The present population of a country estimated to be 10 crores is expected to increase to 13.31 crores during the next three years. The uniform rate of growth is

Options

A. `8%`
B. `10%`
C. `12.7%`
D. `15%`

Answer

To find the uniform rate of growth of the population over three years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial and Final Population:**
- Present population = 10 crores
- Expected population after 3 years = 13.31 crores

2. **Calculate the Total Increase in Population:**
- Increase in population = Final population - Initial population
- Increase = 13.31 crores - 10 crores = 3.31 crores

3. **Calculate the Growth Rate:**
- We need to find the uniform growth rate (r) that results in the population increasing from 10 crores to 13.31 crores over 3 years.
- The formula for population growth can be expressed as:
\[ P = P_0 (1 + r)^n \]
where:
- \( P \) = final population
- \( P_0 \) = initial population
- \( r \) = growth rate (as a decimal)
- \( n \) = number of years

4. **Substituting the Values:**
- Here, \( P = 13.31 \), \( P_0 = 10 \), and \( n = 3 \).
- The equation becomes:
\[ 13.31 = 10 (1 + r)^3 \]

5. **Rearranging the Equation:**
- Divide both sides by 10:
\[ 1.331 = (1 + r)^3 \]

6. **Taking the Cube Root:**
- To find \( 1 + r \), take the cube root of both sides:
\[ 1 + r = \sqrt[3]{1.331} \]

7. **Calculating the Cube Root:**
- Using a calculator, we find:
\[ 1 + r \approx 1.1 \]

8. **Finding the Growth Rate \( r \):**
- Subtract 1 from both sides:
\[ r \approx 1.1 - 1 = 0.1 \]
- Convert \( r \) to a percentage:
\[ r \times 100 \approx 10\% \]

### Final Answer:
The uniform rate of growth is approximately **10% per year**.

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