The population of a town is $8500$. It increases by $20\%$ in the first year and by another $25\%$ in the second year. What would be the population of the town after $2$ years?

### Concept & Strategy When a quantity undergoes consecutive percentage changes at different rates, we apply the successive multiplication method. We multiply the base amount by the fractional equivalent of each year's specific growth factor. $$ \text{Final Amount} = \text{Initial} \times \left(1 + \frac{R_1}{100}\right) \times \left(1 + \frac{R_2}{100}\right) $$ ### Step-by-Step Solution 1. **Given Data:** * Initial Population ($P$) = $8500$ * First Year Increase ($R_1$) = $20\%$ * Second Year Increase ($R_2$) = $25\%$ 2. **Determine Multipliers:** * First year multiplier = $100\% + 20\% = 120\% = \frac{120}{100} = \frac{6}{5}$ * Second year multiplier = $100\% + 25\% = 125\% = \frac{125}{100} = \frac{5}{4}$ 3. **Apply Successive Multiplication:** * $$ \text{Final Population} = 8500 \times \left( \frac{6}{5} \right) \times \left( \frac{5}{4} \right) $$ 4. **Simplify and Solve:** * Notice the $5$ in the numerator and denominator cancel out beautifully: * $$ \text{Final Population} = 8500 \times \left( \frac{6}{4} \right) $$ * Simplify $\frac{6}{4}$ to $1.5$ (or $\frac{3}{2}$): * $$ = 8500 \times 1.5 $$ * $$ = 12750 $$ ### Exam Strategy & Shortcut Instead of chaining multipliers, calculate the single effective percentage increase for the two years using the formula: $X + Y + \frac{X \times Y}{100}$. $$ \text{Net } \% \text{ Increase} = 20 + 25 + \frac{20 \times 25}{100} $$ $$ = 45 + \frac{500}{100} = 45 + 5 = 50\% $$ The population grows by exactly $50\%$ overall. A $50\%$ increase means the final population is $1.5$ times the original. $$ 8500 \times 1.5 = 12750 $$. This mental math calculation bypasses all heavy arithmetic. ### Common Pitfall A very common mistake is simple addition of the percentages: $20\% + 25\% = 45\%$. A student might calculate $145\%$ of $8500$ and get $12325$. This ignores the fundamental nature of compounding—the $25\%$ growth in the second year acts upon the already enlarged population from the first year, creating extra growth. ### Final Answer **Therefore, the correct answer is $12750$.**