A and B are partners in a business. A contributes one fourth of the total capital for 15 months, and B receives two thirds of the total profit. For how many months was B's capital used in the business?

Introduction / Context:
This problem illustrates the idea of capital time in a partnership, where partners invest different amounts of money for different durations. The share of profit is directly proportional to the product of capital and time. Here, we are given that A contributes a known fraction of the total capital for a fixed time and that B receives a specific fraction of the profit. From this, we must deduce how long B's capital remained invested.

Given Data / Assumptions:
- A contributes 1/4 of the total capital for 15 months.
- B contributes the remaining 3/4 of the total capital for some unknown number of months t.
- B receives 2/3 of the total profit, so A receives 1/3 of the profit.
- Profit shares are proportional to capital time, that is, capital multiplied by duration.

Concept / Approach:
Let the total capital be C. Then A's capital is C / 4 and B's capital is 3C / 4. The capital time of A is (C / 4) multiplied by 15 months. The capital time of B is (3C / 4) multiplied by t months. The ratio of their profits is equal to the ratio of these capital times. Since B receives two thirds of the profit and A receives one third, the profit ratio A : B is 1 : 2. Equating the capital time ratio with 1 : 2 gives us an equation in t, which we can solve.

Step-by-Step Solution:
Step 1: Let total capital be C. Then A's capital = C / 4 and B's capital = 3C / 4.Step 2: A uses his capital for 15 months, so A's capital time = (C / 4) * 15 = 15C / 4.Step 3: Let B's capital be used for t months. Then B's capital time = (3C / 4) * t.Step 4: Profit ratio A : B equals capital time ratio, so (15C / 4) : (3C / 4) * t = 1 : 2.Step 5: Simplify the ratio. The C and 1/4 cancel, leaving 15 : 3t = 1 : 2.Step 6: From 15 : 3t = 1 : 2, we get 15 * 2 = 1 * 3t, so 30 = 3t and t = 10 months.

Verification / Alternative check:
Check using the found value t = 10. A's capital time is 15C / 4. B's capital time is (3C / 4) * 10 = 30C / 4. The ratio A : B is (15C / 4) : (30C / 4) = 15 : 30 = 1 : 2. This matches the given profit ratio, where B receives two thirds and A receives one third. Hence t = 10 months is consistent.

Why Other Options Are Wrong:
If B's capital were used for only 3 or 6 months, his capital time would be too small compared to A's, and he could not receive as much as two thirds of the profit. A duration of 12 months would give B a larger share than two thirds because his capital is three times A's and his time would be close to A's. Only 10 months yields the exact required ratio.

Common Pitfalls:
Some learners mistakenly assume that the profit ratio equals the capital ratio directly, ignoring the time factor. Others misinterpret the statement and assume both partners invested for the same period. Another error is in setting up the ratio equation 15 : 3t = 1 : 2, where cross multiplication may be done incorrectly. Writing out the equation carefully and simplifying step by step prevents such mistakes.

Final Answer:
B's capital was used in the business for 10 months.