A and B start a business with their investments in the ratio 4 : 6. After 6 months, B withdraws his entire investment and C joins with twice the amount invested by B. If the total profit at the end of the year is Rs. 3,315, what is the share of C in this profit?

Introduction / Context:
This partnership question involves three partners at different stages: A and B start the business, B withdraws after 6 months and C joins with a different capital. The profit is calculated at the end of one year. We must determine how much of the total profit of Rs. 3,315 goes to C, using the capital * time concept to compute each partner's effective investment.

Given Data / Assumptions:

• Initial investment ratio of A : B = 4 : 6.
• Let the common unit be k so A = 4k and B = 6k.
• B withdraws his capital completely after 6 months.
• After 6 months, C joins with twice B's investment, so C invests 2 * 6k = 12k.
• The business runs for a total of 12 months.
• A remains invested for all 12 months.
• B invests for the first 6 months only.
• C invests for the last 6 months only.
• Total profit at the end of the year is Rs. 3,315.
• Profit is shared in proportion to capital * time.

Concept / Approach:
Because capital contributions change over time, we compute the effective investment for each partner as capital multiplied by the number of months the capital is in the business. These effective investments define the ratio for profit sharing. Using that ratio, we can compute C's share by applying the ratio fraction to the total profit.

Step-by-Step Solution:
Step 1: Effective investment of A: A invests 4k for 12 months. Effective investment of A = 4k * 12 = 48k. Step 2: Effective investment of B: B invests 6k for 6 months. Effective investment of B = 6k * 6 = 36k. Step 3: Effective investment of C: C invests 12k (twice B's capital) for the last 6 months. Effective investment of C = 12k * 6 = 72k. Step 4: Total effective investments in terms of k: A : B : C = 48k : 36k : 72k. Step 5: Simplify by dividing by 12k: A : B : C = 4 : 3 : 6. Step 6: Total ratio parts = 4 + 3 + 6 = 13. Step 7: Each part of profit = 3,315 / 13 = 255. Step 8: C's share = 6 parts = 6 * 255 = Rs. 1,530.

Verification / Alternative check:
We can verify by also computing A's and B's shares. A gets 4 * 255 = 1,020 and B gets 3 * 255 = 765. Adding all three shares gives 1,020 + 765 + 1,530 = 3,315, which equals the total profit. The shares of A, B and C are also proportional to 4 : 3 : 6, confirming that the effective investment method is correctly applied.

Why Other Options Are Wrong:
The other options 450, 1,020 and 765 correspond to 1 part, 4 parts and 3 parts respectively in the 4 : 3 : 6 ratio. They represent possible shares of one of the other partners, not of C, whose share must clearly be the largest because C's effective investment 72k is greater than those of A and B. Hence these options cannot represent C's correct share.

Common Pitfalls:
Learners often forget that B withdraws completely after 6 months and mistakenly treat B as investing for the full year. Another common error is to assume that C invests for 6 months but with the same capital as B, overlooking the phrase "twice the amount of B". Carefully tracking both the amount and the time of each partner's investment is essential for such questions.

Final Answer:
C's share of the profit is Rs. 1,530.