During one year, the population of a town increased by 5% and during the next year, the population decreased by 5%. If the total population is 29925 at the end of the second year, then what was the population size in the beginning of the first year?
30000
To determine the initial population of the town at the beginning of the first year, we need to analyze the changes in population over the two-year period.
- Let the initial population be \(P\).
- During the first year, the population increased by 5%. The population at the end of the first year can be calculated as: \(P_{1} = P + 0.05P = 1.05P\).
- During the second year, the population decreased by 5%. The population at the end of the second year, using the population at the end of the first year, is given by: \(P_{2} = 1.05P - 0.05 \times 1.05P = 1.05P \times (1 - 0.05)\).
- Simplifying, we find: \(P_{2} = 1.05P \times 0.95 = 0.9975P\).
- It is given that the population at the end of the second year is 29925. Therefore, we have the equation: \(0.9975P = 29925\).
- Solving for \(P\): \(P = \frac{29925}{0.9975}\).
- Calculating the value of \(P\) gives: \(P = 30000\).
Therefore, the population at the beginning of the first year was 30,000.
Let's verify the steps:
| Year | Population Change | Calculation | Population |
|---|---|---|---|
| Start of Year 1 | Initial | - | \(30000\) |
| End of Year 1 | 5% Increase | \(30000 \times 1.05 = 31500\) | \(31500\) |
| End of Year 2 | 5% Decrease | \(31500 \times 0.95 = 29925\) | \(29925\) |
Thus, all calculations are consistent, confirming our initial population as 30,000. Therefore, the correct option is 30000.
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