Question

The population of a town is increasing at 5% per annum. What will be the population of the town after 2 years, if the present population is 16000?

• A. 17960
• B. 16560
• C. 16960
• D. 17640

Hint:

1. For small values of \( n \) (like 2 years), doing successive simple interest calculations (year-by-year) is often faster and less prone to decimal multiplication errors than squaring decimals.
2. \( (1.05)^2 \) is easily remembered as \( 1.1025 \) using the algebraic square expansion \( (1 + x)^2 \approx 1 + 2x + x^2 \).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
1. This problem involves calculating the future population of a town under constant annual growth.
2. The population increases by a constant rate of 5% every year, meaning the growth is compounded annually.
3. We are given the initial population, the growth rate, and the duration (2 years).
4. We need to compute the final population after this period.

Step 2: Key Formula or Approach:
1. The compound growth formula for population is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
where:
- \( A \) is the population after \( n \) years.
- \( P \) is the initial (present) population.
- \( R \) is the annual growth rate (percentage).
- \( n \) is the number of years.

Step 3: Detailed Explanation:
1. Firstly, let us identify the given parameters from the problem:
- Present population, \( P = 16000 \)
- Annual growth rate, \( R = 5% \)
- Number of years, \( n = 2 \)
2. Secondly, substitute these values into the compound growth formula:
\[ A = 16000 \left(1 + \frac{5}{100}\right)^2 \]
3. Simplify the fraction inside the parentheses:
\[ 1 + \frac{5}{100} = 1 + 0.05 = 1.05 \]
4. Now, raise this value to the power of 2:
\[ (1.05)^2 = 1.1025 \]
5. Multiply this result by the initial population:
\[ A = 16000 \times 1.1025 \]
6. Performing the multiplication step-by-step:
\[ A = 16 \times 1102.5 = 17640 \]
7. Alternatively, we can calculate the growth year-by-year:
- Year 1 Increase: \( 5% \text{ of } 16000 = \frac{5}{100} \times 16000 = 800 \)
- Population after Year 1: \( 16000 + 800 = 16800 \)
- Year 2 Increase: \( 5% \text{ of } 16800 = \frac{5}{100} \times 16800 = 840 \)
- Population after Year 2: \( 16800 + 840 = 17640 \)

Step 4: Final Answer:
1. The population of the town after 2 years will be exactly 17640.
2. This corresponds directly to Option (D).