A, B and C enter into a partnership. A invests some money at the beginning, B invests double the amount after six months and C invests thrice the amount after eight months. If the annual profit be ₹ 27000; C's share (in ₹) is

### Concept & Proportional Investment When investments are stated in relative terms (double, thrice), express them algebraically using a base variable (e.g., let initial amount be $x$). The equivalent capital is then found by multiplying by the active months. ### Step-by-Step Solution * **Given:** * Annual profit duration = $12 \text{ months}$ * Total Profit = $₹ 27000$ * **Define Investments and Time:** * Let A's initial investment be $x$. A invested for $12 \text{ months}$. * B invests double the amount = $2x$. He joins after 6 months, so his time = $(12 - 6) = 6 \text{ months}$. * C invests thrice the amount = $3x$. He joins after 8 months, so his time = $(12 - 8) = 4 \text{ months}$. * **Calculate Profit Ratio:** * $\text{Ratio} = (A's \text{ capital}) : (B's \text{ capital}) : (C's \text{ capital})$ * $\text{Ratio} = (x \times 12) : (2x \times 6) : (3x \times 4)$ * $\text{Ratio} = 12x : 12x : 12x$ * $\text{Ratio} = 1 : 1 : 1$ * **Calculate C's Share:** * The profit is distributed equally among all three partners because their equivalent capitals are identical. * Sum of ratio parts = $1 + 1 + 1 = 3$ * $\text{C's Share} = \left(\frac{1}{3}\right) \times 27000 = ₹ 9000$ ### Exam Strategy & Shortcut Notice the symmetry immediately: $1 \text{ (base)} \times 12 \text{ (full time)}$ $2 \text{ (double)} \times 6 \text{ (half time)}$ $3 \text{ (triple)} \times 4 \text{ (one-third time)}$ All products equal $12$. Thus, the profit is split equally into three parts. $27000 / 3 = 9000$. ### Common Pitfall Assuming that because someone invested more money later (like C investing thrice the amount), they automatically get a larger share of the profit. Time is just as critical as the investment amount! ### Final Answer Therefore, the correct answer is **₹ 9000**.