Find two numbers whose mean proportion is $24$ and third proportion is $1536$ .

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Solution

Let the numbers= X and Y
$\begin{matrix} X Y = 24^{2} = 576 Y = \frac{576}{x} X : Y y : 1536 Y^{2} = 1536 x 331776 = 1536 x^{3} x^{3} = \frac{331776}{1536} = 216 x = 6 Y = \frac{576}{6} = 96 \end{matrix}$

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