A starts a business with Rs. 3,500. After 5 months, B joins A as a partner. At the end of one year, the profit is divided between A and B in the ratio 2 : 3. What is the amount of capital invested by B?

Introduction / Context:
This is a partnership question with one partner starting the business and a second partner joining later. The final profits are divided in a known ratio. We must use the relation between profit share and capital * time to determine the capital B invested when he joined after 5 months.

Given Data / Assumptions:

• A invests Rs. 3,500 at the start of the business.
• After 5 months, B joins the business as a partner.
• The total duration of the business is 1 year, that is, 12 months.
• A is invested for 12 months.
• B is invested for 12 - 5 = 7 months.
• The profit at the end of the year is divided in the ratio A : B = 2 : 3.
• Profit is proportional to capital * time.
• We must find B's capital.

Concept / Approach:
We compute the effective investments in terms of capital * time. Since the profit ratio equals the ratio of effective investments, we set up a proportion between A's and B's effective investments and the given ratio 2 : 3. Solving this proportion gives us the value of B's capital.

Step-by-Step Solution:
Step 1: Effective investment of A = capital * time = 3,500 * 12 = 42,000 rupee months. Step 2: Let B's capital be x rupees. B invests for 7 months, so effective investment of B = x * 7 = 7x rupee months. Step 3: The ratio of profits A : B is given as 2 : 3, so the ratio of effective investments must also be 2 : 3. Step 4: Therefore, 42,000 : 7x = 2 : 3. Step 5: Write this as a proportion: 42,000 / 7x = 2 / 3. Step 6: Cross multiply: 42,000 * 3 = 2 * 7x. Step 7: This gives 1,26,000 = 14x, so x = 1,26,000 / 14. Step 8: Compute x: x = 9,000. Thus B's capital is Rs. 9,000.

Verification / Alternative check:
Using A's effective investment 42,000 and B's effective investment 7 * 9,000 = 63,000, the ratio becomes 42,000 : 63,000. Dividing by 21,000 gives 2 : 3, which matches the given profit ratio. This confirms that the computed capital of B is consistent with the partnership conditions and the final profit sharing.

Why Other Options Are Wrong:
If B had invested Rs. 7,500, 8,000 or 8,500, the effective investments would be 52,500, 56,000 or 59,500 respectively. None of these yield a ratio of 42,000 : 7x equal to 2 : 3. Instead, the resulting ratios would be different and would not match the given profit division ratio.

Common Pitfalls:
A common mistake is to divide the capitals directly in the ratio 2 : 3 and ignore the fact that B joined late and therefore invested for less time. Another error is miscalculating the number of months B's capital was actually in the business, or setting up the proportion incorrectly. Always compute capital * time for each partner and form the ratio carefully.

Final Answer:
B's capital contribution is Rs. 9,000.