Question Number 191323 by Mingma last updated on 23/Apr/23
Answered by mr W last updated on 23/Apr/23
$$\frac{{A}}{{A}+{B}}=\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\frac{{A}}{{A}+{B}+{C}}=\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{9}} \\ $$$$\Rightarrow{A}=\frac{{A}+{B}+{C}}{\mathrm{9}} \\ $$$$\Rightarrow{A}+{B}=\mathrm{4}{A}\:\Rightarrow{B}=\mathrm{3}{A}=\frac{\mathrm{3}\left({A}+{B}+{C}\right)}{\mathrm{9}} \\ $$$$\Rightarrow{C}=\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{9}}−\frac{\mathrm{3}}{\mathrm{9}}\right)\left({A}+{B}+{C}\right)=\frac{\mathrm{5}\left({A}+{B}+{C}\right)}{\mathrm{9}} \\ $$$$\Rightarrow{A}:{B}:{C}=\mathrm{1}:\mathrm{3}:\mathrm{5} \\ $$
Commented by Mingma last updated on 23/Apr/23
Excellent!