Rahul, Arun and Sumit started a business. Rahul invested $$\frac{1}{2}$$ part, Arun $$\frac{1}{3}$$ part and rest of the capital was invested by Sumit. The ratio of their profits will be = ?
a) 2 : 3 : 1
b) 3 : 2 : 1
c) 2 : 3 : 6
d) 3 : 2 : 5

Answer: b
Explanation: Let total capital be Rs. x
$$\eqalign{ & {\text{Rahul’s share}} = {\text{Rs}}{\text{.}}\frac{x}{2} \cr & {\text{Arun’s share}} = {\text{Rs}}{\text{.}}\frac{x}{3} \cr & {\text{Sumit’s share}} \cr & = {\text{Rs}}{\text{.}}\left[ {x – \left( {\frac{x}{2} + \frac{x}{3}} \right)} \right] \cr & = {\text{Rs}}{\text{. }}\frac{x}{6} \cr & {\text{Required ratio}} \cr & = \frac{x}{2}:\frac{x}{3}:\frac{x}{6} \cr & = \frac{1}{2}:\frac{1}{3}:\frac{1}{6} \cr & = 3:2:1 \cr} $$