Arvind began a business with 550 and was joined afterwards by Brij with 330. When did Brij join, if the profits at the end of the year were divided in the ratio 10 : 3?

Options

A. After 4 months
B. After 6 months
C. After 4.5 months
D. None of these

Answer

To solve the problem, we need to determine when Brij joined the business based on the information given about their investments and the profit-sharing ratio.

### Step-by-Step Solution:

1. **Identify the Investments:**
- Arvind's investment = 550
- Brij's investment = 330

2. **Understand the Profit Sharing Ratio:**
- The profits are shared in the ratio of 10:3 (Arvind:Brij).

3. **Let’s Assume Time Periods:**
- Let the time Arvind invested the money be 12 months (1 year).
- Let the time Brij invested the money be \( x \) months.

4. **Calculate the Effective Capital Contribution:**
- The effective capital contribution is calculated by multiplying the capital invested by the time for which it is invested.
- For Arvind: \( 550 \times 12 = 6600 \)
- For Brij: \( 330 \times x \)

5. **Set Up the Equation Based on Profit Sharing Ratio:**
- According to the profit-sharing ratio:
\[ \frac{6600}{330x} = \frac{10}{3} \]

6. **Cross-Multiply to Solve for \( x \):**
- Cross-multiplying gives:
\[ 6600 \times 3 = 10 \times 330x \]
- Simplifying this:
\[ 19800 = 3300x \]

7. **Solve for \( x \):**
- Dividing both sides by 3300:
\[ x = \frac{19800}{3300} = 6 \]

8. **Interpret the Result:**
- Since \( x = 6 \), this means Brij joined the business 6 months after Arvind started.

### Conclusion:
Brij joined the business after 6 months.