A starts a business with ? 2000 and B joins him after 3 months with ? 8000. Find the ratio of their profits at the end of the year.

Answer

To solve the problem, we need to calculate the profit-sharing ratio between A and B based on their investments and the time they invested in the business.

### Step 1: Calculate the investment of A and B in terms of "months".

- A invests ₹2000 for the entire year (12 months).
- B invests ₹8000 but joins after 3 months, so he invests for only 9 months.

### Step 2: Calculate the total investment in terms of "money-months".

- For A:
\[ \text{A's investment in money-months} = 2000 \times 12 = 24000 \text{ money-months} \]

- For B:
\[ \text{B's investment in money-months} = 8000 \times 9 = 72000 \text{ money-months} \]

### Step 3: Calculate the total investment in money-months.

- Total investment in money-months:
\[ \text{Total} = \text{A's investment} + \text{B's investment} = 24000 + 72000 = 96000 \text{ money-months} \]

### Step 4: Find the ratio of their investments.

- The ratio of A's investment to B's investment is:
\[ \text{Ratio} = \frac{\text{A's investment}}{\text{B's investment}} = \frac{24000}{72000} = \frac{1}{3} \]

### Step 5: Conclusion.

Thus, the ratio of their profits at the end of the year is:
\[ \text{Ratio of profits} = 1:3 \]