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
$\displaystyle \sqrt{a b} = \frac{4}{5} \left( \frac{a + b}{2} \right)$
$\displaystyle \Rightarrow ab = \frac{4}{25} (a + b)^2 \Rightarrow 25 ab = 4a^2 + 8 ab + 4 b^2 \Rightarrow 25 = 4 \left( \frac{a}{b}\right) + 8 + 4 \left(\frac{b}{a}\right)$.
Let the ratio be $\displaystyle x = \frac{a}{b}$. Then:
$\displaystyle 25 = 4x + 8 + \frac{4}{x}$.
Your job is to solve for x.
Hello, Ashish!
My solution is similar to Mr. F's . . .
Let $\displaystyle a$ and $\displaystyle b$ be the two numbers.$\displaystyle A$ 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: .$\displaystyle \begin{Bmatrix}a - 4b \:=\:0 & \Rightarrow & \dfrac{a}{b} \:=\: 4 \\ \\[-2mm] 4a - b \:=\:0 & \Rightarrow & \dfrac{a}{b} \:=\:\dfrac{1}{4} \end{Bmatrix}$