Prakash, Sachin and Anil started a business jointly investing ₹ $11$ lakh, ₹ $16.5$ lakh and ₹ $8.25$ lakh respectively. The profit earned by them in the business at the end of $3$ years was ₹ $19.5$ lakh. What will be $50\%$ of Anil's share in the profit?

### Concept & Equal Time Partnership When investments are made for the same duration, the profit is shared in the ratio of the initial capital investments. $$ \text{Profit Ratio} = \text{Investment}_1 : \text{Investment}_2 : \text{Investment}_3 $$ ### Step-by-Step Solution * **Given:** * Prakash's investment = ₹ $11$ lakh * Sachin's investment = ₹ $16.5$ lakh * Anil's investment = ₹ $8.25$ lakh * Total Profit = ₹ $19.5$ lakh * **Calculation/Deduction:** 1. Find the ratio of their investments: $$\text{Prakash} : \text{Sachin} : \text{Anil} = 11 : 16.5 : 8.25$$ 2. Multiply by $100$ to remove the decimals: $$\text{Ratio} = 1100 : 1650 : 825$$ 3. Simplify the ratio by dividing by their highest common factor ($275$): $$\text{Ratio} = 4 : 6 : 3$$ 4. Sum the parts of the ratio to find the total denominator: $$\text{Total parts} = 4 + 6 + 3 = 13$$ 5. Calculate Anil's total share in the profit (he represents $3$ parts): $$\text{Anil's Share} = 19.5 \times \left(\frac{3}{13}\right)$$ $$\text{Anil's Share} = 1.5 \times 3 = 4.5 \text{ lakh}$$ 6. The question asks for $50\%$ of Anil's share: $$50\% \text{ of } 4.5 \text{ lakh} = \frac{4.5}{2} = 2.25 \text{ lakh}$$ ### Exam Strategy & Shortcut When simplifying $1100 : 1650 : 825$, notice that $825 \times 2 = 1650$, so the ratio between the last two is $2 : 1$ (or $6:3$). Knowing $11 \times 75 = 825$, you can divide by common factors like $25$ and $11$ rapidly. Crucially, don't forget the final step: the question specifically asks for $50\%$ of Anil's share, not his full share. ### Common Pitfall A major trap is calculating Anil's full share ($4.5$ lakh) and immediately selecting option (d). Always re-read the last sentence of the question. ### Final Answer Therefore, the correct answer is **₹ 2.25 lakh**.