Suppose for the principal P, rate R% and time T, the simple interest is S and compound interest is C. Consider the possibilities.
(i) `CgtS` (ii) C=S (iii) `CltS` Then

Options

A. only (i) is correct.
B. either (i) or (ii) is correct.
C. either (ii) or (iii) is correct.
D. only (iii) is correct.

Answer

To solve the problem, we need to analyze the relationship between Simple Interest (S) and Compound Interest (C) based on the given scenarios.

### Step-by-Step Solution:

1. **Understanding Simple Interest (S) and Compound Interest (C)**:
- Simple Interest (S) is calculated using the formula:
\[ S = \frac{P \times R \times T}{100} \]
- Compound Interest (C) is calculated using the formula:
\[ C = P \left(1 + \frac{R}{100}\right)^{T} - P \]

2. **Case (i): C > S**:
- This case occurs when the interest is compounded more frequently than annually, such as quarterly or half-yearly.
- Therefore, if the interest is compounded quarterly (4 times a year) or half-yearly (2 times a year), the compound interest will be greater than the simple interest.

3. **Case (ii): C = S**:
- This case occurs when the interest is compounded annually.
- In this scenario, the amount of interest earned through simple interest and compound interest will be the same because the compounding effect does not take place more than once a year.

4. **Case (iii): C < S**:
- This case is not possible. Simple interest cannot be greater than compound interest when the interest is compounded, as compound interest takes into account the interest on interest, which will always yield a higher amount than simple interest over time.

5. **Conclusion**:
- From the analysis, we can conclude:
- Case (i) is possible when interest is compounded quarterly or half-yearly (C > S).
- Case (ii) is possible when interest is compounded annually (C = S).
- Case (iii) is not possible (C < S).

### Final Answer:
The only valid cases are (i) and (ii). Therefore, the correct option is that C can be greater than S or equal to S, but C cannot be less than S.