At what rate per cent per annum will a sum of Rs. 1,000 amount to Rs. 1,102.50 in 2 years at compound interest?

Options

A. 5
B. 5.5
C. 6
D. 6.5

Answer

To solve the problem of finding the rate of interest per annum at which a sum of Rs. 1,000 amounts to Rs. 1,102.50 in 2 years at compound interest, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Principal (P) = Rs. 1,000
- Amount (A) = Rs. 1,102.50
- Time (T) = 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
- \( P \) = Principal
- \( r \) = Rate of interest per annum
- \( n \) = Number of years

3. **Substitute the Known Values into the Formula:**
\[ 1102.50 = 1000 \left(1 + \frac{r}{100}\right)^2 \]

4. **Divide Both Sides by 1000:**
\[ \frac{1102.50}{1000} = \left(1 + \frac{r}{100}\right)^2 \]
This simplifies to:
\[ 1.1025 = \left(1 + \frac{r}{100}\right)^2 \]

5. **Take the Square Root of Both Sides:**
\[ \sqrt{1.1025} = 1 + \frac{r}{100} \]
Calculating the square root:
\[ 1.05 = 1 + \frac{r}{100} \]

6. **Isolate \( \frac{r}{100} \):**
\[ 1.05 - 1 = \frac{r}{100} \]
This gives:
\[ 0.05 = \frac{r}{100} \]

7. **Multiply Both Sides by 100 to Solve for \( r \):**
\[ r = 0.05 \times 100 \]
Therefore:
\[ r = 5 \]

### Final Answer:
The rate of interest per annum is **5%**.