Rushi and Arun are business partners with initial capitals in the ratio 7 : 9. After 6 months, Ram joins them with a capital equal to two-thirds of Arun's investment. If after one year Rushi's share of the profit is Rs. 5600, what is the total profit of the business at the end of the year?

Introduction / Context:
This problem tests the concept of partnership where profit sharing depends on both the amount of capital invested and the duration for which it is invested. One partner joins after some time, so we must use capital time products to determine the ratio of profits and then find the total profit from one partner's share.

Given Data / Assumptions:

• Initial capital ratio of Rushi and Arun is 7 : 9.
• Ram joins after 6 months with capital equal to two-thirds of Arun's capital.
• Total business duration considered is 12 months.
• Rushi's profit share at the end of one year is Rs. 5600.
• All partners share profit in proportion to capital multiplied by time of investment.

Concept / Approach:
In partnership problems, profit is proportional to capital multiplied by time. So we convert each partner's investment into capital time units, form a ratio, and then use the known profit share of one partner to determine the total profit. Here, Rushi and Arun invest from the start, while Ram comes in after 6 months, so his time period is shorter. Once we have the ratio of their shares, we can treat it as parts of the total profit and calculate the overall profit using Rushi's known share.

Step-by-Step Solution:
Step 1: Let the common capital factor be k. Then Rushi invests 7k and Arun invests 9k for the whole year, that is 12 months each. Step 2: Ram joins after 6 months with capital equal to two-thirds of Arun's capital. Arun's capital is 9k, so Ram's capital is (2/3) * 9k = 6k. Step 3: Convert each partner's investment into capital time units: Rushi: 7k * 12 = 84k Arun: 9k * 12 = 108k Ram: 6k * 6 = 36k Step 4: Their profit shares are in the ratio 84 : 108 : 36. Step 5: Divide by 12 to simplify: 84 : 108 : 36 becomes 7 : 9 : 3. Step 6: Sum of ratio parts is 7 + 9 + 3 = 19 parts. Rushi's share corresponds to 7 parts and is given as Rs. 5600. Step 7: Value of one part is 5600 / 7 = 800. Step 8: Total profit equals 19 * 800 = Rs. 15200.

Verification / Alternative check:
We can also compute individual partner profits using parts. Rushi gets 7 * 800 = 5600, Arun gets 9 * 800 = 7200 and Ram gets 3 * 800 = 2400. Adding them, 5600 + 7200 + 2400 = 15200, which matches the total profit we have calculated. This confirms that the ratio and the value per part have been computed correctly and that the final total profit is consistent with the data.

Why Other Options Are Wrong:
Values like Rs. 26250, Rs. 19200, Rs. 18650 and Rs. 13800 do not produce Rushi's share as exactly Rs. 5600 when the profit is split in the ratio 7 : 9 : 3. For example, if the total profit were Rs. 19200, then one part would be 19200 / 19, which is not an integer, and the share corresponding to 7 parts would not equal Rs. 5600. Hence these options are inconsistent with the required ratio and the given share of Rushi.

Common Pitfalls:
Students often forget that the time period of investment must be included when finding profit shares, especially when a partner joins late. Another common error is taking two-thirds of the total capital instead of two-thirds of Arun's capital. Some candidates also directly try to match Rs. 5600 with a percentage of the total profit, rather than using the proper ratio method. Careful attention to both capital amounts and time is essential in this type of partnership problem.

Final Answer:
The total profit of the business at the end of the year is Rs. 15200, which corresponds to option D.