Find two numbers such that their mean proportional is 6 and third proportional 20.25. · ABC26GN2922

Let the two numbers be $x$ and $y$. Then, $\sqrt{xy}=6$ and $x:y::y:20.25$ $$\Rightarrow xy=(6)^2=36 \text{ and } y^2=20.25x \Rightarrow x=\frac{36}{y} \text{ and } y^2=20.25\times\frac{36}{y}=\frac{729}{y}$$ $$\Rightarrow y^3=729=9^3 \Rightarrow y=9 \Rightarrow x=\frac{36}{9}=4.$$ Hence, the required numbers are 4 and 9.