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Let $a$ and $b$ be two positive numbers such that $a\gt b$. Let $G$ be the geometric mean of $a$ and $b$ (that is, $G=\sqrt{ab}$), and $H$ be the Harmonic mean of $a$ and $b$, that is, $H = \frac{2}{\frac{1}{a}+\frac{1}{b}} = \frac{2ab}{a+b}.$

If $4G = 5H$, what is the value of $a$?

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    To all involved, I cleaned up the comments$a$bit after some of the edits obsoleted them. @anna: LaTeX allows for prettification of the expressions, but if you are only writing arithmetic expressions, you can hit the ground running by just typing the expression as you would on$a$graphing calculator and enclosing them in dollar signs (and stick to the convention that juxtaposed symbols are multiplied, it takes a little bit of getting used to that putting `\times` between `$` signs give $\times$). It may not be pretty, but it will be understandable (provided you use sufficient parentheses).2012-02-24

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The Harmonic Mean of $a$ and $b$ is $\frac{2}{\frac{1}{a}+\frac{1}{b}} = \frac{2ab}{a+b}.$ The Geometric Mean of $a$ and $b$ is $\sqrt{ab}.$ So, to state the problem you have in a way that would be actually intelligible would be:

Let $a$ and $b$ be positive numbers such that $a\gt b$; assume that $4\times\text{geometric-mean(a,b)} = 4\sqrt{ab} = 5\left(\frac{2ab}{a+b}\right) = 5\times\text{harmonic-mean}(a,b).$ What is the value of $a$?

We have $\begin{align*} 4\sqrt{ab} &= \frac{10ab}{a+b}\\ 4(a+b) &= \frac{10ab}{\sqrt{ab}}\\ 2(a+b) &= 5\sqrt{ab}\\ 4(a+b)^2 &= 25ab\\ 4a^2 + 8ab + 4b^2 &= 25ab\\ 4a^2 -17ab + 4b^2 &=0. \end{align*}$ You can view this as a quadratic equation in $a$; the solutions are given by $\frac{17b - \sqrt{(17b)^2 - 64b^2}}{8} = \frac{17b-\sqrt{225b^2}}{8} = \frac{17b-15b}{8} = \frac{b}{4}$ (which is impossible since $a\gt b$) and $\frac{17b + \sqrt{(17b)^2 - 64b^2}}{8} = \frac{17b + \sqrt{225b^2}}{8} = \frac{32b}{8} = 4b.$ So the answer is that $a$ must be $4b$.

You can verify this works: the Geometric Mean of $b$ and $4b$ is $\sqrt{4b^2} = 2b$; the Harmonic mean is $\frac{2(4b)b}{4b+b} = \frac{8b^2}{5b} = \frac{8b}{5}.$ And $4(2b) = 5\left(\frac{8b}{5}\right).$

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    @anna: You also did your algebra incorrectly, since you went from $2(a+b) = 5\sqrt{ab}$ to $4(a^2+b^2)=25ab$; that's wrong, because $(a+b)^2\neq a^2+b^2$.2012-02-24
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According to the given information we have, $4\sqrt{ab} = 5(\frac{2ab}{a+b})$

$(a+b) =\frac{5}{2}\sqrt{ab})$

$\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}} = \frac{5}{2}$

Let , $t$ $ =$ $\sqrt{\frac{a}{b}}$

$[t+\frac{1}{t}=\frac{5}{2}].....................Eq(1)$

A clever person will immediately infer that $t=\frac{1}{2}$

But if its a subjective question we have to justify that also, so

$({t+\frac{1}{t}})^2=\frac{25}{4}$

$t^2 +\frac{1}{t^2} = \frac{17}{4}$ NOW, $(t-\frac{1}{t})^2= t^2+\frac{1}{t^2} -2 =\frac{9}{4}$

$[t-\frac{1}{t}=\frac{3}{2} ] .....................Eq(2)$ neglecting the negative value as we know that L.H.S.>0 , since $,t>0$

From Eq(1) and Eq(2) we have $t=4$, hence $a=4b$