Q) A and B are two numbers such that there G.M is 20% lower than their A.M .Find the ratio between the two numbers.

Thanks,

Ashish

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- Aug 18th 2009, 07:18 AMa69356Arithmetic mean and Geometric meanQ) A and B are two numbers such that there G.M is 20% lower than their A.M .Find the ratio between the two numbers.

Thanks,

Ashish - Aug 18th 2009, 07:38 AMmr fantastic
$\displaystyle \sqrt{a b} = \frac{4}{5} \left( \frac{a + b}{2} \right)$

$\displaystyle \Rightarrow ab = \frac{4}{25} (a + b)^2 \Rightarrow 25 ab = 4a^2 + 8 ab + 4 b^2 \Rightarrow 25 = 4 \left( \frac{a}{b}\right) + 8 + 4 \left(\frac{b}{a}\right)$.

Let the ratio be $\displaystyle x = \frac{a}{b}$. Then:

$\displaystyle 25 = 4x + 8 + \frac{4}{x}$.

Your job is to solve for x. - Aug 18th 2009, 08:06 AMSoroban
Hello, Ashish!

My solution is similar to Mr. F's . . .

Quote:

$\displaystyle A$ and $\displaystyle B$ are two numbers such that there G.M is 20% lower than their A.M.

Find the ratio between the two numbers.

We have: .$\displaystyle \sqrt{ab} \:=\:\frac{4}{5}\left(\frac{a+b}{2}\right) \quad\Rightarrow\quad 5\sqrt{ab} \:=\:2(a+b)$

Square both sides: .$\displaystyle 25ab \;=\;4a^2 + 8ab + 4b^2 \quad\Rightarrow\quad 4x^2 - 17ab + 4b^2 \:=\:0$

. . which factors: .$\displaystyle (a - 4b)(4a - b) \:=\:0$

Hence, we have: .$\displaystyle \begin{Bmatrix}a - 4b \:=\:0 & \Rightarrow & \dfrac{a}{b} \:=\: 4 \\ \\[-2mm] 4a - b \:=\:0 & \Rightarrow & \dfrac{a}{b} \:=\:\dfrac{1}{4} \end{Bmatrix}$