Question

If the ratio of AM to GM of two positive numbers $a$ and $b$ is $5:3$, then $a:b$ is equal to

• A. $3:5$
• B. $2:9$
• C. $9:1$
• D. $5:3$

Answer 1

Answer: (C)

$9:1$

Am of $a$ and $b=\frac{a+b}{2}$

Gm of $a$ and $b=\sqrt{ab}$

Acc to question $\to \frac{a+b}{2\sqrt{ab}}=\frac{5}{3}$

$\Rightarrow \frac{\left(a/b\right)+1}{\sqrt{a/b}}=\frac{10}{3}\Rightarrow {(\frac{a}{b}+1)}^2=\frac{100}{9}\left(\frac{a}{b}\right)$ (squaring both sides)

Let $a/b=x$

$\therefore x^2+1+2x=\frac{100}{9}x\Rightarrow x^2+2x-\frac{100x}{9}+1=0$

$\therefore 9x^2+(18x+-100x)+9=0$

$\Rightarrow 9x^2-82x+9=0\Rightarrow 9x^2-81x-x+9=0$

$\Rightarrow 9x(x-9)-1(x-9)=0$

$\therefore x=\frac{1}{9}$ and $x=9$

$\therefore a:b=9:1$.