To solve the problem step by step, we will analyze the contributions of each partner and their respective time of investment.
### Step 1: Determine the time of investment for each partner
- A starts the business with ₹40,000 and invests for 12 months.
- B joins after 2 months with ₹60,000 and invests for 10 months.
- C joins later with ₹1,20,000. We need to find out how many months after B joined C invested.
### Step 2: Calculate the total capital contribution in terms of months
- A's contribution in terms of capital months = ₹40,000 * 12 months = ₹4,80,000
- B's contribution in terms of capital months = ₹60,000 * 10 months = ₹6,00,000
- C's contribution will be calculated as ₹1,20,000 * X months, where X is the number of months C invested.
### Step 3: Set up the equation for profit sharing
The total profit at the end of the year is ₹3,75,000, and C receives ₹1,50,000 as his share. The profit share is proportional to the investment.
Let’s denote the total contributions:
- Total contribution = A's contribution + B's contribution + C's contribution
- Total contribution = ₹4,80,000 + ₹6,00,000 + ₹1,20,000 * X
### Step 4: Set up the proportion based on profit share
C's share of the profit is ₹1,50,000, which corresponds to his proportion of the total investment:
- C's proportion = (₹1,20,000 * X) / (₹4,80,000 + ₹6,00,000 + ₹1,20,000 * X)
### Step 5: Calculate the total profit proportions
From the total profit of ₹3,75,000, we can express C's share as:
- C's share = (C's proportion of total investment) * Total profit
- ₹1,50,000 = (₹1,20,000 * X) / (₹4,80,000 + ₹6,00,000 + ₹1,20,000 * X) * ₹3,75,000
### Step 6: Solve for X
We can simplify the equation:
- Rearranging gives us:
- ₹1,50,000 * (₹4,80,000 + ₹6,00,000 + ₹1,20,000 * X) = ₹1,20,000 * X * ₹3,75,000
### Step 7: Substitute and simplify
- Substitute the known values and simplify to find X.
### Step 8: Calculate the value of X
After solving the equation, we find that X = 6 months.
### Step 9: Determine when C joined
B joined 2 months after A, which means B joined in March. If C joined 6 months after B, then:
- B joins in March, and C joins in July.
### Step 10: Calculate the duration from B's joining to C's joining
From March to July is 4 months.
### Final Answer
C joined 4 months after B joined the business.
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