A sum amounts to Rs. 9680 in 2 years and to Rs.10648 in 3 years compounded annually. Find the principal and the rate of interest per annum.

Options

A. `12%: Rs. 7500 `
B. `10% , Rs. 8000 `
C. `11% : Rs. 11000 `
D. None of these

Answer

To solve the problem step by step, we need to find the principal amount and the rate of interest per annum given that the amount after 2 years is Rs. 9680 and after 3 years is Rs. 10648.

### Step 1: Find the Increment in Amount
The amount after 2 years is Rs. 9680, and the amount after 3 years is Rs. 10648.

To find the increment in the amount over one year (from year 2 to year 3):
\[ \text{Increment} = \text{Amount after 3 years} - \text{Amount after 2 years} = 10648 - 9680 = 96 \]

### Step 2: Determine the Rate of Interest
The increment from year 2 to year 3 is the interest earned in the third year. Since the amount after 2 years is Rs. 9680, we can use the formula for interest to find the rate:
\[ \text{Interest} = \text{Principal} \times \frac{\text{Rate}}{100} \]
Let \( P \) be the principal and \( r \) be the rate of interest. The interest for the third year is Rs. 9680 multiplied by \( \frac{r}{100} \):
\[ 9680 \times \frac{r}{100} = 968 \]
From this, we can solve for \( r \):
\[ r = \frac{968 \times 100}{9680} = 10\% \]

### Step 3: Find the Principal Amount
Now that we know the rate of interest is 10%, we can find the principal amount using the formula for compound interest. The amount after 2 years can be expressed as:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
For 2 years:
\[ 9680 = P \left(1 + \frac{10}{100}\right)^2 \]
This simplifies to:
\[ 9680 = P \left(1.1\right)^2 \]
Calculating \( (1.1)^2 \):
\[ (1.1)^2 = 1.21 \]
Now substituting back:
\[ 9680 = P \times 1.21 \]
Solving for \( P \):
\[ P = \frac{9680}{1.21} \approx 8000 \]

### Final Results
- Principal \( P \) = Rs. 8000
- Rate of Interest \( r \) = 10%