 ### 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 ?

A. 2 : 1 : 3 B. 1 : 2 : 3 C. 2 : 3 : 1 D. 3 : 2 : 1 Answer: Option A
$\begin{array}{l} \text { Here } \mathrm{p}_{1}: \mathrm{p}_{2}: \mathrm{p}_{3}=5: 3: 12 \\ \text { and } \mathrm{x}_{1}: \mathrm{x}_{2}: \mathrm{x}_{3}=5: 6: 8 \end{array}$ \begin{aligned}&\text{According to the formula}\\ &\text{ Required ratio }\\ &=\frac{\mathrm{p}_1}{\mathrm{x}_1}:\frac{\mathrm{p}_2}{\mathrm{x}_2}:\frac{\mathrm{p}_3}{\mathrm{x}_3}\\ &=\frac{5}{5}:\frac{3}{6}:\frac{12}{8}\\ &=1:\frac{1}{2}:\frac{3}{2}\\ &=2:1:3\end{aligned}
Alternate Solution: \begin{aligned}\text{ Profit }&=\text{ Time }\times\text{ Capital invested }\\ \text{ Time }&=\frac{\text{ Profit }}{\text{ Capital invested }}\\ &\text{Required ratio of time}\\ &=\frac{5}{6}: \frac{3}{6}: \frac{12}{8} \\ &=1: \frac{1}{2}: \frac{3}{2} \\ &=2: 1: 3\end{aligned}