A began business with Rs.45000 and was joined afterwards by B with Rs. 54000. After how many months did B join if the profits at the end of the year were divided in the ratio 2:1?

Options

A. 4
B. 5
C. 6
D. 7

Answer

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

### Step-by-Step Solution:

1. **Identify the Investments**:
- A's investment = Rs. 45,000
- B's investment = Rs. 54,000

2. **Let the number of months A worked be 12 months** (since A started the business at the beginning of the year).
- Let the number of months B worked be \( x \) months.
- Therefore, B joined the business \( 12 - x \) months after A started.

3. **Calculate the Profit Contributions**:
- A's profit contribution = Investment × Time = \( 45000 \times 12 \)
- B's profit contribution = Investment × Time = \( 54000 \times x \)

4. **Set Up the Ratio**:
- According to the problem, the profits are divided in the ratio of 2:1.
- Therefore, we can write the equation:
\[ \frac{45000 \times 12}{54000 \times x} = \frac{2}{1} \]

5. **Cross-Multiply to Solve for x**:
- Cross-multiplying gives:
\[ 45000 \times 12 = 2 \times 54000 \times x \]
- Simplifying this:
\[ 540000 = 108000x \]
- Now, divide both sides by 108000:
\[ x = \frac{540000}{108000} = 5 \]

6. **Conclusion**:
- B joined the business after 5 months.

### Final Answer:
B joined the business after **5 months**.