A and B started a business by investing ₹ 2400 and ₹ 3600 respectively. At the end of 4th month from the start of the business, C joined with ₹ 'X'. After 8 months from the start of the business. B withdrew ₹600. If C's share is ₹ 8000 in the annual profit of ₹ 22500, what was the amount the C invested in the business?

### Concept & Reverse Profit Sharing When the final profit share and total profit are known but an investment is unknown, set up an equation equating the ratio of the unknown equivalent capital to the known equivalent capital with the ratio of their respective profits. $$ \frac{\text{C's Capital}}{\text{(A+B)'s Capital}} = \frac{\text{C's Profit}}{\text{(A+B)'s Profit}} $$ ### Step-by-Step Solution * **Given:** * Business duration is 12 months (annual profit). * A's investment = ₹2400 for 12 months. * B's investment = ₹3600 for 8 months, then withdraws ₹600, meaning (3600 - 600) = ₹3000 for the remaining 4 months. * C's investment = ₹X for 8 months (joined after 4 months). * Total profit = ₹22,500; C's share = ₹8,000. * **Calculation:** * A's equivalent capital = $2400 \times 12 = 28800$ * B's equivalent capital = $(3600 \times 8) + (3000 \times 4) = 28800 + 12000 = 40800$ * C's equivalent capital = $8X$ * Sum of A and B's equivalent capitals = $28800 + 40800 = 69600$ * Sum of A and B's profit = $22500 - 8000 = 14500$ * Equating the ratios: $\frac{8X}{69600} = \frac{8000}{14500}$ * Simplify the profit ratio: $\frac{8000}{14500} = \frac{80}{145} = \frac{16}{29}$ * $\frac{8X}{69600} = \frac{16}{29}$ * $8X = 69600 \times \left(\frac{16}{29}\right)$ * $69600 / 29 = 2400$ * $8X = 2400 \times 16 = 38400$ * $X = \frac{38400}{8} = 4800$ ### Exam Strategy & Shortcut Instead of finding C's proportion over the *total* sum (which requires adding $8X$ to the denominator), compare C directly against the sum of A and B. This keeps the unknown variable $X$ isolated in the numerator, solving the equation faster. ### Common Pitfall A frequent error is calculating C's investment time as 4 months instead of 8 months. "At the end of 4th month... C joined" means C was active for the remaining $12 - 4 = 8$ months. ### Final Answer Therefore, the correct answer is **₹ 4800**.