What will be the amount of the end of 2 years, if the interest is compounded yearly.
I. The simple interest on the same sum for a period of 2 year is Rs.400 at the same rate of `5%` per annum.
II. The difference between the simple interest and the compound interest for 2 years at the rate of `5%` per annum is Rs.100.

Options

A. if the data in statement I alone are sufficient to answer the questin, while the data in statement II alone are not sufficient to answer the question.
B. if the data in statement II alon are sufficient to answer the question, while the data in statement I alone are not sufficient to answer the question.
C. if the data either in statement I alone or in statement II alone in sufficient to answer the question.
D. if the data given in both the statement I and II together are not suffesient to answer the questin, and

Explanation

To solve the problem, we need to determine the amount at the end of 2 years when the interest is compounded yearly. We will analyze the two statements provided to see if they are sufficient to answer the question.

### Step 1: Analyze Statement I
Statement I states that the simple interest on the same sum for a period of 2 years is Rs. 400 at the same rate of 5% per annum.

- We know the formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

- Here, \( SI = 400 \), \( R = 5\% \), and \( T = 2 \).

- Substituting the values into the formula:
\[ 400 = \frac{P \times 5 \times 2}{100} \]
\[ 400 = \frac{10P}{100} \]
\[ 400 = 0.1P \]
\[ P = \frac{400}{0.1} = 4000 \]

Now we have found the principal amount \( P = 4000 \).

### Step 2: Calculate Compound Interest
Next, we calculate the amount at the end of 2 years using the compound interest formula:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where \( A \) is the amount, \( P = 4000 \), \( R = 5\% \), and \( T = 2 \).

Substituting the values:
\[ A = 4000 \left(1 + \frac{5}{100}\right)^2 \]
\[ A = 4000 \left(1 + 0.05\right)^2 \]
\[ A = 4000 \left(1.05\right)^2 \]
\[ A = 4000 \times 1.1025 = 4410 \]

### Conclusion from Statement I
From Statement I alone, we can determine that the amount at the end of 2 years is Rs. 4410.

### Step 3: Analyze Statement II
Statement II states that the difference between the simple interest and the compound interest for 2 years at the rate of 5% per annum is Rs. 100.

- We know from the previous calculations that:
- Simple Interest (SI) for 2 years is \( 400 \).
- Let Compound Interest (CI) for 2 years be \( CI \).

The difference is given by:
\[ CI - SI = 100 \]
Substituting \( SI = 400 \):
\[ CI - 400 = 100 \]
\[ CI = 500 \]

### Step 4: Calculate Principal from Statement II
Using the compound interest formula:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting \( CI = 500 \), \( R = 5\% \), and \( T = 2 \):
\[ 500 = P \left(1.05^2 - 1\right) \]
Calculating \( 1.05^2 - 1 \):
\[ 1.05^2 = 1.1025 \implies 1.1025 - 1 = 0.1025 \]
So,
\[ 500 = P \times 0.1025 \]
\[ P = \frac{500}{0.1025} \approx 4878.05 \]

### Conclusion from Statement II
From Statement II alone, we can also find the principal amount, but we need to calculate the amount using this principal with the compound interest formula:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting \( P \approx 4878.05 \):
\[ A \approx 4878.05 \times 1.1025 \approx 5378.05 \]

### Final Conclusion
Both statements are sufficient to determine the amount at the end of 2 years, but they yield different principals. However, since the question only asks for the amount at the end of 2 years, we can conclude that both statements provide sufficient information to answer the question.