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Key property: If \(\frac{a}{b}\) is a positive fraction, then \(\frac{a + k}{b + k}\) gets closer and closer to 1 as the value of k increases.

Some examples:

EXAMPLE #1: \(\frac{1}{5}=0.2 \)

Add 1 to numerator and denominator to get: \(\frac{1+1}{5+1}=\frac{2}{7}≈0.29 \)

Add 7 to numerator and denominator to get: \(\frac{1+7}{5+7}=\frac{8}{12}≈0.67 \)

Add 30 to numerator and denominator to get: \(\frac{1+30}{5+30}=\frac{31}{35}≈0.89 \)

Add 400 to numerator and denominator to get: \(\frac{1+400}{5+400}=\frac{401}{405}≈0.99 \)

So, when \(\frac{a}{b}\) is less than 1, \(\frac{a}{b} < \frac{a + k}{b + k} <1\) for all positive values of k


EXAMPLE #2: \(\frac{8}{2}=4 \)

Add 1 to numerator and denominator to get: \(\frac{8+1}{2+1}=\frac{9}{3}=3 \)

Add 6 to numerator and denominator to get: \(\frac{8+6}{2+6}=\frac{14}{8}=1.75 \)

Add 100 to numerator and denominator to get: \(\frac{8+100}{2+100}=\frac{108}{102}≈1.06 \)

So, when \(\frac{a}{b}\) is greater than 1, \(1<\frac{a + k}{b + k}<\frac{a}{b} \) for all positive values of k

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Now onto the question!!!!

Since we're told \(x > y > 0\), we know that \(\frac{x}{y}\) is greater than 1, which means \(1<\frac{x + k}{y + k} < \frac{x}{y} \) (for all positive values of k)

So statement III is true.

Answer: C
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