A and B enter into a business by investing Rs 4000 and Rs 5500 respectively. After six months A and B withdraw Rs 1000 and Rs 1500 respectively, and at that time C joins them with a capital of Rs 4x. If after one year three months C receives Rs 2250 as share of profit from a total profit of Rs 12250, what is the value of the investment of C?

Introduction / Context:
This question is a more advanced partnership problem where partners change their capitals during the course of the business and a new partner joins later. The profit share depends on both the amount of capital and the time for which each amount remains invested. We are given the total profit and the share of C, and we need to work backwards to determine the capital of C in terms of rupees, using the algebraic relation involving x.

Given Data / Assumptions:
A invests Rs 4000 initially. B invests Rs 5500 initially. After 6 months, A withdraws Rs 1000 and B withdraws Rs 1500. At the same time, C joins with capital of Rs 4x. Total duration of the business is 15 months (1 year and 3 months). Total profit is Rs 12250 and C share is Rs 2250.

Concept / Approach:
The key concept is money months or capital time product. For each partner we calculate capital * time over each phase, sum these contributions and then share the profit in that resultant ratio. Since C share and total profit are known, C share fraction gives the ratio of C effective contribution to total effective contributions. This leads to an equation in x, from which we calculate 4x and match with the given options to find C capital.

Step-by-Step Solution:
Step 1: Business runs for 15 months. First 6 months have initial capitals, next 9 months have changed capitals and include C. Step 2: For first 6 months, capital of A is 4000 and of B is 5500, C is not yet a partner. Step 3: Money months of A in first phase = 4000 * 6 = 24000. Step 4: Money months of B in first phase = 5500 * 6 = 33000. Step 5: For next 9 months, A capital is 4000 - 1000 = 3000, B capital is 5500 - 1500 = 4000 and C capital is 4x. Step 6: Money months of A in second phase = 3000 * 9 = 27000. Step 7: Money months of B in second phase = 4000 * 9 = 36000. Step 8: Money months of C in second phase = 4x * 9 = 36x. Step 9: Total money months of A = 24000 + 27000 = 51000. Step 10: Total money months of B = 33000 + 36000 = 69000. Step 11: Total money months of C = 36x. Step 12: Let total effective contribution S = 51000 + 69000 + 36x = 120000 + 36x. Step 13: C share fraction = 2250 / 12250. Simplify: 2250 / 12250 = 9 / 49. Step 14: So 36x / (120000 + 36x) = 9 / 49. Step 15: Cross multiply: 36x * 49 = 9 * (120000 + 36x). Step 16: Left side = 1764x, right side = 1080000 + 324x. Step 17: Rearranging gives 1764x - 324x = 1080000, so 1440x = 1080000. Step 18: Therefore x = 1080000 / 1440 = 750. Step 19: C capital = 4x = 4 * 750 = Rs 3000.

Verification / Alternative check:
With x = 750, C effective contribution is 36x = 27000. Total effective contribution S becomes 51000 + 69000 + 27000 = 147000. The fraction of profit for C is 27000 / 147000 = 27 / 147 = 9 / 49. Applying this to total profit 12250 gives 12250 * 9 / 49 = 2250, which matches the stated share. This confirms that C capital must be Rs 3000 and that the algebraic steps are consistent.

Why Other Options Are Wrong:
Rs 3600, Rs 3200 and Rs 4400 correspond to different values of x and would change the fraction 36x / (120000 + 36x). None of those values produce the exact fraction 9 / 49 required to give C share of Rs 2250 from Rs 12250. Therefore those options do not satisfy the effective contribution equation and must be rejected. Only Rs 3000 generates the correct ratio and profit share.

Common Pitfalls:
A frequent error is to ignore the mid year withdrawal and joining, and assume a simple ratio of capital only. Another mistake is to use 12 months instead of 15 months for the duration. Some learners also forget that C contributes only in the second phase, not for the entire period. Setting up money months for each partner separately in each phase and then forming a single equation based on the known profit share is the most reliable method for partnership questions of this style.

Final Answer:
The value of the investment of C is Rs 3000.