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

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.

• Welcome to MathSE. This tutorial explains how to typeset mathematics on this site. – N. F. Taussig Jun 19 at 7:16

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 $$\left(\frac{m}{n}-\sqrt{\frac{m^2}{n^2}-1}\right)^2$$

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 $$n^2t^2+(2n^2-4m^2)t+n^2=0$$

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

• But rohit singh i know that approach to solve this ques ...i wanna to solve using other method – darshh Jun 19 at 6:23
• I am sorry but your answer is too allusive : it doesn'r meet the standards of mathematical writing (see the other answers). – Jean Marie Jun 19 at 6:24
• Then u can check others have also given there methods and I think all are correct – Rohit Singh Jun 19 at 6:25
• See Jean I told u I am a new user and don't know mathjax😔 – Rohit Singh Jun 19 at 6:28
• This tutorial explains how to use MathJax. – N. F. Taussig Jun 19 at 7:17