If a = b/c, is a > b? (1) 0 < c < 1 (2) a > 0
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28 Jan 2013, 03:04
If \(a = \frac{b}{c}\), is \(a > b\)?
Since \(a = \frac{b}{c}\), the question can be rephrased as "Is \(\frac{b}{c} > b\)?", which simplifies to "Is \(b(\frac{1}{c}-1) > 0\)?" Essentially, the question asks whether \(b\) and \(\frac{1}{c}-1\) have the same sign.
(1) \(0 < c < 1\)
Given that \(0 < c < 1\), it follows that \(\frac{1}{c} > 1\) and thus \(\frac{1}{c}-1 > 0\). However, the sign of \(b\) remains unknown. Not sufficient.
(2) \(a > 0\)
Consider the scenario where \(a=b=c=1\); here, \(a=b\), and the answer to the question is NO. Conversely, if \(a=1\) and \(b=c=-1\), then \(a > b\), yielding a YES to the question. Not sufficient.
(1)+(2) From the above, it's established that both \(a\) and \(c\) are positive. Hence, from \(a = \frac{b}{c}\), it must be that \(b\) is also positive. Thus, \(b > 0\) and \(\frac{1}{c}-1 > 0\) (from 1), meaning \(b\) and \(\frac{1}{c} - 1\) have the same sign. Therefore, \(b(\frac{1}{c}-1) > 0\), and we have a definitive YES answer to the question. Sufficient.
Answer: C