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absolute values

by prachi18oct » Tue Jun 23, 2015 12:44 pm
If x and y are non-zero integers and |x| + |y| = 32, what is xy?

(1) -4x - 12y = 0

(2) |x| - |y| = 16

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
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Source: — Data Sufficiency |
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by MartyMurray » Tue Jun 23, 2015 12:59 pm
prachi18oct wrote:If x and y are non-zero integers and |x| + |y| = 32, what is xy?

(1) -4x - 12y = 0

(2) |x| - |y| = 16
(1) Rewrite the equation as -4x = 12y >>> x = -3y

So, |x| = 3|y|.

We can plug this into the original equation to get 4|y| = 32 and |y| = 8 So |x| = 24

Because x = -3y, either x = 24 and y = -8 or y = 8 and and x = -24. Either way, xy = -192

So Statement 1 is sufficient.

(2) This can be added to the equation in the question to get 2|x| = 48 So |x| = 24 and therefore |y| = 8.

This can be satisfied with any combination of x = 24, x = -24, y = 8, or y = -8. In other words, both could be positive, both could be negative or one could be positive and the other negative. So we cannot determine whether xy = 192 or xy = -192

So Statement 2 is insufficient.

Choose A.

One cool takeaway from this question comes from noticing that we didn't need to determine which of x and y is negative and which is positive for Statement 1 to be sufficient. We just needed to know that one has to be negative and the other positive. So we see that sufficiency can be achieved without having the ability to determine actual values for variables.
Last edited by MartyMurray on Tue Jun 23, 2015 1:15 pm, edited 2 times in total.
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by prachi18oct » Tue Jun 23, 2015 1:01 pm
Marty Murray wrote:
prachi18oct wrote:If x and y are non-zero integers and |x| + |y| = 32, what is xy?

(1) -4x - 12y = 0

(2) |x| - |y| = 16
(1) Rewrite the equation as -4x = 12y x = -3y

So, |x| = 3|y|. Therefore 4|y| = 32 and |y| = 8 So |x| = 24

Choose A.
How can be deduce this |x| = 3 |y| ?
Pls explain.
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by MartyMurray » Tue Jun 23, 2015 1:11 pm
prachi18oct wrote:
Marty Murray wrote:
prachi18oct wrote:If x and y are non-zero integers and |x| + |y| = 32, what is xy?

(1) -4x - 12y = 0

(2) |x| - |y| = 16
(1) Rewrite the equation as -4x = 12y x = -3y

So, |x| = 3|y|. Therefore 4|y| = 32 and |y| = 8 So |x| = 24

Choose A.
How can be deduce this |x| = 3 |y| ?
Pls explain.
If x = -3y, we know that one of x and y is positive and the other is negative. If we were to make them both positive then x would be equal to 3y. Right? So that's what absolute value achieves essentially, makes them both the same sign, positive.

Another way to look at it is this. If x = -3y, then the distance between x and 0 is three times the distance between y and 0. Absolute value is the distance between a number and 0. So once again we arrive at |x| = 3|y|.
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by GMATGuruNY » Tue Jun 23, 2015 2:04 pm
If x and y are non-zero integers and |x| + |y| = 32, what is xy?

(1) -4x - 12y = 0

(2) |x| - |y| = 16
Statement 1: -4x - 12y = 0.
-4x = 12y
x = -3y.

Substituting x= -3y into |x| + |y| = 32, we get:
|-3y| + |y| = 32
3|y| + |y| = 32
4|y| = 32
|y| = 8
y = 8 or y = -8.

If y=8, then x = -3*8 = -24, and xy = (-24)(8) = -192.
If y= -8, then x = -3*(-8) = 24, and xy = -8*24 = -192.
Since xy = -192 in each case, sufficient.

Statement 2: |x| - |y| = 16.
Adding this equation to |x| + |y| = 32, we get:
2|x| = 48.
|x| = 24
x=24 or x = -24.

This means:
24 + |y| = 32
|y| = 8.
y = 8 or y = -8.

If x=24 and y=8, then xy = 192.
If x= -24 and y=8, then xy = -192.
Since xy can be different values, insufficient.

The correct answer is A.
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by Brent@GMATPrepNow » Tue Jun 23, 2015 4:11 pm
prachi18oct wrote:If x and y are non-zero integers and |x| + |y| = 32, what is xy?

(1) -4x - 12y = 0
(2) |x| - |y| = 16
The question has already been answered eloquently, so I won't go any further with that.
However, I will mention that this question highlights the importance of making sure you understand what the target question is asking.

Target question: What is the value of xy?

Many students will read this and incorrectly conclude that they must find the individual values of x and y.
So, for statement 1, when they get to y = 8 or y = -8, some will conclude that statement 1 is not sufficient.
However, we're not asked to find the individual values of x and y. We're asked to find the value of xy.
So, even though there are two possible solutions (x = -24, y = 8 AS WELL AS x = 24, y = -8), the value of xy in BOTH cases is -192.

Since we get only ONE answer to the target question, statement 1 must be sufficient.

Cheers,
Brent
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