A sum amount to Rs. 9680 in 2 yr and to Rs. 10648 in 3 yr compounded annually, Then, the sum and rate of interest respectively are

Options

A. Rs. 8000, 3%
B. Rs. 8000, 10%
C. Rs. 9000, 10%
D. Rs. 1064.8, 10%

Answer

To solve the problem, we need to determine the principal amount and the rate of interest based on the given amounts after 2 years and 3 years.

### Step-by-Step Solution:

1. **Identify the Given Amounts:**
- Amount after 2 years (A2) = Rs. 9680
- Amount after 3 years (A3) = Rs. 10648

2. **Calculate the Increase in Amount Over 1 Year:**
- Increase in amount from year 2 to year 3 = A3 - A2
- Increase = 10648 - 9680 = Rs. 96

3. **Determine the Rate of Interest:**
- The increase in amount over 1 year is Rs. 96.
- The amount after 2 years is Rs. 9680.
- Rate of interest (R) can be calculated using the formula:
\[ R = \left( \frac{\text{Increase}}{\text{Amount after 2 years}} \right) \times 100 \]
- Substituting the values:
\[ R = \left( \frac{96}{9680} \right) \times 100 = 0.99\% \]
- Simplifying further, we find:
\[ R \approx 1\% \]

4. **Use the Amount Formula to Find the Principal:**
- The formula for the amount in compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
- For 2 years:
\[ 9680 = P \left(1 + \frac{1}{100}\right)^2 \]
- This simplifies to:
\[ 9680 = P \left(1.01\right)^2 \]
- Calculating \( (1.01)^2 \):
\[ (1.01)^2 = 1.0201 \]
- So, we have:
\[ 9680 = P \times 1.0201 \]
- Rearranging to find P:
\[ P = \frac{9680}{1.0201} \approx 9500 \]

5. **Final Results:**
- Principal (P) = Rs. 9500
- Rate of Interest (R) = 1%

### Conclusion:
The principal amount is Rs. 9500 and the rate of interest is 1%.