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# If the product of X and Y is a positive number, is the sum of X and Y

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Math Expert
Joined: 02 Sep 2009
Posts: 58383
If the product of X and Y is a positive number, is the sum of X and Y  [#permalink]

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31 May 2016, 05:46
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55% (hard)

Question Stats:

64% (01:40) correct 36% (01:51) wrong based on 159 sessions

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If the product of X and Y is a positive number, is the sum of X and Y a negative number?

(1) X > Y^5
(2) X > Y^6

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Re: If the product of X and Y is a positive number, is the sum of X and Y  [#permalink]

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31 May 2016, 19:33
If the product of x & y is a positive number --> we have 2 cases: both x & y are positive or both x & y are negative.

(1) x > y^5
With this information we can't tell the signs of x & y. If x = -1, y = -5 then the sum of x & y is negative. If x = 10 & y = 1 then the sum if positive.
-> Insufficient.
(2) x > y^6
With this information we can tell that x > 0 and hence y > 0 => sum of x & y > 0.
-> Sufficient.

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Re: If the product of X and Y is a positive number, is the sum of X and Y  [#permalink]

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01 Jun 2016, 06:57
If we modify the original condition and the question, there are 2 variables (x and y), and 1 equation (xy>0). Hence, in order to match the number of variables to the number of equations, we need 1 more equation. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that D is the correct answer.
From the condition 1), we can see that x=5, y=1 no, x-5, y=-2 yes. Hence, the condition is not sufficient.
From the condition 2), from x>y^6>=0, we can see x>0 and y>0. Since x+y>0, the answer is no and the condition is sufficient. Hence, the correct answer is B.
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Re: If the product of X and Y is a positive number, is the sum of X and Y  [#permalink]

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03 Nov 2016, 06:59
Bunuel wrote:
If the product of X and Y is a positive number, is the sum of X and Y a negative number?

(1) X > Y^5
(2) X > Y^6

Looks like it's inequalities days for me today....
x and y can be either positive or negative - they need to have the same sign.

1. y can be -2, x can be -1. x+y is negative
y can be 1, x can be 2. x+y is positive.
1 alone is not sufficient. A and D are out.

2. x>y^6.
y^6 is always positive. it means that x is positive. since we know for sure that x and y must have the same sign, we can definitely give an answer to the question: the sum of X and Y is not a negative number!

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Re: If the product of X and Y is a positive number, is the sum of X and Y  [#permalink]

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09 Oct 2018, 11:40
Bunuel wrote:
If the product of X and Y is a positive number, is the sum of X and Y a negative number?

(1) X > Y^5
(2) X > Y^6

$$xy > 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( * \right)\,\,\,\,\left\{ \begin{gathered} \,\,x > 0\,\,{\text{and}}\,\,y > 0\,\,\,\,\left( {{\text{scenario}}\,\,{\text{I}}} \right) \hfill \\ \,\,\,\,OR\,\,\, \hfill \\ \,\,x < 0\,\,{\text{and}}\,\,y < 0\,\,\,\,\left( {{\text{scenario}}\,\,{\text{II}}} \right) \hfill \\ \end{gathered} \right.$$

$$x + y\,\,\mathop < \limits^? \,\,\,0\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\boxed{\,\,\,?\,\,\,:\,\,\,{\text{scenario}}\,{\text{II}}\,\,}$$

$$\left( 1 \right)\,\,\,x > {y^5}\,\,\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {x,y} \right) = \left( {2,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\ \,{\text{Take}}\,\,\left( {x,y} \right) = \left( { - 1, - 2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\ \end{gathered} \right.$$

$$\left( 2 \right)\,\,x > {y^6}\,\, \geqslant 0\,\,\,\, \Rightarrow \,\,\,\,x > 0\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: If the product of X and Y is a positive number, is the sum of X and Y   [#permalink] 09 Oct 2018, 11:40
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