Each of the following questions cousists of a question followed by information in three statements. You have to study the question and the statements and decide which of the statements is/are sufficient to answer the question.
What is the total compound interest earned at the end of 3 years?
I Simple interest earned on that amount at the same rate of interest `2 1/2` years is ₹3000.
II The rate of interest is `5%` per annum.
III The principal sum is ₹24000.

Options

A. Only I
B. Only II and III
C. Only I and II
D. Any two out of the three

Answer

To solve the problem of finding the total compound interest earned at the end of 3 years, we need to analyze the three statements provided and determine which of them are sufficient to answer the question.

### Step-by-Step Solution:

1. **Understanding Compound Interest**:
- The formula for compound interest (CI) is:
\[ \text{CI} = A - P \]
where \( A \) is the total amount after interest and \( P \) is the principal amount.
- The formula for calculating the amount \( A \) is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
where \( r \) is the rate of interest and \( t \) is the time in years.

2. **Analyzing Statement I**:
- Statement I states that the simple interest earned on the amount at the same rate for 2.5 years is ₹3000.
- The formula for simple interest (SI) is:
\[ \text{SI} = \frac{P \times r \times t}{100} \]
- From this, we can set up the equation:
\[ 3000 = \frac{P \times r \times 2.5}{100} \]
- This gives us a relationship between \( P \) and \( r \), but we cannot determine either variable independently.

3. **Analyzing Statement II**:
- Statement II provides the rate of interest as \( 5\% \) per annum.
- This gives us a specific value for \( r \).

4. **Combining Statements I and II**:
- With the rate \( r = 5\% \) from Statement II, we can substitute this into the equation from Statement I:
\[ 3000 = \frac{P \times 5 \times 2.5}{100} \]
- Solving for \( P \):
\[ 3000 = \frac{12.5P}{100} \implies P = \frac{3000 \times 100}{12.5} = 24000 \]
- Now we have both \( P \) and \( r \).

5. **Calculating Amount and Compound Interest**:
- Now we can calculate the amount \( A \) after 3 years using the values of \( P \) and \( r \):
\[ A = 24000 \left(1 + \frac{5}{100}\right)^3 = 24000 \left(1.05\right)^3 \]
- Calculating \( (1.05)^3 \):
\[ (1.05)^3 \approx 1.157625 \]
- Therefore:
\[ A \approx 24000 \times 1.157625 \approx 27783 \]
- Now, calculating the compound interest:
\[ \text{CI} = A - P = 27783 - 24000 = 3783 \]

6. **Analyzing Statement III**:
- Statement III states that the principal sum is ₹24000.
- This confirms the value of \( P \) but is not necessary since we already derived it from the first two statements.

### Conclusion:
- We can conclude that any two of the three statements are sufficient to answer the question. Thus, the answer is that any combination of two statements can be used to find the total compound interest.