The population of a city increases by 6% in the first year, 8% in the second…
2019
The population of a city increases by 6% in the first year, 8% in the second year, and 5% in the third year. What is the total percentage increase in the population at the end of 3 years?
Answer: B. 20.204% — Concept: When a quantity undergoes several successive percentage changes, the changes compound multiplicatively rather than adding directly. If a quantity…
- A.
20.324%
- B.
20.204%
- C.
19%
- D.
19.234%
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Show answer & explanation
Correct answer: B
Concept: When a quantity undergoes several successive percentage changes, the changes compound multiplicatively rather than adding directly. If a quantity grows by r1%, then r2%, then r3% in successive periods, the overall growth factor equals the product (1 + r1/100) × (1 + r2/100) × (1 + r3/100), and the net percentage change equals [(that product) − 1] × 100.
Application: Take the initial population as 100 and apply each year's growth factor to the previous year's result in turn.
Year 1 (6% increase): 100 × 1.06 = 106.
Year 2 (8% increase): 106 × 1.08 = 114.48.
Year 3 (5% increase): 114.48 × 1.05 = 120.204.
Net increase = 120.204 − 100 = 20.204, so the population rises by 20.204% over the three years.
Cross-check: Multiplying the three growth factors directly gives 1.06 × 1.08 × 1.05 = 1.20204, so the percentage increase is (1.20204 − 1) × 100 = 20.204% — the same result. Note that simply adding the rates (6 + 8 + 5 = 19%) undercounts the true increase, because it ignores that each year's growth applies to a population that has already grown in the earlier years.