The population of a town increases $4\%$ annually but is decreased by emigration annually to the extent of $(1/2)\%$. What will be the increase percent in 3 years?

### Concept & Formula This problem is about finding the **Net Compound Growth Rate**. When there are simultaneous increases and decreases happening constantly per period, we first find the net rate of change per period, and then apply the standard compound interest formula over the given time. The net effective annual growth rate is the difference between the increase rate and the decrease rate. ### Step-by-step Solution * **Given:** * Annual increase rate = $4\%$ * Annual decrease (emigration) rate = $\frac{1}{2}\% = 0.5\%$ * Time period ($n$) = $3$ years * **Calculation:** * First, calculate the net annual growth rate: $$\text{Net Rate (R)} = 4\% - 0.5\% = 3.5\%$$ * The population after 3 years will be: $$\text{Final Population} = P\left(1 + \frac{3.5}{100}\right)^3$$ * Let the initial population $P$ be 100 to make percentage calculations easier. $$\text{Final Population} = 100 \times \left(1 + \frac{7}{200}\right)^3 = 100 \times \left(\frac{207}{200}\right)^3$$ * Calculating this explicitly takes time, so let's use the successive percentage change formula or approximation. The formula for successive changes of $x\%$, $y\%$, and $z\%$ is complex. Instead, let's approximate using simple interest for a rough estimate, then add a bit for compounding. * Simple interest over 3 years: $3.5\% \times 3 = 10.5\%$ * Because it's compound growth, the actual increase will be slightly higher than $10.5\%$. * Let's look at the options: (a) 9.8, (b) 10, (c) 10.5, (d) 10.8. * Since it must be greater than $10.5\%$, the only logical answer is 10.8. ### Exam Strategy & Shortcut Use the **Effective Percentage Change Approximation**. For a small rate $R$ over $n$ years, the effective change is roughly $nR + \text{a small compounding factor}$. Here, $R = 3.5\%$ and $n = 3$. $3 \times 3.5\% = 10.5\%$. Because it compounds, the actual value must be strictly greater than $10.5\%$. Looking at the options, $10.8\%$ is the only viable choice. This avoids calculating $1.035^3$ entirely. ### Common Pitfall The most common mistake is simply multiplying the net rate by the number of years ($3.5\% \times 3 = 10.5\%$) and selecting option (c), forgetting that population growth is a compounding process, not a simple linear process. ### Final Answer **Therefore, the correct answer is 10.8.**