If the AM and GM between two numbers are in the ratio $m:n$, then what is the ratio between the two numbers?

Question

If the AM and GM between two numbers are in the ratio $m:n$, then what is the ratio between the two numbers?

I have tried many approach like Let's two number be $a$ and $b$ Then their AM will be $\frac{a+b}{2}$ and their GM will be $(ab)^{1/2}$. But putting these values and after solving equation become much complex.

Please tell me how to solve further.

Answer 1

Let $a$ and $b$ be positives and $a=bx$.

Thus, $$\frac{\frac{a+b}{2}}{\sqrt{ab}}=\frac{m}{n}$$ or $$x+1=\frac{2m\sqrt{x}}{n}.$$ Now, solve this quadratic equation.

Can you end it now?

I got that the needed ratio it's $$\left(\frac{m}{n}+\sqrt{\frac{m^2}{n^2}-1}\right)^2$$ or $$\left(\frac{m}{n}-\sqrt{\frac{m^2}{n^2}-1}\right)^2$$

Answer 2

Assuming $ab>0$. \begin{align*} \frac{a+b}{2\sqrt{ab}} & = \frac{m}{n}\\ n^2(a+b)^2& = 4m^2ab\\ n^2\left(\frac{a}{b}+\frac{b}{a}+2\right) &=4m^2. \end{align*} Let $\frac{a}{b}=t$, then you have a quadratic equation to solve $$n^2t^2+(2n^2-4m^2)t+n^2=0$$

Answer 3

Apply componendo &dividendo in ratio of AM/GM and then u will get root A +root B ka whole square on top and root a -root B ka whole square in bottom,then u can easily get ratio.i am new so I am not able to upload image to show u

Answer 4

Here's my unique solution to this problem. If AM and GM are in the ratio $m:n$, let $A=mk$ and $G=nk$.

This gives $a+b=2A$ and $ab= G^2$. Now create a quadratic equation whose roots are $a$ and $b$. $$x^2-2Ax+G^2=0$$

Solve for $a$ and $b$ using the quadratic formula.