The president of the World Bank is on his way to a meeting with the president of Uruguay. He bumps into you in the hallway and wants to know how long it will take for Uruguayan per capita GDP to increase by a factor of 4. All he knows is that the average growth rate has been about 3.5 percent.

Use the formula for compound growth rates and round your answer to the nearest tenth (of a year).

Question

The president of the World Bank is on his way to a meeting with the president of Uruguay. He bumps into you in the hallway and wants to know how long it will take for Uruguayan per capita GDP to increase by a factor of 4. All he knows is that the average growth rate has been about 3.5 percent.

Use the formula for compound growth rates and round your answer to the nearest tenth (of a year).

BlackTom AI · Accepted Answer

(1) Relevant concepts: Discrete compound growth is described by Future = Present × (1 + g)^t, where g is the growth rate per period and t is the number of periods. To solve for time when the level is a given multiple, take natural logarithms and use t = ln(target multiple) / ln(1 + g).

(2) Step-by-step reasoning: We want per capita GDP to increase by a factor of 4 at an annual growth rate g = 3.5% = 0.035. Set 4 = (1 + 0.035)^t and solve for t by taking natural logs of both sides.

(3) Calculations (all intermediate steps):
- Equation: 4 = (1.035)^t
- Take logs: ln 4 = t · ln 1.035
- Solve: t = ln 4 / ln 1.035
- Compute ln 4 = 1.3862943611198906 (exact to many digits)
- Compute ln 1.035 using a series expansion (or calculator):
  - x = 0.035
  - ln(1 + x) ≈ x − x^2/2 + x^3/3 − x^4/4 ...
  - = 0.035 − 0.001225/2 + 0.000042875/3 − 0.0000015006/4 ...
  - ≈ 0.035 − 0.0006125 + 0.0000142917 − 0.000000375
  - ≈ 0.0344014167
- Divide: t ≈ 1.3862943611 / 0.0344014167 ≈ 40.297...
- Round to nearest tenth: t ≈ 40.3 years

(4) Why this is correct: The logarithmic solution is the direct algebraic inversion of compound growth; it accounts for compounding each year. Using ln isolates the exponent t exactly, giving an accurate time to reach a fourfold increase at 3.5% growth.

(5) Other approaches and common errors: There are no alternative choices here, but common approximations include the Rule of 72 (72/3.5 ≈ 20.6 years per doubling → ~41.2 years for quadrupling), which is only an approximation and slightly overestimates. Treating growth as simple linear (not compounding) would be incorrect.

(6) Conclusion: The calculation yields 40.3 years, which matches the provided correct answer.

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