Manhattan GMAT Challenge Problem of the Week - 2 May 2011

by Manhattan Prep, May 2, 2011

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Question

Is cdef < 0?

1) [pmath]cd^2e^3f^4[/pmath] < 0

2) [pmath]c^2d^3e^4f^5[/pmath] > 0

(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) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Answer

Asking whether the product cdef is less than zero is equivalent to asking whether one of the following cases holds: exactly three variables are positive (and the other one is not), or exactly one of the variables is positive. So the question can be rephrased this way: Is there an odd number of positive variables among c, d, e, and f?

Statement 1: INSUFFICIENT. Even powers of variables reveal nothing about the sign of the variable. Meanwhile, odd powers have the same sign. So this statement can be rephrased as ce < 0, which tells us that either [pmath]c[/pmath] or [pmath]e[/pmath] is negative; the other variable is positive. Exactly one of the variables is positive. However, we dont know anything about the signs of [pmath]d[/pmath] and [pmath]f[/pmath] .

Statement 2: INSUFFICIENT. This statement can be rephrased as [pmath]df[/pmath] > 0, which tells us that either both variables are positive, or neither variable is positive. However, we dont know anything about the signs of [pmath]c[/pmath] and [pmath]e[/pmath] .

Statements 1 and 2 together: SUFFICIENT. Together, we know that either 1 or 3 of the variables are positive. Thus, we can answer the question with a definite Yes.

The correct answer is C.

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