Question

If two positive numbers are in the ratio of $3+2\sqrt{2}:3-2\sqrt{2},$ then the ratio between their AM and GM is

• A. $6:1$
• B. $3:2$
• C. $2:1$
• D. $3:1$

Answer 1

The correct option is D

$3:1$

Explanation for the correct answer:

Find the ratio between their AM and GM:

Let $a$ and $b$ be two numbers, then $a:b=3+2\sqrt{2}:3–2\sqrt{2}$

$\begin{array}{rcl}AM& =& \frac{(a+b)}{2}\\ & =& \frac{\left(\right(3+2\sqrt{2})x+(3–2\sqrt{2}\left)x\right)}{2}\\ & =& \frac{6x}{2}\\ & =& 3x\end{array}$

$\begin{array}{rcl}GM& =& \sqrt{\left(ab\right)}\\ & =& \sqrt{(3+2\sqrt{2})x(3–2\sqrt{2})x}\\ & =& \sqrt{(9-8)x^2}\\ & =& x\end{array}$

$AM:GM=3:1$

Hence, the correct option is D.