Determine the ratio of two numbers if the ratio of their arithmetic mean to geometric mean is $25: 24$.

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![](https://cdn.mathpix.com/cropped/2024_05_06_fcac0c1ff1d265557ffdg-12.jpg?height=49&width=1992&top_left_y=1319&top_left_x=16)
\(\{\mathrm{y}\}\}+\mid \operatorname{sqrt}\{\backslash f r a c\{\mathrm{y}\}\{\mathrm{x}\}\}=\backslash \operatorname{frac}\{25\}\{12\} \). Let \(\$ \mathrm{q}=\mid \operatorname{sqrt}\{\backslash \operatorname{frac}\{\mathrm{x}\}\{\mathrm{y}\}\} \$$, we get the quadratic equation \(\$ \mathrm{q} \wedge 2-\frac \{25\}\{12\} q+1=0 \). Its roots are \(\$q_1=|frac\{4\}\{3\}\$ and \(\$q_2=|frac34\$). Accordingly, \(\$x \colon y = 16 \colon 9 \$$ or \(\$ 9 \colon 16 \$$.

## Answer

\(16: 9\) or \(9: 16\)