Question

During one year, the population of town increased by 5%. If the total population is 9975 at the end of the second year, then what was the population size in the beginning of the first year?

• A. 1000
• B. 10000
• C. 8500
• D. 9000
• E.

Hint:

For population growth problems with percentage increases, apply the growth rate iteratively and use the formula for compound growth: \[ \text{Population} = \text{Initial Population} \times (1 + \text{Rate})^n \]

Answer 1

The Correct Option is D

Let the population at the beginning of the first year be \( P \). Step 1: At the end of the first year, the population increases by 5\%, so the new population is: \[ P \times 1.05 \] Step 2: At the end of the second year, the population again increases by 5\%, giving: \[ P \times 1.05 \times 1.05 = P \times 1.05^2 \] Step 3: We know the population at the end of the second year is 9975. Therefore, \[ P \times 1.05^2 = 9975 \] Step 4: Solving for \( P \): \[ P = \frac{9975}{1.05^2} = \frac{9975}{1.1025} \] \[ P = 9000 \] Thus, the population at the beginning of the first year was 9000.