The current population of a city is 1,10,250. If it has been increasing at the rate of 5% per annum, what was its population 2 years ago?

Population Problem Analysis

This question asks us to find the population of a city two years ago, given its current population and a consistent annual growth rate. This is a classic problem involving compound growth, similar to compound interest calculations.

Here's what we know:

- **Current Population ($P_n$): 1,10,250
- Annual Growth Rate (r): 5%
- Time Period (n): 2 years

We need to find the population 2 years ago ($P_0$).

Growth Rate Formula Explained**

The formula for population growth (or compound growth) is used to calculate the future value of a quantity when it grows at a constant percentage rate over time. The formula is:

$$P_n=P_0{\left(1+\frac{r}{100}\right)}^n$$

Where:

- ${\mathbf{P}}_{\mathbf{n}}$ is the population after 'n' years (the current population).
- ${\mathbf{P}}_0$ is the initial population (the population 'n' years ago).
- $\mathbf{r}$ is the annual rate of increase (in percent).
- $\mathbf{n}$ is the number of years.

In our case, we need to find $P_0$. So, we can rearrange the formula to solve for $P_0$:

$$P_0=\frac{P_n}{{\left(1+\frac{r}{100}\right)}^n}$$

Past Population Calculation Step-by-Step

Let's substitute the given values into the rearranged formula to find the population 2 years ago:

- Current Population ($P_n$) = 1,10,250
- Rate of increase (r) = 5%
- Time (n) = 2 years

Step 1: Convert the percentage rate to a decimal.

$$r=5\%=\frac{5}{100}=0.05$$

Step 2: Calculate the growth factor.

$$1+\frac{r}{100}=1+0.05=1.05$$

Step 3: Raise the growth factor to the power of 'n' (number of years).

$${\left(1+\frac{r}{100}\right)}^n=(1.05{)}^2$$

$$(1.05{)}^2=1.05\times 1.05=1.1025$$

**Step 4: Substitute all values into the formula for $P_0$.**

$$P_0=\frac{P_n}{{\left(1+\frac{r}{100}\right)}^n}$$

$$P_0=\frac{110250}{1.1025}$$

**Step 5: Perform the division to find $P_0$.**

$$P_0=100000$$

Summary of CalculationsTherefore, the population of the city 2 years ago was 1,00,000.