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

### Concept & Partnership Ratio When investments are made for the exact same time period, the profit sharing ratio is exactly equal to the ratio of their invested capitals. $$ \text{Profit Ratio} = \text{Investment Ratio} $$ ### Step-by-Step Solution * **Given:** * Rahul's investment = $\frac{1}{2}$ of total * Arun's investment = $\frac{1}{3}$ of total * **Calculation/Deduction:** 1. Find Sumit's investment by subtracting Rahul's and Arun's shares from the whole ($1$): $$\text{Sumit's share} = 1 - \left( \frac{1}{2} + \frac{1}{3} \right)$$ $$\text{Sumit's share} = 1 - \left( \frac{3 + 2}{6} \right) = 1 - \frac{5}{6} = \frac{1}{6}$$ 2. Set up the ratio of investments for Rahul : Arun : Sumit: $$\text{Ratio} = \frac{1}{2} : \frac{1}{3} : \frac{1}{6}$$ 3. Multiply the entire ratio by the Least Common Multiple (LCM) of the denominators (which is $6$) to convert it into whole numbers: $$\text{Ratio} = \left(\frac{1}{2} \times 6\right) : \left(\frac{1}{3} \times 6\right) : \left(\frac{1}{6} \times 6\right)$$ $$\text{Ratio} = 3 : 2 : 1$$ ### Exam Strategy & Shortcut To quickly clear fractions in a ratio like $\frac{1}{2} : \frac{1}{3} : \frac{1}{6}$, simply multiply by the LCM of $2, 3,$ and $6$. The LCM is $6$, immediately yielding $3 : 2 : 1$. ### Common Pitfall A frequent mistake is to leave the ratio as fractions and incorrectly map the numerators without establishing a common denominator. Always normalize the ratio using the LCM. ### Final Answer Therefore, the correct answer is **3 : 2 : 1**.