With an annual compound rate of $r = 6\%/\text{year}$, the approximate doubling time is:
Underlying Mathematics: The exact equation for compound growth is $(1 + r)^t = 2 \implies t = \frac{\ln(2)}{\ln(1 + r)}$. Since $\ln(2) \approx 0.693$ and $\ln(1+r) \approx r$ for small values of $r$, we have $t \approx \frac{69.3}{r}$. Financial mathematicians round $69.3$ to $72$ because $72$ is highly composite (divisible by $2, 3, 4, 6, 8, 9, 12$), making accurate mental estimations effortless in real life.
Each flip of a fair coin is a strictly independent event:
Practical Application: Nature does not "balance out" random streaks by altering the odds of future independent events. Recognizing this prevents costly psychological traps in investing, gambling, and risk analysis.
Average speed over equal distances is given by the Harmonic Mean, not the arithmetic mean:
Key Insight: Because lower speed requires more elapsed time, it receives a higher "time weight" in the overall velocity calculation, pulling the final average closer to $30\text{ km/h}$.
Percentages must always be evaluated relative to their specific baseline:
General Formula: Whenever an asset increases by $x\%$ and subsequently decreases by $x\%$, the net outcome is always a loss of $\left(\frac{x}{100}\right)^2$. For $x = 20$, the net loss is always $(0.2)^2 = 0.04 = 4\%$.
You have successfully mastered four essential quantitative reasoning principles: the Rule of 72 in finance, independent events in probability, harmonic mean in velocity, and compounded percentage asymmetry.