Practice GMAT Data Sufficiency Question

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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?

X > Y⁵
X > Y⁶

A)	Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)	Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)	BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)	EACH statement ALONE is sufficient.
E)	Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct Answer: B

There are two possible cases (or conditions) under which the product of X and Y could be positive:
Case (1): Positive(Positive) = Positive
Case (2): Negative(Negative) = Positive
Evaluate Statement (1) alone.
1. Since Y is raised to an odd exponent, the sign of the base (i.e., the sign of Y) is the same as the sign of the entire expression (i.e., the sign of Y⁵).
2. There is no way of distinguishing whether we are in Case (1) or (2) and the answer to the resulting question of whether X + Y is negative can be different depending on the chosen values of X and Y. Consider two examples, one from each case.
3. Case (1):
  X = 100
  Y = 2
  XY = (100)(2) = 200 = Positive
  X > Y⁵
  X + Y = 100 + 2 = 102 = Positive
4. Case (2):
  X = -10
  Y = -2
  XY = (-10)(-2) = 20 = Positive
  X > Y⁵
  X + Y = (-10) + (-2) = -12 = Negative
5. Since there is no way to determine whether X + Y is positive, Statement (1) is not sufficient.
6. Statement (1) is NOT SUFFICIENT.
Evaluate Statement (2) alone.
1. Y⁶ must be a positive number since, even if Y were negative, raising it to an even exponent would make the entire quantity positive.
2. Substituting this into the information given in Statement (2):
  X > Y⁶
  X > (positive number)
  X must be positive since any number that is larger than a positive number is itself positive.
3. Since X is positive, in order for XY to be positive, Y must also be positive (i.e., we are dealing with Case (1) from above). Consequently, a positive number (i.e., X) plus a positive number (i.e., Y) must itself be positive.
  X + Y = ?
  Positive + Positive = Positive
  We can definitively answer "no" to the original question.
4. Statement (2) is SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT but Statement (2) alone is SUFFICIENT, answer B is correct.

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