Bunuel wrote:
If \(x > 0\), \(y > 0\), and \(a ≠ y\), is \(\frac{x − a}{y − a} < \frac{x}{y}\) ?
(1) \(x > y\)
(2) \(a > 0\)
Solution: Before jumping into the statements, lets do some pre-analysis:
We have: \(\frac{x − a}{y − a} < \frac{x}{y}\)
\(\Rightarrow \frac{x − a}{y − a} - \frac{x}{y}<0\)
\(\Rightarrow \frac{(xy-ay-xy+ax)}{(y(y-a))}<0\)
\(\Rightarrow \frac{-a(y-x)}{y(y-a)}<0\)
\(\Rightarrow \frac{a(y-x)}{y(y-a)}>0\)
So, we need to find if \(\frac{a(y-x)}{y(y-a)}\) is positive or not.
Statement 1: \(x > y\)
Thus we get \(x-y>0\) or \(y-x<0\). But we get nothing on value/nature if \(a\).
Thus, statement 1 alone is not sufficient. We can eliminate options A and D.
Statement 2: \(a > 0\)
This gives us that \(a\) is positive. But we get nothing on value or relative nature of \(x,y\).
Thus, statement 2 alone is also not sufficient. We can eliminate option B.
Combinning: We have \(y-x<0\) and \(a > 0\). These make the numerator of \(\frac{a(y-x)}{y(y-a)}\) negative.
However, we cant say anything on the denominator \(y(y-a)\) or more precisely about \(y-a\).
Hence the right answer is
Option E.
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