The population of a town 2 years ago was 62,500. Due to migration to big cities, it decreases every year at the rate of 4%. The present population of the town is

### Concept & Formula When a value undergoes a constant percentage decrease successively, it behaves like compound interest in reverse (compound depreciation). The formula for successive decrease is: $$ P_{final} = P_{initial} \times \left(1 - \frac{R}{100}\right)^n $$ To calculate faster, we convert the percentage decrease directly into a fraction multiplier. ### Step-by-Step Solution * **Given:** Initial population = $62,500$. Rate of decrease = $4\%$ per year. Time = $2$ years. * **Fractional Multiplier:** A decrease of $4\%$ means the population becomes $96\%$ of its previous value each year. Fraction equivalent: $4\% = \frac{4}{100} = \frac{1}{25}$. The multiplying factor for a decrease is $1 - \frac{1}{25} = \frac{24}{25}$. * **Apply the multiplier twice (for 2 years):** $$ P_{present} = 62500 \times \frac{24}{25} \times \frac{24}{25} $$ * **Simplify:** $$ P_{present} = 62500 \times \frac{576}{625} $$ * Notice that $62500 \div 625 = 100$. * **Final Calculation:** $$ P_{present} = 100 \times 576 = 57600 $$ ### Exam Strategy & Shortcut **Divisibility Rule of 9:** The multiplying factor is $\frac{24}{25}$. Since $24 = 8 \times 3$, multiplying by $24$ twice means we are multiplying by $3 \times 3 = 9$. Therefore, the final answer must be a multiple of 9. Check the sum of digits of the options: (a) $5+6+7+0+0 = 18$ (Possible) (b) $5+7+6+0+0 = 18$ (Possible) (c) $5+8+8+0+0 = 21$ (Not a multiple of 9) (d) $6+0+0+0+0 = 6$ (Not a multiple of 9) Since two options are multiples of 9, the fractional method above is the most reliable and takes barely 10 seconds to execute because $625$ cancels out perfectly. ### Common Pitfall A frequent mistake is calculating $4\%$ of $62,500$ (which is $2,500$), doubling it for two years ($5,000$), and subtracting it to get $57,500$. This calculates simple decrease instead of successive (compound) decrease. ### Final Answer **Therefore, the correct answer is 57,600.**