A invested ¥ 76,000 in a business. After few months, B joined him with ¥ 57,000. At the end of the year, the total profit was divided between them in the ratio 2:1. After how many months did B join?

- A invested ¥ 76,000. A's investment was for the full year, i.e., 12 months. - B invested ¥ 57,000. Let B have joined after 'x' months. This means B's investment was for (12 - x) months. - In a partnership, the profit is divided in the ratio of the product of the investment and the time period for which the investment was made. - Ratio of profits (A : B) = (A's Investment A's Time) : (B's Investment B's Time). - The given profit ratio is 2:1. - So, (76,000 12) : (57,000 (12 - x)) = 2 : 1. - We can simplify the investment amounts by dividing by 1000: (76 12) : (57 (12 - x)) = 2 : 1. - The ratio can be written as a fraction: (76 12) / (57 (12 - x)) = 2/1. - We can simplify 76 and 57. Both are divisible by 19. 76 = 19 4, and 57 = 19 3. - (4 19 12) / (3 19 (12 - x)) = 2. - Cancel out 19: (4 12) / (3 (12 - x)) = 2. - Simplify 12 and 3: (4 4) / (12 - x) = 2. - 16 / (12 - x) = 2. - 16 = 2 (12 - x). - 8 = 12 - x. - x = 12 - 8 = 4. - B joined after 4 months.