Question

If ratio of H.M. and G.M. between two positive numbers $a$ and $b(a>b)$ is $4:5$, then $a:b$ is.

• A. $1:1$
• B. $2:1$
• C. $4:1$
• D. $3:1$

Answer 1

The correct option is C $4:1$

H.M $=\frac{2ab}{a+b}$, G.M$=\sqrt{ab}$
Using, $\frac{H.M}{G.M}=\frac{4}{5}\Rightarrow \frac{2\sqrt{ab}}{a+b}=\frac{4}{5}$
$\Rightarrow 2a+2b=5\sqrt{ab}$
$\Rightarrow 2\frac{a}{b}-5\sqrt{\frac{a}{b}}+2=0$
$\Rightarrow \sqrt{\frac{a}{b}}=\frac{5\pm 3}{4}=2,\frac{1}{2}$
But $a>b$
$\sqrt{\frac{a}{b}}=2\Rightarrow \frac{a}{b}=4$
Hence, option 'C' is correct.