The population of a village was 1,80,000. It increased by 15% in the first…

Concept: A percentage change always acts on the amount present at the start of that period, not on the original amount. A rise of p% multiplies a quantity by (1 + p/100), so two successive rises of p% and q% multiply the starting quantity by (1 + p/100) × (1 + q/100). Because the two rises are measured on different bases, the percentage figures of successive periods cannot simply be added.

Application: the starting population is 1,80,000, the first year's rise is 15% and the second year's rise is 30%.

1. First-year multiplier: 1 + 15/100 = 1.15.

2. Population at the end of the first year: 1,80,000 × 1.15 = 2,07,000, since 15% of 1,80,000 is 27,000 and 1,80,000 + 27,000 = 2,07,000.

3. Second-year multiplier: 1 + 30/100 = 1.30, applied to the first year's population, which is the base standing at the start of the second year.

4. Population at the end of the second year: 2,07,000 × 1.30 = 2,69,100, since 30% of 2,07,000 is 62,100 and 2,07,000 + 62,100 = 2,69,100.

Cross-check: the two multipliers combine into a single factor 1.15 × 1.30 = 1.495, that is a net rise of 49.5%, and 1,80,000 × 1.495 = 2,69,100 — the same figure. Contrast this with adding the two rates: 15% + 30% = 45% would give 1,80,000 × 1.45 = 2,61,000, which falls short by 8,100, and that gap is exactly 30% of the 27,000 added during the first year — the growth on which the second year's rise also acts.

Population after two years = 2,69,100.