‘D’ started a business with a capital of Rs. 15000. After 4 months, ‘E’ joined the business with a capital of Rs. 10000. 6
months after ‘E’, ‘F’ joined the business with a capital of Rs. 25000. If the profit received by ‘E’ at the end of the year
is Rs. 5000, then find the total profit received by all three at the end of the year.

To solve this problem, we can use the concept of profit sharing based on the ratio of the capital invested and the duration for which it was invested.

Let's break it down:

1. **Investment and Time Periods**:
   - **D** invested Rs. 15,000 for the entire year (12 months).
   - **E** invested Rs. 10,000 for 8 months (since he joined 4 months after D).
   - **F** invested Rs. 25,000 for 6 months (since he joined 6 months after E).

2. **Calculate the Ratio of Investments**:
   The effective capital contribution (investment × time) for each person is calculated as follows:

   - **D's contribution**: \( 15,000 \times 12 = 180,000 \)
   - **E's contribution**: \( 10,000 \times 8 = 80,000 \)
   - **F's contribution**: \( 25,000 \times 6 = 150,000 \)

   Now, the ratio of their contributions is:
   \[
   \text{Ratio of D : E : F} = 180,000 : 80,000 : 150,000 = 180 : 80 : 150 = 9 : 4 : 7.5
   \]

3. **Finding the Total Profit**:
   Let's assume the total profit at the end of the year is \( x \).

   Since E's share of the profit is Rs. 5,000, we can set up the equation based on the ratio of their shares:
   \[
   \text{E's share} = \frac{4}{9 + 4 + 7.5} \times x = \frac{4}{20.5} \times x
   \]
   We know E's share is Rs. 5,000, so:
   \[
   \frac{4}{20.5} \times x = 5,000
   \]
   Solving for \( x \):
   \[
   x = \frac{5,000 \times 20.5}{4} = \frac{102,500}{4} = 25,625
   \]

   So, the total profit received by all three at the end of the year is **Rs. 25,625**.

Would you like more details, or do you have any questions?

Here are 5 questions related to the problem:
1. How is profit sharing calculated in a partnership?
2. What happens if all partners invest different amounts but for the same duration?
3. How does the timing of investment affect profit distribution in a business?
4. Can you solve a similar problem where all partners invest the same amount but at different times?
5. How would the solution change if one partner withdraws their investment before the year ends?

**Tip**: When dealing with partnership problems, always consider both the amount of investment and the duration for which it was invested to calculate the profit ratio accurately.

Learn how to calculate total profit in a partnership problem where partners invest different amounts at different times. Understand the concept of profit sharing based on capital and duration invested. Includes a step-by-step solution to find total profit received by all partners at the end of the year. Suitable for students and professionals dealing with partnership and profit sharing calculations.

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