# Arithmetic mean and Geometric mean

• August 18th 2009, 07:18 AM
a69356
Arithmetic mean and Geometric mean
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
• August 18th 2009, 07:38 AM
mr fantastic
Quote:

Originally Posted by a69356
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

$\sqrt{a b} = \frac{4}{5} \left( \frac{a + b}{2} \right)$

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

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

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

Your job is to solve for x.
• August 18th 2009, 08:06 AM
Soroban
Hello, Ashish!

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

Quote:

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

Let $a$ and $b$ be the two numbers.

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

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

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

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