$\displaystyle \frac{4a-9b}{4a} = \frac{a-2b}{b}$
$\displaystyle \frac{4a-9b}{4a}\cdot \frac{b}{b} = \frac{a-2b}{b} \cdot \frac{4a}{4a}$
$\displaystyle \frac{4ab-9b^2}{4ab} = \frac{4a^2-8ab}{4ab}$
$\displaystyle \frac{4a^2-8ab}{4ab} - \frac{4ab-9b^2}{4ab} = 0$
$\displaystyle \frac{4a^2-8ab-4ab+9b^2}{4ab}=0$
$\displaystyle \frac{4a^2-12ab+9b^2}{4ab}=0$
$\displaystyle 4a^2-12ab+9b^2=0$
$\displaystyle (2a-3b)(2a-3b)=0$
$\displaystyle 2a=3b\neq0$
$\displaystyle a=1.5b$
$\displaystyle \frac{a}{b} = \frac{1.5b}{b} = 1.5$