Population Change — A village population increases by 5% in the first year and then decreases by 5% in the second year. If the population after the second year is 47,880, what was the population at the beginning of the first year?

Introduction / Context:
Successive percentage changes are multiplicative. An increase followed by a decrease of the same rate does not return to the initial value because the base changes after the first year. We exploit multiplicative factors to back-compute the original population from the final value.

Given Data / Assumptions:

• Year 1: +5% ⇒ multiply by 1.05
• Year 2: −5% ⇒ multiply by 0.95
• Final population after two years = 47,880

Concept / Approach:
Let initial population be P0. After two years: P2 = P0 * 1.05 * 0.95 = P0 * 0.9975. Therefore P0 = P2 / 0.9975.

Step-by-Step Solution:
P0 = 47,880 / 0.99751 / 0.9975 ≈ 1.002506…P0 ≈ 47,880 * 1.002506… ≈ 48,000

Verification / Alternative check:
Forward check: 48,000 * 1.05 = 50,400; then 50,400 * 0.95 = 47,880 ✔

Why Other Options Are Wrong:
45,500 and 43,500 are too low; 53,000 is too high; 47,760 is a rounding distraction not matching the exact reversal.

Common Pitfalls:
Adding percentages (+5 − 5 = 0) and assuming no change, which is incorrect because the 5% decrease applies to the larger base after the increase.

Final Answer:
48,000