A begins a business with an investment of Rs. 85,000. After some time, B joins the business with an investment of Rs. 42,500. If at the end of the year the profits are divided in the ratio 3 : 1 between A and B, for how many months was B's capital invested in the business?

Introduction / Context:
This question is about a partnership where one partner starts the business and the other partner joins later. Profit is shared according to capital time, which is the product of capital invested and the duration of investment. Knowing the ratio of final profits and the amounts invested allows us to work backward to find how long the second partner's capital was actually used.

Given Data / Assumptions:
- A invests Rs. 85,000 at the beginning of the year and keeps it invested for the entire 12 months.
- B invests Rs. 42,500 but joins after some time, so his capital is invested for t months only.
- At the end of the year, profits are distributed between A and B in the ratio 3 : 1.
- We assume that there are no additional changes in capital or time beyond what is stated.

Concept / Approach:
Let the profit shares of A and B be proportional to their respective capital times. A's capital time is 85,000 multiplied by 12 months. B's capital time is 42,500 multiplied by t months. The ratio of these capital times must equal the profit ratio 3 : 1. This gives a simple equation in t that we can solve. The key is to cancel common factors and handle the ratio properly.

Step-by-Step Solution:
Step 1: Compute A's capital time: 85,000 * 12.Step 2: Let B's capital time be 42,500 * t, where t is the number of months B's money stays in the business.Step 3: Profit ratio A : B = 3 : 1, so capital time ratio is 85,000 * 12 : 42,500 * t = 3 : 1.Step 4: Divide both terms by 42,500. We get (85,000 / 42,500) * 12 : t = 3 : 1.Step 5: Note that 85,000 / 42,500 = 2. So the ratio becomes 2 * 12 : t = 3 : 1, that is 24 : t = 3 : 1.Step 6: From 24 : t = 3 : 1, we have 24 * 1 = 3 * t, so t = 24 / 3 = 8 months.

Verification / Alternative check:
Using t = 8, B's capital time is 42,500 * 8. A's capital time is 85,000 * 12. Their ratio is (85,000 * 12) : (42,500 * 8). Cancelling 42,500, we get (2 * 12) : (1 * 8) = 24 : 8 = 3 : 1, which matches the given profit ratio. This confirms that B's investment duration of 8 months is correct.

Why Other Options Are Wrong:
If B had invested for only 4 or 5 months, his capital time would be too small and the resulting ratio would not reach 3 : 1. If B had invested for 6 months, the ratio becomes 24 : 12 = 2 : 1, which is different from the required 3 : 1. A duration of 8 months is the only option that produces the correct ratio.

Common Pitfalls:
Some students wrongly set up the ratio as 85,000 : 42,500 = 3 : 1 and ignore time completely, which is not correct when investments are made for different durations. Another frequent error is to assume that B joins exactly at the half year mark without calculation, or to misunderstand that the 3 : 1 ratio already includes time. Always explicitly relate profit ratio to capital multiplied by time to avoid such mistakes.

Final Answer:
B's capital was invested in the business for 8 months.