How this calculator works
The rule of 72 is a mental-math shortcut for estimating how long it takes an investment to double at a given annual compounding rate, or conversely, what rate is needed to double in a given number of years. Divide 72 by the rate (as a whole number, not a decimal) to get years, or divide 72 by years to get the required rate.
It is an approximation of the exact formula t = ln(2) ÷ ln(1 + r), which has no convenient mental-math shortcut of its own. This calculator shows both numbers side by side, in both directions (rate to years, and years to rate), so you can see exactly how close the shortcut is for your own numbers.
The formulas
Rule of 72 approximation: years ≈ 72 ÷ rate (rate as a whole number, e.g. 8 for 8%)Exact years to double: t = ln(2) ÷ ln(1 + r), where r is the rate as a decimalExact required rate for a target number of years: r = 2^(1/years) − 172 was chosen historically because it divides evenly by many small numbers (1, 2, 3, 4, 6, 8, 9, 12...), making it easy to compute by hand — not because it is the mathematically exact constant. The exact constant, ln(2) × 100 ≈ 69.3, is sometimes used instead ('rule of 69.3') for continuous compounding.
Worked example: 8% annual return
- Rule of 72 estimate: 72 ÷ 8 = 9 years.
- Exact calculation: t = ln(2) ÷ ln(1.08) = 0.693147 ÷ 0.076961 ≈ 9.0065 years.
- The approximation is off by only about 0.0065 years (roughly 2 days) — an error of about 0.07%, because 8% sits right in the rule's most accurate range.
Frequently asked questions
How accurate is the rule of 72?
It is very accurate between roughly 6% and 10%, where the error is well under 1%. Outside that range the error grows steadily in both directions — it is also slightly less accurate for very low rates than for its 6-10% sweet spot, though the effect is far smaller than at high rates.
Why does the rule break down at high interest rates?
The rule of 72 is a linear approximation of a logarithmic relationship. At low-to-moderate rates the two curves stay close together, but at high rates (think 30%, 50%, 100%) they diverge sharply — at 72%, for example, the rule says 1 year while the exact answer is closer to 1.28 years.
Can I use the rule of 72 for inflation or debt instead of investment growth?
Yes — the math is the same for anything that grows or shrinks by a fixed percentage each period. Divide 72 by an inflation rate to estimate how long prices take to double, or by a credit card's APR to estimate how fast an unpaid balance would double if left completely untouched.
What is the 'rule of 70' or 'rule of 69.3'?
They are the same idea with a different numerator. Rule of 69.3 (or sometimes 69) is closer to the true mathematical constant ln(2) × 100 and is more accurate for continuously compounded growth; 72 remains the most commonly taught version because it has more small integer divisors.
How many times will my money double over a long time horizon, like 30 years?
Divide the number of years by the doubling time from the rule of 72. At 8%, this calculator's worked example shows a doubling time of about 9 years, so over 30 years the money would double roughly 30 ÷ 9 ≈ 3.3 times — meaning it multiplies by about 2^3.3, or roughly 10 times its starting value, since each full doubling multiplies the total by 2 again.