The population of a village was 20,000 and after 2 years it became 22050. What is the rate of increase per annum?

Options

A. 0.1
B. 0.08
C. 0.05
D. 0.06

Answer

To find the rate of increase per annum for the population of the village, we can use the compound interest formula:

\[ A = P \left(1 + \frac{R}{100}\right)^n \]

Where:
- \( A \) = the amount after time \( n \) (final population)
- \( P \) = the principal amount (initial population)
- \( R \) = rate of interest (rate of increase per annum)
- \( n \) = number of years

### Step-by-step Solution:

1. **Identify the values:**
- Initial population (\( P \)) = 20,000
- Final population (\( A \)) = 22,050
- Time period (\( n \)) = 2 years

2. **Substitute the values into the formula:**
\[ 22,050 = 20,000 \left(1 + \frac{R}{100}\right)^2 \]

3. **Divide both sides by 20,000 to isolate the compound factor:**
\[ \frac{22,050}{20,000} = \left(1 + \frac{R}{100}\right)^2 \]
\[ 1.1025 = \left(1 + \frac{R}{100}\right)^2 \]

4. **Take the square root of both sides:**
\[ \sqrt{1.1025} = 1 + \frac{R}{100} \]
\[ 1.05 = 1 + \frac{R}{100} \]

5. **Subtract 1 from both sides:**
\[ 1.05 - 1 = \frac{R}{100} \]
\[ 0.05 = \frac{R}{100} \]

6. **Multiply both sides by 100 to find \( R \):**
\[ R = 0.05 \times 100 \]
\[ R = 5 \]

### Conclusion:
The rate of increase per annum is **5%**.