If xy = - 6, what is the value of xy(x+y)?

known xy=-6, unknown xy(x+y)

statement 1) x-y=5; we know that xy=-6, we can substitute x=-(6/y) in the given equation, which becomes
-(6/y)-y=5; y^2+5y+6=0; (y+3)(y+2)=0; hence y=-3 or y=-2,
when y=-3, x=2. thus x+y=-1 and xy(x+y)=-(6)(-1)=6
when x=-2, y=3 thus x+y=1 and xy(x+y)= (-6)(1)= -6 hence 1 is not sufficient

statement 2) xy^2=18; can be written as xy(y)=18,
-6y=18;
y=-3; also we know from the question statement that xy=-6; thus x=2 hence x+y=2-3=-1, hence xy(x+y) = -6(-1)= 6

hence sufficient. Therefore answer should be B

If xy = - 6, what is the value of xy(x+y)?

(1) x - y = 5
(2) xy^2 = 18

the question asks what is the value of -6(x+y)?

Statement I is insufficient:

xy = -6
x = y + 5
-6 = y^2 + 5y
y^2 + 5y + 6 = 0
y = -3, y = -2
y = -3, x = 2, x + y = -1
y = -2, x = 3, x + y = 1
Since we have two different answers hence statement I is insufficient

Statement II
x(y)(y) = 18
-6(y) = 18
y = -3 and since xy = -6, x = 2.

Hence answer is B

Hi All,

There's a subtle, but important lesson in how this question is designed: you MUST make sure that you finish the work and answer the question that is asked.....



This poster correctly figured out the value of Y from Fact 2, but then stopped working....by plugging the value of Y back into either the starting equation or the one in Fact 2, we have the value of X.....AND we can answer the question.
Fact 2 is SUFFICIENT.

It's important to remember that DS questions have NO "safety net" - the moment you make a little mistake, you're probably going to get the question wrong and you won't even realize it. It's important to make sure that you have PROOF of what you believe. In this case, if we really had no idea about the value of X, then we should be able to quickly come up with 2 different possibilities that fit the given restrictions and PROVE that the answer to the question changes. When Quant scores take a significant drop, it's almost always due to some type of little mistakes. Practice putting ALL of the work that yo do on the pad and you'll be far more likely to remove those silly mistakes from your approaches.

GMAT assassins aren't born, they're made,
Rich

In the Option 1 , cant we write X - Y =5 as (X - Y)^2 = 25 ?? as in this way we can get answer by A also.

Hi rahulhkdwivedi,

While you could square both sides of that equation, you haven't shown any of the steps that you would do next... so how do you know that it's sufficient? I suggest that you attempt what you've suggested, and then answer try to answer the GIVEN question: What is the value of XY(X+Y)?

You should be able to prove that it's insufficient.

GMAT assassins aren't born, they're made,
Rich

I agree with EMPOWERgmatRichC. Additionally, if you do square the expression to get \((x-y)^2=25\) --->\(x^2+y^2-2xy=25\), you still wont get any answers for \(xy(x+y)\) or \(x^2y+xy^2\)

Also, when you square any expression you end up increasing the number of solutions by a factor of 2. For example, you are given x-y=5 but when you square the expression,

\((x-y)^2=25\) ---> this is quadratic equation, you end up getting, \(x-y=\pm 5\), thus giving you an additional equation in \(x-y=-5\)

Do not square or cube any given expression unless absolutely needed to. Even when you do make sure to check whether all solutions given by the squared or the cubed equations satisfy the given requirements.

Hope this helps.

I got to one set of values of x and y in B, but then I rejected this option on the basis that nowhere it is Given that X and y are integers. What about real nos.

Please help.
Bunuel ScottTargetTestPrep

Posted from my mobile device

As show above, xy = -6 and xy^2 = 18 has only one real solution: x = 2 and y = -3. If you check above, you'll see that while solving, we nowhere assumed that x and y are integers. We just got that they are after we solved.

Why couldn't we solve it by plugging in the (xy=-6) in xy(x+y) first? So it would look like this:
-6(x+y)
-6x-6y
(and then plug in our first statement)
-6(y+5) - 6y
-6y -30 - 6y
-12y -30
-12y=30
y=-30/12 and so forth

Hi NandishSS

I totally advocate the use of smart numbers for solving such problems.

The generalization of any method (ALgebraic or number plugging) is difficult as it works based on your first instinct.

P.S. When I solved this problem for teh first time, then I used numbers instead of algebraic approach

However, a good suggestion will be to simplify the expressions algebraically if it seems possible. When stcuk in solving/inferring it further then start using numbers

My flow of thought is as follows:

Given: If xy = - 6 Some possible values of (x,y) are (2, -3) (-2, 3) (3, -2) (-3, 2) but the values ,may be Non-integers too

Question: xy(x + y) = ? so we need only (x+y) as we know the value of x*y

Statement 1: \(x - y = 5\)

(2, -3) and (3, -2) satisfy but the value of x+y may be +1 or -1 hence

NOT SUFFICIENT

Statement 2: \(xy^2 = 18\)

i.e. Y must be negative and x must be positive
Also, \(18 = 2*3^2\) therefore (x, y) = (2, -3)
i.e. x+y = -1

SUFFICIENT

Answer: Option B

GMATinsight Why you didn't choose (6,1)?

What are the smart numbers?

There might be many possibilities as well.

NandishSS

Yes, it was possible but I had to stop somewhere realizing infinite possibilities in non-integer solutions. So after thinking a few possibilities I looked at statements and found my answer rather than summarising all possible solutions beforehand.

This is what I did too. Is this approach correct?

Solution:

We need to determine the value of xy(x + y) given that xy = -6.


Statement One Alone:

We can re-express x - y = 5 as x = y + 5 and substitute it into the equation provided in the question stem:

xy = -6

(y + 5)(y) = -6

y^2 + 5y + 6 = 0

(y + 3)(y + 2) = 0

y = -3 or y = -2

If y = -3, then x = -6/y = 2. In this case, x + y = -1 and xy(x + y) = (-6)(-1) = 6. However, if y = -2, then x = -6/y = 3. In this case, x + y = 1 and xy(x + y) = (-6)(1) = -6. Since there are two possible values for xy(x + y), statement one is not sufficient.

Statement Two Alone:

We can re-express xy^2 = 18 as (xy)(y) = 18. Since we are given that xy = -6, we have:

(xy)(y) = 18

-6y = 18

y = 18/-6

y = -3

Thus, the value of x can be determined:

xy^2 = 18

x(9) = 18

x = 2

We know the values of both x and y, so the value of xy(x + y) can be determined. Statement two is sufficient.

Answer: B

ScottTargetTestPrep

For statement 1 the quadratic route is good, however, can we use the following logic?
(1) x−y=5

(x - y)^2 = 25

x^2 - 2xy + y^2 = 25
x^2 + 12 + y^2 = 25
x^2 + y^2 = 13 Since the question stem does not mention that x and y are integers, there are many possible values for x and y. Therefore, not sufficient.

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