Aman started a business investing Rs. 70,000. Rakhi joined him after six months with an amount of Rs. 1,05,000 and Sagar joined them with Rs. 1.4 lakhs after another six months. The amount of profit earned should be distributed in what ratio among Aman, Rakhi and Sagar respectively , 3 years after started the business?

Options

A. 7:6:10
B. 12:15:16
C. 42:45:56
D. Cannot be determined

Answer

To solve the problem of distributing the profit among Aman, Rakhi, and Sagar based on their investments and the time they invested, we will follow these steps:

### Step 1: Determine the Investment Amounts
- Aman invested Rs. 70,000.
- Rakhi joined after 6 months with Rs. 1,05,000.
- Sagar joined after another 6 months (1 year from the start) with Rs. 1,40,000.

### Step 2: Determine the Duration of Investment
- Aman invested for 3 years.
- Rakhi invested for 2.5 years (3 years - 6 months).
- Sagar invested for 2 years (3 years - 1 year).

### Step 3: Calculate the Capital Contribution
To find the capital contribution, we multiply the investment amount by the time period of investment.

- **Aman's Contribution**:
\[ 70,000 \times 3 = 2,10,000 \]

- **Rakhi's Contribution**:
\[ 1,05,000 \times 2.5 = 2,62,500 \]

- **Sagar's Contribution**:
\[ 1,40,000 \times 2 = 2,80,000 \]

### Step 4: Calculate Total Contributions
Now, we can summarize the contributions:
- Aman: Rs. 2,10,000
- Rakhi: Rs. 2,62,500
- Sagar: Rs. 2,80,000

### Step 5: Find the Ratio of Contributions
To find the ratio of their contributions, we can express it as:
\[ \text{Aman's Contribution} : \text{Rakhi's Contribution} : \text{Sagar's Contribution} = 2,10,000 : 2,62,500 : 2,80,000 \]

### Step 6: Simplify the Ratio
To simplify the ratio, we can divide each term by 10,000:
\[ 21 : 26.25 : 28 \]

To eliminate the decimal, we can multiply each term by 4:
\[ 21 \times 4 : 26.25 \times 4 : 28 \times 4 = 84 : 105 : 112 \]

Now, we can simplify this further by dividing by 21:
\[ 4 : 5 : \frac{112}{21} \approx 5.33 \]

To express this in whole numbers, we can multiply by 3:
\[ 12 : 15 : 16 \]

### Final Ratio
Thus, the final ratio of the profit distribution among Aman, Rakhi, and Sagar is:
\[ 12 : 15 : 16 \]

### Conclusion
The profits should be distributed in the ratio of 12:15:16 among Aman, Rakhi, and Sagar respectively.

---