A district has 64,000 inhabitants. If the population increases at the rate of 2.5% per annum, what will be the number of inhabitants at the end of 3 years?

Introduction:
This is another compound growth problem related to population. The annual increase is given as a constant percentage rate over multiple years. The goal is to correctly apply the compound interest type formula for three successive years to determine the future population.

Given Data / Assumptions:
Initial population of the district = 64000 inhabitants.
Annual growth rate = 2.5% per year, assumed constant.
Time period = 3 years.
We must find the population after 3 years of growth.

Concept / Approach:
For steady percentage growth over multiple periods, the new population is given by P * (1 + r)^n, where P is the initial population, r is the growth rate expressed as a decimal, and n is the number of years. Here, r = 2.5 / 100 = 0.025 and n = 3. We then multiply 64000 by (1.025)^3 to get the population after three years.

Step-by-Step Solution:
Initial population P0 = 64000. Annual growth rate r = 2.5% = 0.025. Number of years n = 3. Compound growth formula: Pn = P0 * (1 + r)^n. Compute (1 + r) = 1 + 0.025 = 1.025. Now compute (1.025)^3. (1.025)^2 ≈ 1.050625. Multiply again by 1.025: 1.050625 * 1.025 ≈ 1.076953125. So growth factor over three years is approximately 1.076953125. Population after 3 years P3 = 64000 * 1.076953125 ≈ 68920.9999. Rounding to the nearest whole number gives 68921.
Verification / Alternative check:
We can compute year by year for an alternative check. After first year: 64000 * 1.025 = 65600. After second year: 65600 * 1.025 = 67240. After third year: 67240 * 1.025 ≈ 68921. This stepwise calculation agrees with the direct compound formula and confirms that 68921 is the correct population after 3 years.

Why Other Options Are Wrong:
65,380 is only a small increase and would correspond to a much lower annual growth rate. 70,987 and 72,345 are too high for a 2.5% growth over three years, while 75,000 is even higher and unrealistic for such a modest rate. Only 68,921 matches the value obtained by proper application of the compound growth formula.

Common Pitfalls:
A typical pitfall is to treat the growth as simple interest and add 2.5% three times to get 7.5% and then compute 64000 * 1.075. This ignores compounding and gives a slightly different answer. Another mistake is rounding excessively at intermediate steps, which can cause deviation from the accurate answer. Keeping at least four decimal places in intermediate calculations is safer for such percentage problems.

Final Answer:
The number of inhabitants at the end of 3 years will be approximately 68,921.