Rule of 72

The Rule of 72 is a mental math shortcut: the time for an investment to double at a fixed annual rate $r\%$ is approximately: $$t_{\text{double}} \approx \frac{72}{r}$$ Where does 72 come from? We need $(1 + r/100)^t = 2$, so $t = \frac{\ln 2}{\ln(1 + r/100)}$. For small $r$, $\ln(1+r/100) \approx r/100$, giving $t \approx \frac{100 \ln 2}{r} = \frac{69.3}{r}$. We round to 72 because it has many divisors (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72), making mental division easy. Intuition: Compound growth is exponential, and humans are bad at intuiting exponentials. The Rule of 72 converts "exponential growth at rate $r$" into the concrete question "how long until it doubles?" — which is much easier to grasp. Concrete examples: - At 6%: doubles in $72\mathrm{/}6=12$$\frac{72}{6}$ = 12 years (exact: 11.9 years) - At 8%: doubles in $72\mathrm{/}8=9$$\frac{72}{8}$ = 9 years (exact: 9.01 years) - At 12%: doubles in $72\mathrm{/}12=6$$\frac{72}{12}$ = 6 years (exact: 6.12 years) - At 1%: doubles in $72\mathrm{/}1=72$$\frac{72}{1}$ = 72 years (exact: 69.7 years — the approximation is less accurate for very low rates) Works in reverse too: If prices double in 10 years, inflation is roughly $72\mathrm{/}10=7.2\mathrm{\%}$$\frac{72}{10}$ = 7.2\%. When to use: Quick compound interest estimates in interviews, back-of-envelope financial calculations, and sanity-checking discounted cash flow models. Interviewers love it because it tests whether you can think quantitatively without a calculator. Alternative approach: For precision, use $t = \ln 2 / \ln(1 + r/100)$. The Rule of 72 is accurate to within 1% for rates between 2% and 20%.