Aman started a business investing ₹ 70000. Rakhi joined him after six months with an amount of ₹ 105000 and Sagar joined them with ₹ 1.4 lakhs after another six months. The amount of profit earned should be distributed in what ratio among Aman, Rakhi and Sagar respectively, 3 years after Aman started the business?

### Concept & Formula The core concept in partnership problems is that the profit is divided in the ratio of the product of the investment amount and the time period for which the investment was active. $$ \text{Ratio of Profit} = (\text{Investment}_1 \times \text{Time}_1) : (\text{Investment}_2 \times \text{Time}_2) : \dots $$ ### Step-by-Step Solution * **Given:** * Aman's investment = $₹ 70000$ * Rakhi's investment = $₹ 105000$ * Sagar's investment = $₹ 1.4 \text{ lakhs} = ₹ 140000$ * Total duration of business = $3 \text{ years} = 36 \text{ months}$ * **Determine Time Periods:** * Aman invested for the full duration: $36 \text{ months}$ * Rakhi joined after 6 months, so her money was invested for $36 - 6 = 30 \text{ months}$ * Sagar joined after another 6 months (12 months from start), so his money was invested for $36 - 12 = 24 \text{ months}$ * **Calculate Profit Ratio:** * $\text{Ratio} = (70000 \times 36) : (105000 \times 30) : (140000 \times 24)$ * Simplify by cancelling out common zeros (divide by $1000$): * $\text{Ratio} = (70 \times 36) : (105 \times 30) : (140 \times 24)$ * Divide all terms by $35$: * $\text{Ratio} = (2 \times 36) : (3 \times 30) : (4 \times 24)$ * $\text{Ratio} = 72 : 90 : 96$ * Divide all terms by $6$: * $\text{Ratio} = 12 : 15 : 16$ ### Exam Strategy & Shortcut Instead of multiplying large numbers, cancel out the common factors (like thousands and 35) from the investments *before* multiplying by the time periods. This significantly speeds up calculation. ### Common Pitfall Forgetting to convert years to months or miscalculating the time Sagar's money was invested ("another six months" means $6+6=12$ months late, so $36-12=24$ months active). ### Final Answer Therefore, the correct answer is **12 : 15 : 16**.