Understanding the Population Growth Problem

This question involves calculating the difference between the population of a town at two different points in the past, given its current population and a constant annual growth rate. We need to determine the population figures for 3 years ago and 2 years ago and then find the difference between them.

Calculating Past Population Figures

The formula to calculate the population in the past, given the current population and a constant growth rate, is derived from the compound growth formula:

$$ P_{past} = \frac{P_{current}}{(1 + r)^t} $$

Where:

• $P_{past}$ is the population in the past
• $P_{current}$ is the current population
• $r$ is the annual growth rate
• $t$ is the number of years in the past

Population 2 Years Ago

Given:

• Current Population ($P_{current}$) = 2,50,000
• Annual Growth Rate ($r$) = 25% = 0.25
• Time ($t$) = 2 years

Using the formula:

$$ P_{-2 \text{ years}} = \frac{2,50,000}{(1 + 0.25)^2} $$

$$ P_{-2 \text{ years}} = \frac{2,50,000}{(1.25)^2} $$

$$ P_{-2 \text{ years}} = \frac{2,50,000}{1.5625} $$

$$ P_{-2 \text{ years}} = 1,60,000 $$

So, the population 2 years ago was 1,60,000.

Population 3 Years Ago

Using the same formula with $t=3$ years:

$$ P_{-3 \text{ years}} = \frac{2,50,000}{(1 + 0.25)^3} $$

$$ P_{-3 \text{ years}} = \frac{2,50,000}{(1.25)^3} $$

$$ P_{-3 \text{ years}} = \frac{2,50,000}{1.953125} $$

$$ P_{-3 \text{ years}} = 1,28,000 $$

Therefore, the population 3 years ago was 1,28,000.

Calculating the Difference in Population

The question asks for the difference between the population 3 years ago and 2 years ago. This means we need to subtract the population from 3 years ago from the population 2 years ago.

Difference = Population 2 years ago - Population 3 years ago

$$ \text{Difference} = P_{-2 \text{ years}} - P_{-3 \text{ years}} $$

$$ \text{Difference} = 1,60,000 - 1,28,000 $$

$$ \text{Difference} = 32,000 $$

Conclusion

The difference in the town's population between 3 years ago and 2 years ago is 32,000.