The Rule of 72, and Where It Actually Breaks Down
A plain explanation of the Rule of 72 shortcut for doubling time, with the real math showing exactly how far off it gets at higher rates.
The Rule of 72 is a mental-math shortcut for estimating how long it takes money to double at a given annual growth rate. Divide 72 by the interest rate, and the result is roughly the number of years to double. The U.S. Securities and Exchange Commission's investor education site, Investor.gov, uses this exact shortcut as teaching material in its compound interest resources, alongside its official Compound Interest Calculator.
The appeal is that it turns a logarithm into a division problem you can do in your head. But "roughly" is the important word. The rule is an approximation, and the size of the error changes depending on the rate you plug in.
The actual math behind doubling time
The real formula for how long it takes an amount to double at a fixed annual compounding rate r (as a decimal) is:
years to double = ln(2) / ln(1 + r)
The Rule of 72 shortcut approximates this as 72 divided by the interest rate expressed as a whole number (for example, 6 for 6%). Here is what happens when you compute both and compare them side by side:
| Annual rate | Rule of 72 estimate (years) | Actual years to double | Difference |
|---|---|---|---|
| 2% | 36.00 | 35.00 | +1.00 |
| 3% | 24.00 | 23.45 | +0.55 |
| 4% | 18.00 | 17.67 | +0.33 |
| 6% | 12.00 | 11.90 | +0.10 |
| 8% | 9.00 | 9.01 | -0.01 |
| 10% | 7.20 | 7.27 | -0.07 |
| 12% | 6.00 | 6.12 | -0.12 |
| 15% | 4.80 | 4.96 | -0.16 |
| 20% | 3.60 | 3.80 | -0.20 |
Two things stand out. First, the rule is remarkably accurate in the middle of the table, especially between 6% and 10%, which happens to cover the range most often used in long-term investment illustrations. Second, the rule drifts in opposite directions depending on which side of that sweet spot you're on: at low rates it overestimates the doubling time (so your money actually doubles a bit sooner than the rule says), and at high rates it underestimates the doubling time (so your money actually takes a bit longer than the rule says).
Why the error appears at all
The Rule of 72 is a linear approximation of a naturally logarithmic relationship. Compounding is multiplicative: each year's growth applies to a bigger base than the year before. The Rule of 72 treats the relationship between rate and doubling time as if it were a flat, evenly-spaced tradeoff, when in reality the curve bends. That bend is barely noticeable in the single-digit percentage range, which is why the rule feels almost exact for ordinary savings and investment rates. The bend becomes more visible as the rate climbs into double digits, which is more common in examples involving credit card interest or high-growth investment pitches than in typical savings accounts.
A useful way to think about it: at 2% the rule is off by a full year on a 35-year timeline, which is a small percentage error. At 20% the rule is off by 0.2 years on a horizon of under 4 years, which is close to a 5% error. The absolute error shrinks as rates rise, but the relative error grows, because the whole timeline is shrinking too.
When the shortcut is good enough
For everyday financial planning, where the expected annual return is somewhere between about 4% and 10%, the Rule of 72 gets you within a few tenths of a year of the real answer. That is close enough to compare two savings scenarios or sanity-check a projection someone else gave you. It is not precise enough to use as the final number in a spreadsheet you are relying on for a real decision, and it should never substitute for running the actual compound interest formula when the stakes are higher, such as comparing loan terms or retirement projections with decades at stake.
It's also worth remembering what the rule does not account for: irregular contributions, taxes, fees, and rates that change year to year. The 72-divided-by-rate shortcut assumes a single fixed rate applied every year with nothing added or withdrawn in between. Real accounts rarely behave that cleanly, which is exactly why tools like the SEC's own Compound Interest Calculator let you plug in variable contributions and compounding frequency rather than relying on a one-line estimate.
Key takeaways
- The Rule of 72 (divide 72 by the annual rate) estimates years to double; the actual formula is ln(2) divided by ln(1 plus the rate).
- The shortcut is most accurate between roughly 6% and 10%, where the estimate is within a tenth of a year of the real answer.
- Below that range the rule slightly overestimates doubling time; above it, the rule underestimates it, and the gap widens as the rate climbs.
- The rule assumes one fixed rate with no added or withdrawn money along the way, which is rarely how real accounts work.
- Use the shortcut for quick comparisons, but switch to a full compound interest calculation for any decision with real money on the line.
Treat 72 divided by the rate as a starting estimate, not a final answer, especially outside the single-digit percentage range where it was built to work best.