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Math Help - which number is closer to sqrt(5): (a/b) or (2a+5b)/(a+2b)

  1. #1
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    which number is closer to sqrt(5): (a/b) or (2a+5b)/(a+2b)

    a, b are positive integers and \sqrt5 lies inbetween \frac{a}{b} and \frac{2a+5b}{a+2b}. How would I go about showing which number is closer to \sqrt5?
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  2. #2
    MHF Contributor
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    Hi

    You must compare \frac{2a+5b}{a+2b}-\sqrt{5} and \sqrt{5}-\frac{a}{b}

    Using the same denominator and rearranging a little bit

    \frac{2a+5b}{a+2b}-\sqrt{5} = \frac{b}{a+2b}(\sqrt{5}-2) \left(\sqrt{5}-\frac{a}{b}\right)

    Then you can show that

    \frac{2a+5b}{a+2b}-\sqrt{5} \leq \frac{1}{8} \left(\sqrt{5}-\frac{a}{b}\right)
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