Of three positive numbers, the ratio of first and second is 8 : 9, that of second and third is 3 : 4. The product of first and third is 2400. The sum of the three number is = ?

$$\eqalign{ & {\text{First}}:{\text{Second}}:{\text{Third}} \cr & \,\,\,\,8\,\,\,\,\,\,:\,\,\,\,\,\,9 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,:\,\,\,\,\,4 \cr & \frac{{\overline {\,\,24\,\,\,:\,\,\,\,\,27\,\,\,\,\,\,\,\,:\,\,\,\,36} }}{{\underline {\,8\,\,\,\,\,\,\,\,:\,\,\,\,\,\,\,\,9\,\,\,\,\,\,\,\,:\,\,\,\,12} }} \cr & {\text{Let 8x}}:{\text{9x}}:{\text{12x}} \cr & \therefore {\text{First}} \times {\text{Third}} \cr & 8x \times 12x = 2400 \cr & \Rightarrow 96{x^2} = 2400 \cr & \Rightarrow {x^2} = \frac{{2400}}{{96}} = 25 \cr & \Rightarrow x = 5 \cr & \therefore {\text{Sum of three numbers }} \cr & {\text{First}} + {\text{Second}} + {\text{Third}} \cr & 8x + 9x + 12x = 29x \cr & \therefore 29 \times 5 = 145 \cr} $$