Q.5 The population of a new city is 5 million and is growing at 20% annually. How many years would
it take to double at this growth rate?
(A) 3-4 years (B) 4-5 years (C) 5-6 years (D) 6-7 years
Years to Double Population at 20% Growth: 3-4 Years Explained
A city’s population starting at 5 million grows at 20% annually through compound growth, doubling when it reaches 10 million. Precise calculation shows this happens in approximately 3.8 years, fitting the 3-4 years range.
✅ Correct Answer
The correct option is (A) 3-4 years. Using the doubling time formula \( t = \frac{\ln(2)}{\ln(1.2)} \approx 3.8 \) years or Rule of 70 (\( 70 / 20 = 3.5 \) years), the population exceeds 10 million between 3 and 4 years.
📊 Calculation Breakdown
Start with P = 5 million, growth rate r = 0.20. Future population: P(1 + r)^t = 10.
Simplifies to (1.2)^t = 2, so t = \frac{\log(2)}{\log(1.2)} \approx \frac{0.3010}{0.0792} = 3.8 years.
Rule of 70 provides quick estimate: t ≈ 70 / 20 = 3.5 years, close to exact value for rates around 20%.
📈 Year-by-Year Growth
- Year 1: 5 × 1.2 = 6 million
- Year 2: 6 × 1.2 = 7.2 million
- Year 3: 7.2 × 1.2 = 8.64 million
- Year 4: 8.64 × 1.2 = 10.368 million (doubles)
At end of year 3, still under 10 million; surpasses during year 4.
❌ Option Explanations
| Option | Range | Why Incorrect/Correct | Population at End |
|---|---|---|---|
| (A) 3-4 years | 3-4 | Correct: Hits ~10 million mid-year 4 (3.8 years total) | 10.368 million |
| (B) 4-5 years | 4-5 | Too high: Already doubled by end of year 4 | Already >10M |
| (C) 5-6 years | 5-6 | Far too slow for 20% rate (~12.5% rate needed) | ~13M (excessive) |
| (D) 6-7 years | 6-7 | Matches ~10-12% rate via Rule of 70 (70/10=7) | ~16M (way over) |
