A began a business with Rs. 85,000. He was joined afterwards by B with Rs. 42,500. For how much period does B join, if the profits at the end of the year are divided in the ratio of 3:1?

Options

A. 4 months
B. 5 months
C. 6 months
D. 8 months

Answer

To solve the problem step by step, we need to find out for how long B joined the business based on the given profit-sharing ratio of A and B.

### Step-by-Step Solution:

1. **Identify the Capital and Time for A:**
- A's capital = Rs. 85,000
- A's time = 12 months (since he started the business at the beginning of the year)

2. **Identify the Capital for B:**
- B's capital = Rs. 42,500
- Let B's time in the business be \( x \) months.

3. **Calculate the Effective Capital Contribution:**
- A's effective capital contribution = Capital × Time = \( 85,000 \times 12 \)
- B's effective capital contribution = Capital × Time = \( 42,500 \times x \)

4. **Set Up the Profit Ratio:**
- According to the problem, the profits are divided in the ratio of 3:1 (A:B).
- This means:
\[ \frac{A's \, effective \, capital}{B's \, effective \, capital} = \frac{3}{1} \]
- Substituting the effective capital contributions:
\[ \frac{85,000 \times 12}{42,500 \times x} = \frac{3}{1} \]

5. **Cross-Multiply to Solve for \( x \):**
- Cross-multiplying gives:
\[ 85,000 \times 12 = 3 \times 42,500 \times x \]
- Simplifying the left side:
\[ 1,020,000 = 127,500 \times x \]

6. **Solve for \( x \):**
- Divide both sides by 127,500:
\[ x = \frac{1,020,000}{127,500} \]
- Calculating \( x \):
\[ x = 8 \]

7. **Conclusion:**
- B joined the business for **8 months**.