[tex]a+b+c+d+e=218 \\ \frac{a}{5}=\frac{b}{6}=\frac{c}{10}\\ 3c=8d=12e; \\ a=\frac{5c}{10}=\frac{c}{2}; \\ b=\frac{6c}{10}=\frac{3c}{5}; \\ d=\frac{3c}{8}; \\ e=\frac{3c}{12}=\frac{c}{4}; \\ \Rightarrow \frac{c}{2}+\frac{3c}{5}+c+\frac{3c}{8}+\frac{c}{4}=218 \\ \frac{20c}{40}+\frac{24c}{40}+\frac{40c}{40}+\frac{15c}{40}+\frac{10c}{40}=218 \\ \frac{109c}{40}=218 \ \Leftrightarrow 109c=218 \cdot 40=8 \ 720 \\ \Rightarrow c=\frac{8 \ 720}{109} \ \Longleftrightarrow \boxed{c=80};[/tex]
[tex]a=\frac{c}{2}=\frac{80}{2} \ \Longleftrightarrow \boxed{a=40}; \\ b=\frac{3c}{5}=\frac{3 \cdot 80}{5}=3 \cdot 16 \ \Longleftrightarrow \boxed{b=48}; \\ d=\frac{3c}{8}=\frac{3 \cdot 80}{8}=3 \cdot 10 \ \Longleftrightarrow \boxed{d=30}; \\ e=\frac{c}{4}=\frac{80}{4}=\frac{20}{1} \ \Longleftrightarrow \boxed{e=20}.[/tex]
