The population of a city 3 years ago was 40960. If the present population of that city is 49130, then the percentage annual increase is :

Question

Accepted Answer

To determine the percentage annual increase in the population of the city, we need to calculate the compound annual growth rate (CAGR) over the 3-year period. The formula for CAGR is:

$$ \text{CAGR} = \left( \frac{\text{Final Population}}{\text{Initial Population}} \right)^{\frac{1}{n}} - 1 $$

where:
- Final Population = 49130
- Initial Population = 40960
- n = number of years = 3

Plugging in the values, we get:

$$ \text{CAGR} = \left( \frac{49130}{40960} \right)^{\frac{1}{3}} - 1 $$

First, calculate the ratio:

$$ \frac{49130}{40960} \approx 1.2 $$

Now, calculate the cube root of 1.2:

$$ 1.2^{\frac{1}{3}} \approx 1.0625 $$

Subtract 1 to find the growth rate:

$$ 1.0625 - 1 = 0.0625 $$

Convert this to a percentage:

$$ 0.0625 \times 100 = 6.25\% $$

This is equivalent to 6$\frac{1}{4}$%, which matches Option 3.

**Explanation for why Option 3 is correct:**
Option 3 is correct because the calculated compound annual growth rate over the 3-year period is approximately 6.25%, which is expressed as 6$\frac{1}{4}$%. This matches the given correct answer.

**Explanation for why other options are incorrect:**

1. **Option 1 (2$\frac{1}{4}$%)**: This option is incorrect because the calculated growth rate of 6.25% is significantly higher than 2.25%. A growth rate of 2.25% would result in a much smaller increase in population over 3 years.

2. **Option 2 (5$\frac{1}{4}$%)**: This option is incorrect because the calculated growth rate of 6.25% is higher than 5.25%. A growth rate of 5.25% would not account for the observed increase in population from 40960 to 49130 over 3 years.

3. **Option 4 (7$\frac{1}{4}$%)**: This option is incorrect because the calculated growth rate of 6.25% is lower than 7.25%. A growth rate of 7.25% would result in a larger increase in population than what was observed.

Thus, Option 3 is the only choice that accurately reflects the calculated annual percentage increase in population.