Given the following statements about investments and profits:

**Column I:**
1. Profit of A and B are equal and profit of C is greater than profit of D. Investment period of A and C are equal.
2. Investment of C is double the investment of A and $ \frac{10}{9} $ of the investment of D.
3. Profit of B is Rs. 6000.
4. Total profit is less than or equal to Rs. 15000.

**Column II:**
I) Both A and B either invested for 6 months or 9 months.
II) Investment of B is Rs. 5000.

Which combination of statements from Column I ensures that both statements in Column II are correct?

_Note:_
- Maximum time period of investment is 12 months.
- Profit share of D is Rs. 4500 and he invested for 12 months.
- The time period/investment done by all four is an integer.

Question

Given the following statements about investments and profits:

**Column I:**
1. Profit of A and B are equal and profit of C is greater than profit of D. Investment period of A and C are equal.
2. Investment of C is double the investment of A and $ \frac{10}{9} $ of the investment of D.
3. Profit of B is Rs. 6000.
4. Total profit is less than or equal to Rs. 15000.

**Column II:**
I) Both A and B either invested for 6 months or 9 months.
II) Investment of B is Rs. 5000.

Which combination of statements from Column I ensures that both statements in Column II are correct?

_Note:_
- Maximum time period of investment is 12 months.
- Profit share of D is Rs. 4500 and he invested for 12 months.
- The time period/investment done by all four is an integer.

Accepted Answer

**Solution:**

1. From statement (1), profit of A and B are equal, and profit of C is greater than D. Also, A and C invested for the same time period.

2. Statement (3) gives the profit of B as Rs. 6000.

3. Given that D's profit is Rs. 4500 and he invested for 12 months.

4. Since profit is proportional to investment × time, and A and B have equal profits, their investment × time products are equal.

5. Using statement (1) and (3), and the given data, we can deduce the investment and time periods for A and B.

6. It turns out that for both A and B to have equal profits and B's profit to be Rs. 6000, they must have invested either for 6 or 9 months, satisfying statement I in Column II.

7. Also, the investment of B comes out to Rs. 5000, satisfying statement II in Column II.

8. Statements (1) and (3) together satisfy both statements in Column II.

9. Other combinations either do not satisfy the integer time period condition or the profit/investment constraints.

**Answer:** (e) 1, 3

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