Detailed Solution for Initial Town Population Calculation

Understanding the Population Change Scenario

This problem involves calculating the original population of a town based on percentage changes over three consecutive years. We are told the population experienced specific changes linked to the Corona virus waves.

• In the first year, the population decreased by 5%.
• In the second year, the population again decreased by 5%.
• In the third year, the population increased by 5%.
• We know the final population at the end of the third year is 9,47,625.
• Our goal is to find the population at the beginning of the first year.

We need to reverse the percentage changes to find the initial population. Let's denote the initial population as \(P_0\).

Step-by-Step Calculation of Initial Population

We will calculate the population year by year, setting up an equation based on the final population.

• Step 1: Calculate the population after the first year.

  The population decreased by 5%. This means the remaining population is \(100\% - 5\% = 95\%\) of the population at the start of the year. Population after Year 1, \(P_1 = P_0 \times (1 - \frac{5}{100})\) \(P_1 = P_0 \times (1 - 0.05)\) \(P_1 = P_0 \times 0.95\)

• Step 2: Calculate the population after the second year.

  The population again decreased by 5% from the end of Year 1 population (\(P_1\)). Population after Year 2, \(P_2 = P_1 \times (1 - \frac{5}{100})\) \(P_2 = P_1 \times 0.95\) Substituting the value of \(P_1\): \(P_2 = (P_0 \times 0.95) \times 0.95\) \(P_2 = P_0 \times (0.95)^2\) \(P_2 = P_0 \times 0.9025\)

• Step 3: Calculate the population after the third year.

  The population increased by 5% from the end of Year 2 population (\(P_2\)). Population after Year 3, \(P_3 = P_2 \times (1 + \frac{5}{100})\) \(P_3 = P_2 \times (1 + 0.05)\) \(P_3 = P_2 \times 1.05\) Substituting the value of \(P_2\): \(P_3 = (P_0 \times 0.9025) \times 1.05\) \(P_3 = P_0 \times 0.947625\)

• Step 4: Set up the equation using the final population.

  We are given that the population at the end of the third year (\(P_3\)) is 9,47,625. So, \(P_0 \times 0.947625 = 9,47,625\)

• Step 5: Solve for the initial population (\(P_0\)).

  To find the initial population, we need to isolate \(P_0\). \(P_0 = \frac{9,47,625}{0.947625}\) To simplify the division, we can write 0.947625 as a fraction: \(0.947625 = \frac{947625}{1000000}\) \(P_0 = \frac{9,47,625}{\frac{947625}{1000000}}\) \(P_0 = 9,47,625 \times \frac{1000000}{947625}\) \(P_0 = 1,000,000\)

Conclusion on Initial Population

After performing the calculations step-by-step, we found that the population at the beginning of the first year was 1,000,000. This is the original number of people in the town before the described changes occurred.