Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
In words, we are asked whether the absolute value of w is less than the absolute value of v. We could rephrase this geometrically: is w closer to 0 on the number line than v is? However, we should also retain the original phrasing, which uses concise algebra. (By the way, the condition that v does not equal 0 is only there to prevent division by 0 in the statements.)
Statement (1): INSUFFICIENT. We should not cross-multiply this statement, since we don’t know whether v is positive or negative. That is, we cannot claim that w < v results from w/v < 1, since we can’t multiply both sides of an inequality by a variable and preserve the inequality sign as is, unless we know that the variable is positive.
Let’s instead plug a couple of quick pairs of numbers, trying for a “Yes” case and a “No” case. (The “Yes” case would be one in which |w| is in fact less than |v|, and the “No” case would be one in which |w| is not less than |v|.) Remember to try negative numbers!
Yes case: w = 2, v = 3. 2/3 is indeed less than 1, satisfying the statement.
As for the question, we get |2| < |3|, giving us a “Yes” answer to the question.
No case: w = -5, v = 3. -5/3 is indeed less than 1, satisfying the statement.
But |-5| is NOT less than |3|, so we would answer the question “No.”
Since we have both a Yes case and a No case, this statement is insufficient.
Statement (2): SUFFICIENT. Here, we CAN cross-multiply, since v2 is definitely positive. Thus, we can rephrase the statement as follows:
w2/v2 < 1
w2 < v2
Now, since both sides are positive (or at least not negative), we can take the positive square root of both sides. The positive square root of a square is in fact the absolute value, so we get
|w| < |v|
If you like, test numbers that satisfy w2 < v2. In every case, you will find that |w| < |v|.
The correct answer is (B).