The population of a village has increased
annually at the rate of 25%. If at the end of 3 years
it is 10,000, the population in the beginning of the
first year

To find the population at the beginning of the first year, we can use the formula for compound interest, which is applicable to population growth:

\[
P = \frac{A}{(1 + r)^n}
\]

Where:
- \( P \) is the initial population (what we are solving for).
- \( A \) is the population at the end of \( n \) years.
- \( r \) is the annual growth rate (expressed as a decimal).
- \( n \) is the number of years.

Given:
- \( A = 10,000 \) (population at the end of 3 years),
- \( r = 25\% = 0.25 \),
- \( n = 3 \) years.

Now, substitute the values into the formula:

\[
P = \frac{10,000}{(1 + 0.25)^3}
\]

Calculate the denominator:

\[
(1 + 0.25)^3 = (1.25)^3 = 1.953125
\]

Thus:

\[
P = \frac{10,000}{1.953125} \approx 5,119.87
\]

So, the population at the beginning of the first year was approximately **5,120** people.

Would you like any further details or have any questions?

Here are some related questions to consider:

1. What would the population be at the end of 4 years with the same growth rate?
2. How does the population change if the growth rate drops to 20% annually?
3. What is the formula for continuous population growth?
4. How would the population change if instead of increasing, it decreased by 10% annually?
5. How long would it take for the population to double with a 25% annual growth rate?
6. What is the effect of compound interest in financial contexts compared to population growth?
7. How can you model a population that grows at different rates in different years?
8. What are some real-life factors that might cause a population to grow at a constant annual rate?

**Tip:** When dealing with percentage increases or decreases, always convert the percentage to a decimal before performing calculations.

Learn how to calculate the initial population of a village using compound interest formulas. Given a 25% annual growth rate and a population of 10,000 after 3 years, find the initial population at the start of the first year. Explore related questions and applications of compound interest in population growth.

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