A, B and C enter into a partnership with capitals in the ratio 5 : 6 : 8. At the end of the business term, they received the profit in the ratio 5 : 3 : 12. Find the ratio of time for which they contributed their capitals ?

$$\eqalign{ & {\text{Here}}\,{{\text{p}}_1}:{{\text{p}}_2}:{{\text{p}}_3} = 5:3:12 \cr & {\text{and}}\,{{\text{x}}_1}:{{\text{x}}_2}:{{\text{x}}_3} = 5:6:8 \cr & {\text{According}}\,{\text{to}}\,{\text{the}}\,{\text{formula}} \cr & {\text{Required}}\,{\text{ratio}} \cr & = \frac{{{{\text{p}}_1}}}{{{{\text{x}}_1}}}:\frac{{{{\text{p}}_2}}}{{{{\text{x}}_2}}}:\frac{{{{\text{p}}_3}}}{{{{\text{x}}_3}}} \cr & = \frac{5}{5}:\frac{3}{6}:\frac{{12}}{8} \cr & = 1:\frac{1}{2}:\frac{3}{2} \cr & = 2:1:3 \cr & \cr & {\bf{Alternate:}} \cr & {\text{Profit}} = {\text{Time}} \times {\text{Capital invested}} \cr & {\text{Time}} = \frac{{{\text{Profit}}}}{{{\text{Capital invested}}}} \cr & {\text{Required ratio of time}} \cr & = \frac{5}{6}:\frac{3}{6}:\frac{{12}}{8} \cr & = 1:\frac{1}{2}:\frac{3}{2} \cr & = 2:1:3 \cr} $$