X and Y commenced a business with investments of Rs. 20,000 and Rs. 25,000 respectively. After several months, Z invested Rs. 30,000 into the business. If the total profit for the year is Rs. 50,000 and Z's portion of the profit is Rs. 14,000, how many months after the start did Z become a partner?

Explanation

The investments of X, Y, and Z are Rs. 20,000, Rs. 25,000, and Rs. 30,000 respectively. Let Z's investment duration be x months. The ratio of their annual investments is (20,000 × 12) : (25,000 × 12) : (30,000 × x), which simplifies to 240 : 300 : 30x or 8 : 10 : x. Given Z's profit share is Rs. 14,000 out of Rs. 50,000 total, we set up the equation (x / (18 + x)) × 50,000 = 14,000. Solving yields x = 7 months. However, since the total year is 12 months, Z joined 5 months after the business started, which is not listed among the options.