Scope and assumptions
Doubling time t ≈ 72 ÷ (rate%). Exact: t = ln(2) ÷ ln(1 + r). Illustrative.
Most growth questions run forwards: rate in, future value out. The rule of 72 runs the other way. Give it a rate and it returns a time — specifically, how long until the quantity doubles — using nothing more than one division.
At 8% a year, 72 ÷ 8 = 9 years. The exact answer is 9.01. For a mental calculation, that's close enough to be genuinely useful.
The rule
Doubling time ≈ 72 ÷ rate (in percent)
The rate goes in as a percent, not a decimal. 8%, not 0.08.
| Rate | Rule of 72 | Exact |
|---|---|---|
| 4% | 18 yr | 17.67 |
| 6% | 12 yr | 11.90 |
| 8% | 9 yr | 9.01 |
| 12% | 6 yr | 6.12 |
| 24% | 3 yr | 3.22 |
Where 72 comes from
Doubling time solves (1 + r)^t = 2, so t = ln 2 ÷ ln(1 + r). For small r, ln(1 + r) ≈ r, giving t ≈ 0.693 ÷ r — about 69.3 ÷ rate as a percent.
72 is a practical substitute: it slightly overestimates at low rates and sits close to exact through the middle range, and it divides cleanly by 2, 3, 4, 6, 8, 9, and 12, which makes the mental arithmetic easy.
When the 72 Rule Drifts
The rule is tightest between roughly 6% and 12%. Outside that:
- Very low rates (1–2%) — 72 overstates the time slightly; 70 is closer.
- High rates (above 15%) — 72 underestimates noticeably; use the exact formula.
| Rate | Rule of 72 | Exact | Error |
|---|---|---|---|
| 2% | 36 yr | 35.0 | +1.0 |
| 30% | 2.4 yr | 2.64 | −0.24 |
The same rule, any quantity
Nothing in the rule is specific to money. The only requirement is a steady compound rate:
- A city growing 3% a year doubles in about 24 years.
- A user base growing 10% a month doubles in about 7 months.
- A 7% annual decline halves in about a decade.
The timescale follows the rate's period. A monthly rate yields a monthly answer.
Misusing Doubling Time in Practice
Entering the rate as a decimal (72 ÷ 0.08 returns 900); expecting precision; applying it to simple, non-compounding growth; forgetting that the answer inherits the rate's time unit; using it for halving without adjusting the sign; and trusting it at rates far from the middle range.
Related growth & decay guides
For the rest of the growth-and-decay cluster, start at the hub: Population Growth Percentage. Adjacent: Annual Percentage Rate Explained and Compound Growth Percentage.
FAQ
Why do some sources use 70 or 69.3 instead of 72? Because the exact constant is 100 × ln(2) ≈ 69.3. Using 70 suits continuous compounding; 72 is preferred for annual compounding near 8% because it has more whole-number divisors.
How do I find halving time instead? Same division. A 6% annual decline halves in about 72 ÷ 6 = 12 years, because halving is just doubling with a negative rate.
Can I use the rule for inflation? Yes — and it's sobering. At 3% inflation, purchasing power halves in roughly 24 years; at 7%, in about a decade.
What if my rate is monthly rather than annual? The answer comes out in months. A 1% monthly rate doubles in about 72 months — six years, not six months.
Sources: Investopedia, Rule of 72; Khan Academy.
Disclaimer: Illustrative. The rule is an approximation. Informational only.
Sources and references
- Investopedia — Rule of 72https://www.investopedia.com/terms/r/ruleof72.asp
- Khan Academy — Exponential growthhttps://www.khanacademy.org/math
Maintenance and corrections
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