Discussion
Which one of the following fractions will have minimum change in its value if 2 is added to both numerator and the denominator?
Answer: B
The fraction that changes least is 5/8.
Adding 2 to both parts of a/b changes it by
\[ \begin{aligned} \frac{a + 2}{b + 2} - \frac{a}{b} \\ = \frac{2(b - a)}{b(b + 2)} \end{aligned} \]
\[ \begin{aligned} 1/5 &: \tfrac{8}{35} \approx 0.229 \\ 5/8 &: \tfrac{6}{80} = 0.075 \\ 1/2 &: \tfrac{2}{8} = 0.25 \\ 3/4 &: \tfrac{2}{24} \approx 0.083 \end{aligned} \]
5/8 moves to 7/10, a change of 0.075, the smallest. 3/4 is the near miss: it goes to 5/6, a change of about 0.083. The rule of thumb is that the change is small when the gap \(b - a\) is small relative to \(b^2\).
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