# If the harmonic mean H and geometric mean G of two positive numbers a and b are in ratio 3:5, then a and b are in the ratio

$\begin{array}{1 1} 1:9 \\ 1:4 \\ 3:5 \\ 1:3 \end{array}$

Explanation :$H=\large\frac{2ab}{a+b}$
$G=\sqrt{a}{b}$
$\frac{H}{G}=\large\frac{2ab/(a+b)}{\sqrt{a}{b}}=\large\frac{3}{5}$
$\large\frac{2\sqrt{a}{b}}{a+b}=\large\frac{3}{5}$
$100\;ab=9(a^2+2ab+b^2)$
$9a^2-82ab+9b^2=0$
$9a^2-81ab-ab+9b^2=0$
$9a(a-9b)-b(a-9b)=0$
$(9a-b)(a-9b)=0$
$9a=b\qquad\;a=9b$
$\frac{a}{b}=1:9\quad\;or\;\quad9:1.$
edited Jan 24, 2014