A invested Rs. 76000 in a business. After few months, B joined him with Rs. 57000. At the end of the year, the total profit was divided between them in the ratio 2 : 1. After how many months did B join ?

Answer

To solve the problem, we need to determine after how many months B joined A in the business, given that the total profit was divided in the ratio of 2:1.

### Step-by-Step Solution:

1. **Understanding the Investment:**
- A invested Rs. 76,000 for the entire year (12 months).
- B invested Rs. 57,000 but for an unknown number of months, which we will denote as \( x \).

2. **Calculating the Contribution of Each Partner:**
- A's contribution to the profit can be calculated as:
\[ \text{A's Contribution} = \text{Capital} \times \text{Time} = 76000 \times 12 \]
- B's contribution to the profit can be calculated as:
\[ \text{B's Contribution} = \text{Capital} \times \text{Time} = 57000 \times x \]

3. **Setting Up the Profit Ratio:**
- According to the problem, the profit ratio of A to B is 2:1. Therefore, we can set up the equation:
\[ \frac{A's Contribution}{B's Contribution} = \frac{2}{1} \]
- Substituting the contributions:
\[ \frac{76000 \times 12}{57000 \times x} = \frac{2}{1} \]

4. **Cross-Multiplying to Solve for x:**
- Cross-multiplying gives us:
\[ 76000 \times 12 = 2 \times 57000 \times x \]
- Simplifying this:
\[ 912000 = 114000 \times x \]

5. **Solving for x:**
- Now, divide both sides by 114000:
\[ x = \frac{912000}{114000} = 8 \]

6. **Conclusion:**
- B joined A after 8 months.

### Final Answer:
B joined after 8 months.