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Math Help - Arithmetic mean and Geometric mean

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by a69356 View Post
    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.
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  3. #3
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    Hello, Ashish!

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


    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}

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