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
City Population Growth: How Many Years to Double at 20% Annual Rate?
5 million population growing 20% annually – Calculate doubling time using Rule of 72
Population doubles from 5M to 10M in approximately 3.8 years
Understanding Population Doubling Time
A city with 5 million people growing at 20% annually reaches double population (10 million) in 3-4 years. This demonstrates the power of compound growth, where how many years would it take to double at this growth rate is solved using the Rule of 72 or logarithmic calculations.
Rule of 72: Quick Doubling Estimate
72 / 20 = 3.6 yearsWorks best for 5-25% growth rates
Exact Mathematical Calculation
Population formula: P_t = P_0 × (1 + r)^t
t = ln(2) / ln(1 + r) = 0.6931 / ln(1.2) ≈ 3.80 yearsYear-by-Year Population Growth
| Year | Population (millions) | % of Original |
|---|---|---|
| 0 | 5.00 | 100% |
| 1 | 6.00 | 120% |
| 2 | 7.20 | 144% |
| 3 | 8.64 | 173% |
| 3.8 | ~10.00 | 200% ✓ |
| 4 | 10.37 | 207% |
Option Analysis
3.8 years falls perfectly in this range. Rule of 72 gives 3.6 years.
Would require ~15% growth rate (72/15 = 4.8 years)
Matches ~12% growth rate (72/12 = 6 years)
Corresponds to ~10.5% growth (72/10.5 ≈ 6.9 years)
Real-World Applications
- Finance: Investment doubling time
- Biology: Bacterial population growth
- Urban Planning: City expansion forecasts
- India Context: New smart cities often exceed 20% growth
