In a partnership, A invests $$\frac{1}{6}$$ of the capital $$\frac{1}{6}$$ for of the time, B invests $$\frac{1}{3}$$ of the capital for $$\frac{1}{3}$$ of the time and C, the rest of the capital for the whole time. Out of a profit of Rs. 4600, B's share is ?

Explanation:-

$$\eqalign{ & {\text{Suppose, }} \cr & {\text{A invests Rs}}{\text{.}}\frac{x}{6}{\text{ for }}\frac{y}{6}{\text{ months}} \cr & {\text{Then,}} \cr & {\text{ B invests Rs}}{\text{.}}\frac{x}{3}{\text{ for }}\frac{y}{3}{\text{ months}} \cr & {\text{C invests}}\left[ {x - \left( {\frac{x}{6} + \frac{x}{3}} \right)} \right]i.e.,{\text{ Rs}}{\text{.}}\frac{x}{2}{\text{ for }}y{\text{ months}} \cr & \therefore {\text{A}}:{\text{B}}:{\text{C}} \cr & {\text{ = }}\left( {\frac{x}{6} \times \frac{y}{6}} \right):\left( {\frac{x}{3} \times \frac{y}{3}} \right):\left( {\frac{x}{2} \times y} \right) \cr & = \frac{1}{{36}}:\frac{1}{9}:\frac{1}{2} \cr & = 1:4:18 \cr & {\text{Hence, B's share}} \cr & = {\text{Rs}}{\text{. }}\left( {4600 \times \frac{4}{{23}}} \right) \cr & = {\text{Rs}}{\text{. 800}} \cr}$$