<s>[INST] Find two numbers whose $AM+...$ Find two numbers whose $AM + GM =25$ and $AM:GM=5:3$.
My Attempt;
Given,
$\frac {AM}{GM}=\frac {5}{3} = k (let) $
$\AM=5k,
GM=3k$.
Also,
$AM+GM=25$
$5k+3k=25$
$8k=25$
$k=\frac {25}{8}$.
Am I going right? Or, is there any other simple alternative.?
 [/INST] Let the numbers be $a^2$ and $b^2$. 
Then we have :
$$
\frac{a^2+b^2}{2} +ab=25
\implies (a+b)^2=50
\implies a+b=5\sqrt{2}
$$
Let $AM$ be $5x$ and $GM$ be $3x$. Then, $x=\frac{25}{8}$ and the $AM$ is $\frac{125}{8}$ and the $GM$ is $\frac{75}{8}$. 
Hence 
$$
a^2+b^2=2AM=\frac{125}{4} \\
2ab=2GM=\frac{75}{4} \\
a^2+b^2-2ab=\frac{125-75}{4}\\
a-b=\sqrt{\frac{50}{4}}=\frac{5\sqrt{2}}{2} 
$$
Find $a$ and $b$ from the two equations and $a^2$ and $b^2$ are your answers. 
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