'A' invests Rs.10,000 for 3 years at a certain rate of interest. At the end of the second year, it amounts ta Rs. 11,461. Calculate the rate of interest per annum if interest is compounded annually .

Options

A. 7.9
B. 8.5
C. 8.7
D. `8.0`

Answer

To solve the problem, we need to find the rate of interest per annum when 'A' invests Rs. 10,000 for 3 years and at the end of the second year, it amounts to Rs. 11,461 with interest compounded annually.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Principal (P) = Rs. 10,000
- Amount after 2 years (A) = Rs. 11,461
- Time (N) = 2 years

2. **Use the Compound Interest Formula:**
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^N \]
where:
- A = Amount after N years
- P = Principal amount
- R = Rate of interest per annum
- N = Number of years

3. **Substitute the Known Values into the Formula:**
\[ 11,461 = 10,000 \left(1 + \frac{R}{100}\right)^2 \]

4. **Divide Both Sides by 10,000:**
\[ \frac{11,461}{10,000} = \left(1 + \frac{R}{100}\right)^2 \]
\[ 1.1461 = \left(1 + \frac{R}{100}\right)^2 \]

5. **Take the Square Root of Both Sides:**
\[ \sqrt{1.1461} = 1 + \frac{R}{100} \]
\[ 1.070 = 1 + \frac{R}{100} \]

6. **Isolate R:**
\[ \frac{R}{100} = 1.070 - 1 \]
\[ \frac{R}{100} = 0.070 \]

7. **Multiply by 100 to Find R:**
\[ R = 0.070 \times 100 = 7 \]

8. **Conclusion:**
The rate of interest per annum is **7%**.