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

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If √(xy) = xy, what is the value of x + y? [#permalink]

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If √(xy) = xy, what is the value of x + y? [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Algebra.



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

(1) x = -1/2
(2) y is not equal to 0



[Reveal] Spoiler:
Answer: C


Statement 1: \(X=-1/2\).... substitute in main equation

Scenario A: \(Y = -2/1\) is a reciprocal
\(\sqrt{xy}=xy\)
LHS =\(\sqrt{-1/2*-2/1}= \sqrt{1}= 1\)
RHS =\(-1/2*-2/1= -1*-1 = 1\)
Therefore, LHS = RHS

Scenario B: \(Y = 0\)
LHS =\(\sqrt{-1/2*-0}= \sqrt{0}= 0\)
RHS =\(-1/2*0= 0\)
Therefore again, LHS = RHS

Hence, we have two possible solutions, therefore Statement 1 is insufficient



Statement 2 : \(Y\) is not equal to zero
But, X could be equal to zero or be the reciprocal of Y, therefore, statement 2 would fall under the same scenarios as statement 1

Hence, we have two possible solutions, therefore Statement 2 is insufficient



Both Statement 1 & 2 together confirms that \(X & Y\) both are not equal to zero, therefore, they have to be reciprocals and since statement gives us the value of \(X=-1/2\), we can also the value of \(Y=-2/1\) (Reciprocal of X) and thereafter we can find the value of \(X + Y = -1/2 + -2/1 = -5/2\)

Hence, both together are sufficient, therefore C is the correct Answer!


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Re: If √(xy) = xy, what is the value of x + y? [#permalink]

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New post 15 Oct 2014, 21:26
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Bunuel wrote:

Tough and Tricky questions: Algebra.



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

(1) x = -1/2
(2) y is not equal to 0



I did something similar to DMMK, except for one thing:
I know \(\sqrt{n}\) = \(n\) only when \(n\) = 0 or \(n\) = 1. For all other values of \(n\), the two would not be equal. Knowing this made it much simpler to plug in the statements. I just had to determine if from the statement(s) provided, can I rule out xy = 0 or xy=1.

Stament 1:
Knowing that x = -1/2, y could equal 0 or -2, and still make the premise true. Either value of y would make x + y a different value. Therefore, it is insufficient.

Statement 2:
Knowing that y is not equal to 0, x could equal zero or the inverse of y, and still make the premise true. Either value of x would make x + y a different value. Therefore, it is insufficient.

Now to evaluate both statements together.

The reason we rejected statement 1 by itself was because y could equal one of two possible values. Statement 2 eliminates one of those options. Therefore y must equal -2. And therefore both statements together are sufficient to answer "what is the value of x+y."

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Re: If √(xy) = xy, what is the value of x + y? [#permalink]

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New post 15 Oct 2014, 22:12
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Bunuel wrote:

Tough and Tricky questions: Algebra.



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

(1) x = -1/2
(2) y is not equal to 0


let xy=k, thus
k^(1/2) =k;
squaring both sides we have
k=k^2
k(k-1)=0
k=0 or 1
i.e. xy=0 or xy=1

st.1

for x=-1/2
both y=0 and y=-2 satisfy the possible value of xy i.e. 0 or 1
hence not sufficient

st.2

y is not equal to zero. clearly not sufficient.
as nothing is said about x, therefore x can take any value it can be zero, fraction, integer etc.


combining st.1 and st.2

we know that y cannot be equal to zero. thus y=-2 and x=-1/2
and x+y= -2-(1/2)=-5/2

hence sufficient.

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Re: If √(xy) = xy, what is the value of x + y? [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Algebra.



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

(1) x = -1/2
(2) y is not equal to 0


Given: √(xy) = xy
(xy)^2 - xy = 0
xy = 0 or 1 - (i)

Required: x + y = ?

Statement 1: x = -1/2
From the given equation (i), we can have two different values of y.
Hence two different values of x + y
INSUFFICIENT

Statement 2: y is not equal to 0
No information about x
INSUFFICIENT

Combining Statement 1 and Statement 2: x = -1/2 and y is not equal to 0
From (i), xy cannot be 0 since both x and y are not = 0
Hence xy = 1
y = -2
x + y = \(-\frac{5}{2}\)
SUFFICIENT

Option C
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If √(xy) = xy, what is the value of x + y? [#permalink]

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New post 14 Apr 2017, 07:13
Bunuel wrote:

Tough and Tricky questions: Algebra.



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

(1) x = -1/2
(2) y is not equal to 0


The trick here is from \(\sqrt{xy}=xy\) anyone could easily deduce that \(xy=1\) or \(xy=0\) and just consider one of these cases.

Here is my solution:
\(\sqrt{xy}=xy \implies \sqrt{xy}-xy=0 \implies \sqrt{xy}(\sqrt{xy}-1)=0\)
Hence we have \(xy=0\) or \(xy=1\).

(1) If \(x=-\frac{1}{2}\), we need to consider two cases:

Case 1: If \(xy=0\implies y=0 \implies x+y = -\frac{1}{2}\)

Case 2: If \(xy=1 \implies y=-2 \implies x+y = -\frac{5}{2}\)

It's clear that (1) is insufficient.

(2) If \(y \neq 0\), we still need to consider two cases:

Case 1: If \(xy=0\implies x=0 \implies x+y = y\).

If \(y=1 \implies x+y = 1\)
If \(y=2 \implies x+y = 2\)

Hence, (2) is insufficient.


Now combine (1) and (2):
Since \(x \neq 0\) and \(y \neq 0\) hence \(xy \neq 0 \implies xy=1\)

Since \(x= -\frac{1}{2} \implies y = -2 \implies x+y = -\frac{5}{2}\)

Hence (1) & (2) are sufficient.
The answer is C.
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Re: If √(xy) = xy, what is the value of x + y? [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Algebra.



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

(1) x = -1/2
(2) y is not equal to 0



Bunuel / moderators

please move the question to DS section


thanks

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Re: If √(xy) = xy, what is the value of x + y? [#permalink]

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Re: If √(xy) = xy, what is the value of x + y?   [#permalink] 14 Apr 2017, 07:54
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