Partnership

Step 1: Calculate the investment period for each partner. The business runs for a year (12 months).

A's investment period = 12 months.
B left after 5 months, so B's investment period = 5 months.
C joined after 5 months, so C's investment period = 12 - 5 = 7 months.

Step 2: Calculate the capital for each partner.

A's capital = Rs.90,000.
B's capital = Rs.80,000.
C's capital = B's capital - Rs.60,000 = Rs.80,000 - Rs.60,000 = Rs.20,000.

Step 3: Find the ratio of their effective investments (Capital × Time).

A's share ratio = 90,000 × 12 = 1,080,000
B's share ratio = 80,000 × 5 = 400,000
C's share ratio = 20,000 × 7 = 140,000
The profit-sharing ratio is A : B : C = 1,080,000 : 400,000 : 140,000. Simplifying by dividing by 20,000, we get the ratio: 54 : 20 : 7.

Step 4: Calculate the total profit based on C's share. The total parts in the ratio are 54 + 20 + 7 = 81 parts. We are given that C's share of the profit (which is 7 parts) is Rs.42,000. So, 7 parts = Rs.42,000. 1 part = Rs.42,000 / 7 = Rs.6,000. The total profit is 81 parts. Total Profit = 81 × Rs.6,000 = Rs.4,86,000.
Note: The calculated total profit of Rs.4,86,000 does not match any of the given options. There might be an error in the question's data or the options provided.