The population of a village is 1,00,000. If the population in creases at `10%` per annum, then the population at the start of the third year is

Options

A. 1,33,100
B. 1,21,000
C. 1,20,000
D. 1,10,000

Answer

To find the population at the start of the third year given that the population of a village is 100,000 and increases at a rate of 10% per annum, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Population and Growth Rate:**
- Initial population (P) = 100,000
- Growth rate (r) = 10% per annum

2. **Convert the Growth Rate to Decimal:**
- Convert the percentage to a decimal for calculations:
\[ r = \frac{10}{100} = 0.10 \]

3. **Calculate the Population After 2 Years:**
- The formula for population growth is given by:
\[ P(t) = P \times (1 + r)^t \]
- Here, \(t\) is the number of years. For the start of the third year, we need to calculate the population after 2 years (\(t = 2\)):
\[ P(2) = 100,000 \times (1 + 0.10)^2 \]

4. **Calculate \( (1 + 0.10)^2 \):**
- First, calculate \(1 + 0.10\):
\[ 1 + 0.10 = 1.10 \]
- Now square it:
\[ (1.10)^2 = 1.21 \]

5. **Multiply by the Initial Population:**
- Now, substitute back into the population formula:
\[ P(2) = 100,000 \times 1.21 \]

6. **Calculate the Final Population:**
- Perform the multiplication:
\[ P(2) = 121,000 \]

### Conclusion:
The population at the start of the third year is **121,000**.