Two years ago the population of a village was 12,500. Due to migration towards…
2025
Two years ago the population of a village was 12,500. Due to migration towards cities, it decreases at the rate of 16% annually. In the last 2 years, by what percentage (rounded off to 1 decimal place) did the population decrease?
- A.
29.4%
- B.
25.6%
- C.
24.9%
- D.
26.5%
Attempted by 35 students.
Show answer & explanation
Correct answer: A
CONCEPT: When a quantity falls by the same r% every year, the fall compounds — each year's r% is removed from that year's OWN (already-reduced) base, not from the original figure. So after n years the quantity becomes the original value times (1 − r/100)n, and the overall percentage fall measured against the original value is 100 × [1 − (1 − r/100)n]. For exactly two years this matches the successive-percentage-change shortcut: net fall% = 2r − r2/100.
APPLICATION:
Population two years ago, P0 = 12,500; annual fall rate r = 16%, so the yearly retention factor is 1 − 16/100 = 0.84.
After the first year: 12,500 × 0.84 = 10,500.
After the second year, the same 16% is removed from this new base of 10,500: 10,500 × 0.84 = 8,820.
Total fall over the two years = 12,500 − 8,820 = 3,680.
Percentage fall on the original population = (3,680 / 12,500) × 100 = 29.44%, which rounds to 29.4%.
CROSS-CHECK: Using the successive-percentage-change shortcut directly: net fall% = 2(16) − (162)/100 = 32 − 2.56 = 29.44%, matching the step-by-step result independently — confirming the two-year decrease is 29.4%.