If A : B : C = 2 : 3 : 4, then $$\frac{{\text{A}}}{{\text{B}}}{\text{:}}\frac{{\text{B}}}{{\text{C}}}{\text{:}}\frac{{\text{C}}}{{\text{A}}}$$   is equal to?

$$\eqalign{ & {\text{A}}:{\text{B}}:{\text{C}} \cr & {\text{ }}2:\,\,3\,\,:4 \cr & \therefore \frac{{\text{A}}}{{\text{B}}}:\frac{{\text{B}}}{{\text{C}}}:\frac{{\text{C}}}{{\text{A}}} \cr & \left( {{\text{multiply with ABC}}} \right) \cr & \therefore \frac{{{\text{A}} \times {\text{ABC}}}}{{\text{B}}}{\text{:}}\frac{{{\text{B}} \times {\text{ABC}}}}{{\text{C}}}{\text{:}}\frac{{{\text{C}} \times {\text{ABC}}}}{{\text{A}}} \cr & \Rightarrow {{\text{A}}^2}{\text{C}}:{{\text{B}}^2}{\text{A}}:{\text{B}}{{\text{C}}^2} \cr & \Rightarrow {\left( 2 \right)^2} \times 4:{\left( 3 \right)^2} \times 2:{\left( 4 \right)^2} \times 3 \cr & \Rightarrow 16:18:48 \cr & \Rightarrow 8:9:24 \cr & \cr & {\bf{Alternate:}} \cr & \frac{{\text{A}}}{{\text{B}}}:\frac{{\text{B}}}{{\text{C}}}:\frac{{\text{C}}}{{\text{A}}} = \frac{2}{3}:\frac{3}{4}:\frac{4}{2} \cr & \Rightarrow \frac{2}{3} \times 12:\frac{3}{4} \times 12:\frac{4}{2} \times 12 \cr & \Rightarrow 8:9:24 \cr} $$