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26 May 2017, 04:41
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If \(c > 0\), is \(\frac{a+c}{b+c} > \frac{a}{b}\) ?
(1) \(0 < a < b\)
(2) \(a < b\)
(1) \(0 < a < b\)
(2) \(a < b\)
26 May 2017, 04:41
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Official Solution:
You can take the 1st step of the variable approach and modify the original condition and the question. In general, when the numbers increase (as shown below, both numerators and denominators are increasing),
(1) The fraction gets smaller if it's bigger than 1
(eg) \(\frac{2}{1} > \frac{3}{2} > \frac{4}{3} > \frac{5}{4} > \frac{6}{5} > \ldots \ldots > 1\)
(2) The fraction gets bigger if it's smaller than 1
(eg) \(\frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{4}{5} < \frac{5}{6} < \ldots \ldots < 1\)
If so,
in order for the question to be "is \(\frac{a + c}{b + c} > \frac{a}{b}\) ?", in other words, in order for the value of the fraction to increase as both the numerators and denominators increase, \(\frac{a}{b}\) has to be smaller than 1 (by (1) shown above). If you modify the question, you get "\(\frac{a}{b} < 1\)?".
In the case of con 1), if you divide both sides of \(0 < a < b\) by \(b\), (\(b > 0\), so the sign of inequality does not change even if it is divided), you get \(\frac{a}{b} < 1\), hence yes, it is sufficient.
In the case of con 2), you cannot determine the sign of inequality, hence it is not sufficient. The answer is A.
Answer: A
You can take the 1st step of the variable approach and modify the original condition and the question. In general, when the numbers increase (as shown below, both numerators and denominators are increasing),
(1) The fraction gets smaller if it's bigger than 1
(eg) \(\frac{2}{1} > \frac{3}{2} > \frac{4}{3} > \frac{5}{4} > \frac{6}{5} > \ldots \ldots > 1\)
(2) The fraction gets bigger if it's smaller than 1
(eg) \(\frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{4}{5} < \frac{5}{6} < \ldots \ldots < 1\)
If so,
in order for the question to be "is \(\frac{a + c}{b + c} > \frac{a}{b}\) ?", in other words, in order for the value of the fraction to increase as both the numerators and denominators increase, \(\frac{a}{b}\) has to be smaller than 1 (by (1) shown above). If you modify the question, you get "\(\frac{a}{b} < 1\)?".
In the case of con 1), if you divide both sides of \(0 < a < b\) by \(b\), (\(b > 0\), so the sign of inequality does not change even if it is divided), you get \(\frac{a}{b} < 1\), hence yes, it is sufficient.
In the case of con 2), you cannot determine the sign of inequality, hence it is not sufficient. The answer is A.
Answer: A
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