If the population of a town increases by 25% in the first year and increases by 36% in the second year, then what is the average rate (in percentage) of increase in the population?

Options

A. 26
B. `30.5`
C. `31.66`
D. 35

Answer

To find the average rate of increase in the population over two years, we can follow these steps:

### Step 1: Understand the Initial Population
Let’s assume the initial population of the town is \( P \).

### Step 2: Calculate the Population After the First Year
The population increases by 25% in the first year.
- Increase in population after the first year = \( 25\% \) of \( P \) = \( \frac{25}{100} \times P = 0.25P \).
- Therefore, the population at the end of the first year = \( P + 0.25P = 1.25P \).

### Step 3: Calculate the Population After the Second Year
The population then increases by 36% in the second year based on the new population.
- Increase in population after the second year = \( 36\% \) of \( 1.25P \) = \( \frac{36}{100} \times 1.25P = 0.45P \).
- Therefore, the population at the end of the second year = \( 1.25P + 0.45P = 1.70P \).

### Step 4: Calculate the Total Increase in Population
The total increase in population over the two years is:
- Total increase = Final population - Initial population = \( 1.70P - P = 0.70P \).

### Step 5: Calculate the Average Rate of Increase
The average rate of increase over the two years can be calculated using the formula:
\[ \text{Average Rate of Increase} = \left( \frac{\text{Total Increase}}{\text{Initial Population}} \right) \times 100 \]
Substituting the values:
\[ \text{Average Rate of Increase} = \left( \frac{0.70P}{P} \right) \times 100 = 70\% \]
However, since this is over two years, we need to find the average per year:
\[ \text{Average Rate per Year} = \frac{70\%}{2} = 35\% \]

### Final Answer
The average rate of increase in the population over the two years is **35%**.

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