Bunuel wrote:
Is (a - k)/(b - k) > (a + k)/(b + k) ?
(1) a > b > k
(2) k > 0
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:In rephrasing this question, we should recall that we do not know the sign of b – k and b + k. Thus, after cross-multiplying, we should set up
Flow Charts to evaluate two different questions: one for the case in which (b – k) and (b + k) have the same sign and another for the case in which (b – k) and (b + k) have different signs:
Statement (1) tells us that a > b > k. This is not sufficient. We do not know whether (b – k)(b + k) is positive, so we do not know which question to answer. Even if we did, we could get different results. For example, if a and b are positive and k is negative, then (b – k)(b + k) could be positive. Thus the relevant question would be “Is ak > bk?” Because k is negative, ak < bk. By contrast, if a, b, and k are all positive, then (b – k)(b + k) is positive. Thus the relevant question would be “Is ak > bk?” Because a > b > k, we would know ak > bk. We get two different answers depending on whether k is positive. INSUFFICIENT.
Statement (2) tells us that k > 0. This is not sufficient, because the statement tells us nothing about a and b. INSUFFICIENT.
Statements (1) and (2) combined are sufficient, because if a > b > k > 0, then (b – k)(b + k) > 0, so the relevant question is “Is ak > bk?” We know that a > b, and k is positive, so ak > bk, and the answer to the question is a definite “YES.”
Notice the use of the
Scenario Chart—specifically,
Flow Charts—to handle the different versions of the question depending on the sign of (b – k)(b + k). Additionally, we were careful to
Beware of Inequalities—the inequalities in this problem make it easy to make a mistake in rephrasing the question in the
Flow Chart or in evaluating the statements.
Notice also that if the algebra and thought process became too complicated, we could
Cross-Multiply Inequalities and guess between C and E. Because the combined statements tell us that a > b > k > 0, we would know a lot about the relative values of the variables in the problems, and it might be reasonable to choose C as the best answer.
The correct answer is C.Attachment:
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