The Rule of 70: A Comprehensive Guide to Estimating Doubling Times and Compound Growth

The Rule of 70 is defined as a "quick and easy method" designed to determine the speed at which a growing variable will double in size over time.
This rule is universally applicable to any variable undergoing growth, regardless of the subject matter. Specific examples cited include:
- Personal savings accounts.
- Global population levels.
- Computing power (technological advancement).
- Biological growth, such as the number of bacteria in a petri dish.
- National economic indicators, such as a country's economy or GDP.
The Rule of 70 functions as an approximation tool that allows for the rapid estimation of doubling time, which can be used to "impress your friends and confound your enemies."

The core calculation for the Rule of 70 is based on the following formula:
- $\text{Doubling Time} = \frac{70}{x}$
In this formula, the variable $x$ represents the growth rate per period (usually expressed as a percentage).
Crucially, when using this rule, you do not use the decimal form of the growth rate; you use the integer or whole number percentage value (e.g., for $5\%$ , you divide by $5$ , not $0.05$ ).

The Rule of 70 is an approximation of the mathematical reality of compound interest.
In a scenario with a growth rate of $5\%$ per year, compounding works as follows:
- Year 1: The investment grows by $5\%$ of the original principal.
- Year 2: The investment grows by $5\%$ of the new total (the original principal plus the growth captured in Year 1).
This exponential process means the variable grows on top of previous growth, accelerating the doubling process compared to simple interest models.

To demonstrate the accuracy of the rule, the narrator provides a comparative example using a $5\%$ annual growth rate:
- Rule of 70 Approximation:
  - $\frac{70}{5} = 14\,\text{years}$
- Exact Mathematical Calculation:
  - $13.86\,\text{years}$
The proximity of these two results— $14$ versus $13.86$ —illustrates that the Rule of 70 is a highly accurate tool for practical estimation.

The Rule of 70 is particularly useful for comparing the evolution of living standards across different nations.
The narrator identifies specific "growth miracles" in countries such as Korea, China, and Japan, where annual growth rates have reached levels of $7\%$ to $10\%$ .
Case Study: China’s Economic Expansion
- If a country maintains a $10\%$ annual growth rate, its living standards double every $7\,\text{years}$ ( $\frac{70}{10} = 7$ ).
- China sustained high growth for approximately $35\,\text{years}$ .
- To calculate the total growth over that period:
  - Number of doubling periods: $\frac{35}{7} = 5$
  - Cumulative growth calculation: $2 \times 2 \times 2 \times 2 \times 2 = 2^5 = 32$
- Result: After $35\,\text{years}$ , the GDP per capita in China became $32\,\text{times}$ larger than its initial starting point.
The primary benefit of the Rule of 70 in this context is that it allows observers to visualize the "power of compounding" without performing complex exponential equations.

The Rule of 70 highlights the dramatic long-term impact of small differences in growth rates.
China (Growth Miracle): At high growth rates, an economy becomes $32\,\text{times}$ bigger over $35\,\text{years}$ .
Developed Countries: Most developed nations typically experience a significantly lower growth rate, often around $2\%$ .
- Calculation for a $2\%$ growth rate over $35\,\text{years}$ :
  - Doubling time: $\frac{70}{2} = 35\,\text{years}$ .
  - Over a $35\text{-year}$ period, the economy doubles only once ( $2^1 = 2$ ).
The Contrast: The difference between a $32\text{-fold}$ increase and a $2\text{-fold}$ increase illustrates the profound effect that sustained high growth rates have on national prosperity.

The rule can be mathematically manipulated to find the growth rate if the doubling time is already known.
Equation: $x = \frac{70}{\text{Doubling Time}}$
Example: Real Estate Market
- Observed Fact: House prices doubled between the years $2000$ and $2006$ .
- Known Doubling Time ( $T$ ): $6\,\text{years}$ .
- Calculation: $\frac{70}{x} = 6\,\text{years}$ .
- Solving for $x$ : $x = \frac{70}{6} = 11.6$ .
- Conclusion: House prices increased at an average rate of approximately $11.6\%$ per year during that period.

The Rule of 70 serves as a handy tool to approximate doubling time based on a known annual growth rate.
The material concludes by suggesting further investigation into why different countries grow at such disparate speeds, pointing toward the "extreme examples" of North Korea and South Korea as a Case Study for understanding economic divergence.