Is \( x^2*y^5*z>0\)? GMAT Problem Solving

Question: Is \(x^2*y^5*z>0\)?

1. \(\frac{xz}{y}>0\)
2. \(\frac{y}{z}>0\)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“Is \(x^2*y^5*z>0\) ?”– is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken f0rom the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

For \(x^2*y^5*z>0\) to hold true:

1. X must not be zero. Because if x = 0, then \(x^2*y^5*z \) will be equal to 0, not more than it.
   AND
2. y and z must be either both positive or both negative. Because if y and z have different signs (or if either of them is 0), then yz will be negative (or 0), not positive as required.

So, only if BOTH conditions are met we’d have \((x^2)*(y^5*z)\) = (positive)*(positive) = positive

1. \(\frac{xz}{y}>0\) .The first condition is satisfied: \(x\neq0\), but we don’t know about the second one: \(\frac{xz}{y}>0\) means that either all of them are positive (answer YES) ar ANY two are negative and the third one is positive, so it’s possible y and z have opposite signs (answer NO).
   Hence this statement is not sufficient.
2. \(\frac{y}{z}>0\). From this statement it follows that y and z have opposite signs, hence the second condition is already violated,, so the answer to the question is NO.
   Hence this statement is sufficient.

Side note for (2): \(\frac{y}{z}>0\) does not mean that \(x^2*y^5*z<0\), it means that \(x^2*y^5*z\leq0\) because it’s possible for x to be equal to zero and in this case\(x^2*y^5*z=0\). But in any case \(x^2*y^5*z \) is not MORE than zero, so we can answer NO to the question.

Correct Answer: B

Approach Solution 2:

S1: with this fact. The problem can be re-written as:

\(\frac{xz}{y}*\frac{xz}{y}*\frac{y^7}{z}\)

+ve * +ve * -ve or +ve

Since the last term can be either +ve or –ve, it is not sufficient

S2: with this fact. The problem can be re-written as:

\(x^2*\frac{y^5}{z^5}*z^6\)

+ve * +ve * +ve

The product is +ve, it is sufficient

Correct Answer: B

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