A district has 64000 inhabitants. If the population increases at the rate of $2 \frac{1}{2}\%$ per annum, then the number of inhabitants at the end of 3 years will be

### Concept & Formula Population growth over multiple years is calculated using the compound interest formula, because the growth rate applies to an expanding base population each year. $$ A = P \left(1 + \frac{R}{100}\right)^T $$ Where $P$ is the initial population, $R$ is the annual growth rate, and $T$ is the number of years. ### Step-by-Step Solution 1. **Given Data:** * Initial population ($P$) = $64000$ * Rate ($R$) = $2 \frac{1}{2}\% = 2.5\% = \frac{5}{2}\%$ * Time ($T$) = $3$ years 2. **Find the Fractional Multiplier:** * First, express the rate as a fraction of $100$: $\frac{2.5}{100} = \frac{1}{40}$. * The multiplier for each year is $(1 + \frac{1}{40}) = \frac{41}{40}$. 3. **Apply the Compound Growth Formula:** * $$ \text{Population after 3 years} = 64000 \times \left( \frac{41}{40} \right)^3 $$ * $$ = 64000 \times \left( \frac{41}{40} \right) \times \left( \frac{41}{40} \right) \times \left( \frac{41}{40} \right) $$ * $$ = 64000 \times \frac{41 \times 41 \times 41}{40 \times 40 \times 40} $$ * $$ = 64000 \times \frac{68921}{64000} $$ 4. **Simplify:** * The $64000$ in the numerator and denominator perfectly cancel each other out. * $$ = 68921 $$ ### Exam Strategy & Shortcut Memorize standard fractional equivalents and their cubes. A $2.5\%$ rate is exactly $\frac{1}{40}$. Over 3 years, the denominator will be $40^3 = 64000$. Since the initial population is exactly $64000$, it will completely cancel out the denominator. Thus, the final answer is simply the numerator cubed: $41^3$. If you know $40^3 = 64000$, $41^3$ must end in a $1$ (since $1 \times 1 \times 1 = 1$). Looking at the options, only $68911$ and $68921$ end in $1$. $41^3$ calculation reveals $68921$. ### Common Pitfall Students often waste immense amounts of time calculating the decimal values sequentially (e.g., finding $2.5\%$ of $64000$, adding it, then finding $2.5\%$ of the new number, etc.). Always convert unusual percentage rates like $2.5\%$, $12.5\%$, or $16.66\%$ into simple fractions before applying the power. ### Final Answer **Therefore, the correct answer is $68921$.**