A begins a business with a capital of Rs. 85,000. After some time, B joins the business with a capital of Rs. 42,500. If at the end of the year the profits of A and B are in the ratio 3 : 1, for how many months did B invest his capital in the business?

Introduction / Context:
This is a partnership problem in which one partner, A, starts the business and another partner, B, joins later. The profits are divided at the end of the year in a known ratio. To determine the duration of B's investment, we use the principle that profits in partnership are proportional to capital multiplied by time.

Given Data / Assumptions:

• A invests Rs. 85,000 for the whole year (12 months).
• B invests Rs. 42,500 but joins after some time, so his investment is for t months.
• At the end of the year, the ratio of their profits A : B is 3 : 1.
• Profit is shared in proportion to capital multiplied by time.
• We must find how many months B keeps his capital invested, that is the value of t.

Concept / Approach:
Let the profit share of A be proportional to 85000 * 12, and the profit share of B be proportional to 42500 * t. Given that their profit ratio is 3 : 1, we equate the ratio of capital time products with 3 : 1 and solve for t. The resulting value of t gives the number of months B's capital was employed in the business.

Step-by-Step Solution:
Step 1: Compute A's capital time units. A invests 85000 for 12 months, so A's units = 85000 * 12. Step 2: Let B invest 42500 for t months, so B's capital time units = 42500 * t. Step 3: The ratio of profits is given as A : B = 3 : 1. Step 4: Therefore, 85000 * 12 : 42500 * t = 3 : 1. Step 5: Simplify the ratio on the left side. First divide both capital amounts by 42500. Then 85000 / 42500 = 2. Step 6: So (85000 * 12) / (42500 * t) = (2 * 12) / t = 24 / t. Step 7: Set 24 / t = 3 / 1 (because profit ratio is 3 : 1). Step 8: Cross multiply to get 24 * 1 = 3 * t, so 24 = 3t. Step 9: Solve for t: t = 24 / 3 = 8 months.

Verification / Alternative check:
Check with capital time products. A has 85000 * 12 = 1020000 capital time units. B has 42500 * 8 = 340000 capital time units. The ratio 1020000 : 340000 simplifies by dividing both by 340000 to 3 : 1, which matches the given profit ratio. This confirms that B must have invested for exactly 8 months.

Why Other Options Are Wrong:
If B had invested for 6 months, his capital time units would be 42500 * 6 = 255000 and the ratio 1020000 : 255000 would be 4 : 1, not 3 : 1. For 7, 9 or 5 months, similar mismatches occur. Only when t = 8 months does the capital time ratio equal the profit ratio of 3 : 1. Therefore, options B, C, D and E cannot be correct.

Common Pitfalls:
One typical mistake is to assume that the ratio of profits is the same as the ratio of capitals, ignoring the time factor. Another error is mismanaging the ratio equation, such as writing 3 : 1 = 85000 : 42500 and solving without including the months. To avoid such pitfalls, always remember that in partnership problems profits are proportional to capital multiplied by time, and construct the equation accordingly.

Final Answer:
B invests his capital in the business for 8 months, which is option A.