Rule of 72: Estimating Doubling Time

The shortcut

Rule of 72: $\text{doubling time (years)} \approx \dfrac{72}{r_{\%}}$

where $r_{\%}$ is the annual rate as a percentage (so 6%, not 0.06). This is a quick mental estimate — no logarithms required.

The exact formula

Doubling means $(1 + r)^t = 2$, so $t_{\text{exact}} = \dfrac{\ln 2}{\ln(1 + r)}$

Since $\ln 2 \approx 0.6931$ and $\ln(1 + r) \approx r$ for small $r$, this becomes approximately $\dfrac{0.6931}{r}$. Multiplying numerator and denominator by 100 gives $\dfrac{69.31}{r_{\%}}$, which is very close to 72 — the rounding to 72 comes from choosing a nice-dividing number with more useful factors.

Step-by-step check at 6%

Step 1 — Rule of 72: $t \approx \dfrac{72}{6} = 12 \text{ years}$

Step 2 — Exact (annual compounding): $t = \dfrac{\ln 2}{\ln 1.06} = \dfrac{0.69315}{0.05827} \approx 11.896 \text{ years}$

Step 3 — Compare: $\text{Error} = 12 - 11.896 = 0.104 \text{ years} \approx 5 \text{ weeks}$

That's roughly a 0.9% error — acceptable for back-of-envelope work.

Accuracy by rate

The Rule of 72 is most accurate near 8%. Error grows at extremes:

• 2%: Rule says 36 yrs, actual ~35.0 yrs (1.1 yr high)
• 6%: Rule says 12 yrs, actual ~11.9 yrs (0.1 yr high)
• 8%: Rule says 9 yrs, actual ~9.0 yrs (spot on)
• 12%: Rule says 6 yrs, actual ~6.12 yrs (0.12 yr low)
• 20%: Rule says 3.6 yrs, actual ~3.80 yrs (0.20 yr low)

Rule of 69.3 and Rule of 70

• Rule of 69.3 is exact for continuous compounding (since $\ln 2 \approx 0.693$).
• Rule of 70 is used in demography and epidemiology.
• Rule of 72 is preferred because 72 has divisors 2, 3, 4, 6, 8, 9, 12 — great for mental math.

Variations

• Tripling time: $\dfrac{\ln 3}{\ln(1+r)} \approx \dfrac{115}{r_{\%}}$ → "Rule of 115."
• 10× time: $\dfrac{\ln 10}{\ln(1+r)} \approx \dfrac{230}{r_{\%}}$ → "Rule of 230."
• Halving time (decay): same formula with negative rate.

Common mistakes

• Plugging in the decimal. $72/0.06 = 1200$ years — use the percentage integer 6, not 0.06.
• Applying it to very high rates. At 25%, Rule of 72 says 2.88 years; actual is 3.11 years — 8% off. Beyond ~20% the rule loses bite; use logs directly.
• Confusing with simple interest. The rule assumes compounding. Simple interest doubles in exactly $1/r$ years ($100/r_{\%}$).

Try it in the visualization

A curve of exact doubling time vs. the Rule of 72 approximation, shaded by error. Slide the rate to see where the two curves hug closest (around 8%) and where they diverge.