The population of a town at the beginning of the year 1986 was 2,65,0000. if the rate of increase be 52 per thousand of the population. Find the population at the beginning of the year 1991.

Options

A. 3,40,400
B. 3,41,400
C. 3,42,400
D. 3,43,400

Answer

To find the population of the town at the beginning of the year 1991, we will follow these steps:

### Step 1: Identify the initial population and the rate of increase
- The initial population at the beginning of the year 1986 is given as **2,650,000**.
- The rate of increase is **52 per thousand** of the population.

### Step 2: Convert the rate of increase to a percentage
- To convert the rate of increase from per thousand to a percentage, we use the formula:
\[ \text{Rate of increase} = \frac{52}{1000} \times 100 = 5.2\% \]

### Step 3: Determine the time period for the population increase
- The time period from the beginning of 1986 to the beginning of 1991 is **5 years**.

### Step 4: Use the population growth formula
- The formula to calculate the future population is:
\[ P = P_0 \times \left(1 + \frac{R}{100}\right)^t \]
where:
- \( P_0 \) = initial population
- \( R \) = rate of increase (in percentage)
- \( t \) = time period (in years)

### Step 5: Substitute the values into the formula
- Substituting the values we have:
\[ P = 2,650,000 \times \left(1 + \frac{5.2}{100}\right)^5 \]
- This simplifies to:
\[ P = 2,650,000 \times \left(1 + 0.052\right)^5 \]
\[ P = 2,650,000 \times (1.052)^5 \]

### Step 6: Calculate \( (1.052)^5 \)
- Using a calculator, we find:
\[ (1.052)^5 \approx 1.294618 \]

### Step 7: Calculate the future population
- Now substitute back to find \( P \):
\[ P \approx 2,650,000 \times 1.294618 \approx 3,434,000 \]

### Step 8: Round off the population to the nearest whole number
- The population at the beginning of the year 1991 is approximately **3,434,000**.

### Final Answer
- The population at the beginning of the year 1991 is **3,434,000**.

---