufficient.

(2) x/y < 1. Again we cant say about the signs.
Both x&y could be positive, in which case: x<y
Both x&y could be negative, in which case: x>y
One out of x&y could be positive, and other negative; in which case x/y will be negative and hence < 1
Insufficient.

Combining the two statements,
Both x&y cannot be positive, because in that case x-y=1/3 and x/y < 1 both cannot be true
Both x&y CAN be negative, eg; x=-1, y=-4/3. In this case: x-y = 1/3 and x/y < 1
We CAN also have x positive, y negative, eg: x=1/6, y=-1/6. In this case: x-y = 1/3 and x/y < 1

So there is no surety whether both x&y are negative or only one out of them is negative. Insufficient. Hence E answer
Re: Are x and y both negative? (1) 3x - 3y = 1 (2) x/y < 1 &nbs [#permalink] 28 Nov 2017, 23:24
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