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

• Mar 28th 2011, 01:25 PM
chewitard
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 $\sqrt5$?
• Mar 29th 2011, 09:59 AM
running-gag
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

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