Question

Find two numbers such that the mean proportional between them is $18$ and the third proportional to them is $144$.

Answer 1

Let the required number be $x\&y$.

$\therefore 14$is the mean proportional between $x\&y$.

Then,

$\frac{x}{18}=\frac{18}{y}$

$\Rightarrow xy=18\times 18=324$

$\Rightarrow x=\frac{324}{y}.......\left(1\right)$

And given that, $144$ is the third proportional to $x\&y$.

Then, $x:y::y:144$

$i.e.\frac{x}{y}=\frac{y}{144}$

$\Rightarrow y^2=144x.......\left(2\right)$

Putting the value of $x$ by equation (1) to (2) and we get,

$\Rightarrow y^2=144\times \frac{324}{y}$

$\Rightarrow y^3=144\times 324$

$\Rightarrow y^3=2\times 2\times 2\times 2\times 3\times 3\times 2\times 3\times 3\times 2\times 3\times 3$

$\Rightarrow y^3=36\times 36\times 36$

$\Rightarrow y^3={36}^3$

$\Rightarrow y=36$

Put the value of y in equation (1), and we get

$x=\frac{324}{36}$

$x=9$

Hence, this is the answer.