Question

A started a business with a capital of 24,000.B joined him after a month with a capital of 22,000 and C joined A and B with a capital of `20,000 and new partners joined in this manner so on till L joined in the last month with a capital of 2,000.What is the total profit if D got 72,000 more than L?

• A. 2,42,500
• B. 3,40,000
• C. 5,85,000
• D. 7,25,000
• E. 10,80,000

Answer 1

The Correct Option is C

Step 1: Understand the problem.
A started a business with a capital of Rs. 24,000. B joined after a month with a capital of Rs. 22,000, and C joined after another month with a capital of Rs. 20,000. This pattern continues, with new partners joining each month with decreasing capital, such that L joined in the last month with a capital of Rs. 2,000.
The total profit is to be determined, with the additional information that D received Rs. 72,000 more than L.

Step 2: Calculate the capitals of each partner.
The capital of each partner decreases by Rs. 2,000 every month. Therefore, the capitals for each partner are:
- A: Rs. 24,000 (since A started the business)
- B: Rs. 22,000 (joined after 1 month)
- C: Rs. 20,000 (joined after 2 months)
- D: Rs. 18,000 (joined after 3 months)
- E: Rs. 16,000 (joined after 4 months)
- F: Rs. 14,000 (joined after 5 months)
- G: Rs. 12,000 (joined after 6 months)
- H: Rs. 10,000 (joined after 7 months)
- I: Rs. 8,000 (joined after 8 months)
- J: Rs. 6,000 (joined after 9 months)
- K: Rs. 4,000 (joined after 10 months)
- L: Rs. 2,000 (joined after 11 months)
Therefore, there are 12 partners, each joining at a different time, with the capital reducing by Rs. 2,000 every month.

Step 3: Use the profit-sharing rule based on capital invested and time.
The profit is divided based on the capital invested and the time for which each partner invests that capital. The total capital-time product for each partner is the capital multiplied by the time for which they invested that capital.
For each partner, the capital-time product is calculated as follows:
- A: \( 24,000 \times 12 = 288,000 \)
- B: \( 22,000 \times 11 = 242,000 \)
- C: \( 20,000 \times 10 = 200,000 \)
- D: \( 18,000 \times 9 = 162,000 \)
- E: \( 16,000 \times 8 = 128,000 \)
- F: \( 14,000 \times 7 = 98,000 \)
- G: \( 12,000 \times 6 = 72,000 \)
- H: \( 10,000 \times 5 = 50,000 \)
- I: \( 8,000 \times 4 = 32,000 \)
- J: \( 6,000 \times 3 = 18,000 \)
- K: \( 4,000 \times 2 = 8,000 \)
- L: \( 2,000 \times 1 = 2,000 \)

Step 4: Calculate the total capital-time product.
The total capital-time product is the sum of the individual capital-time products:
\[ \text{Total capital-time product} = 288,000 + 242,000 + 200,000 + 162,000 + 128,000 + 98,000 + 72,000 + 50,000 + 32,000 + 18,000 + 8,000 + 2,000 = 1,700,000 \]

Step 5: Find the profit share for each partner.
The profit share for each partner is based on their capital-time product. The total profit is to be divided in the ratio of the capital-time product of each partner to the total capital-time product.
We are told that D got Rs. 72,000 more than L. From the previous step, we know that:
- D’s capital-time product = 162,000
- L’s capital-time product = 2,000
The profit share ratio of D to L is:
\[ \text{Profit ratio} = \frac{162,000}{2,000} = 81 \] Therefore, D’s share is 81 times L’s share. The profit difference of Rs. 72,000 is distributed in this ratio. So, the total profit is:
\[ \text{Total profit} = \frac{72,000}{81} \times (81 + 1) = 72,000 \times \frac{82}{81} = 585,000 \] Thus, the total profit is Rs. 585,000.

Step 6: Conclusion.
The total profit in the business is Rs. 585,000.

Final Answer:
The correct answer is (C): 5,85,000.