What the rule of 72 is
Divide 72 by a yearly rate, and you get roughly how many years that rate takes to double an amount, if the rate stays fixed and the growth stays in.
Example: 6%. At an example rate of 6% a year, 72 ÷ 6 = 12.00 years. The exact answer is 11.90 years, so the rule is off by 0.10 years.
Example: 9%. At an example rate of 9% a year, 72 ÷ 9 = 8.00 years. The exact answer is 8.04 years.
The exact answer. For a fixed rate added once a year, the exact doubling time is:
years = ln 2 ÷ ln(1 + r)
Here ln is the natural logarithm, a button on most scientific calculators. r is the rate as a decimal, so an example rate of 6% is 0.06. ln 2 is about 0.693147.
At an example rate of 6%: ln 1.06 = 0.058269, and 0.693147 ÷ 0.058269 = 11.90 years.
For education only, not financial advice.
To see the full growth, year by year, use the compound interest calculator. The compound interest formula page explains the math behind it.
How accurate it is
The rule is an estimate. Here is how close it comes at every whole example rate from 1% to 20%, against the exact answer for a rate added once a year.
| Example rate | Rule of 72 (years) | Exact (years) | Rule minus exact (years) | % off |
|---|---|---|---|---|
| 1% | 72.00 | 69.66 | 2.34 | +3.4 |
| 2% | 36.00 | 35.00 | 1.00 | +2.8 |
| 3% | 24.00 | 23.45 | 0.55 | +2.3 |
| 4% | 18.00 | 17.67 | 0.33 | +1.9 |
| 5% | 14.40 | 14.21 | 0.19 | +1.4 |
| 6% | 12.00 | 11.90 | 0.10 | +0.9 |
| 7% | 10.29 | 10.24 | 0.05 | +0.4 |
| 8% | 9.00 | 9.01 | −0.01 | −0.1 |
| 9% | 8.00 | 8.04 | −0.04 | −0.5 |
| 10% | 7.20 | 7.27 | −0.07 | −1.0 |
| 11% | 6.55 | 6.64 | −0.09 | −1.5 |
| 12% | 6.00 | 6.12 | −0.12 | −1.9 |
| 13% | 5.54 | 5.67 | −0.13 | −2.3 |
| 14% | 5.14 | 5.29 | −0.15 | −2.8 |
| 15% | 4.80 | 4.96 | −0.16 | −3.2 |
| 16% | 4.50 | 4.67 | −0.17 | −3.6 |
| 17% | 4.24 | 4.41 | −0.17 | −4.1 |
| 18% | 4.00 | 4.19 | −0.19 | −4.5 |
| 19% | 3.79 | 3.98 | −0.19 | −4.9 |
| 20% | 3.60 | 3.80 | −0.20 | −5.3 |
For education only, not financial advice.
Reading the table.
- Rule minus exact is the rule's figure minus the exact figure, both as shown. A positive number means the rule gives more years than the exact answer. A negative number means fewer.
- % off is that gap as a share of the exact doubling time.
As a share of the doubling time, the rule of 72 is within about 2% of the exact answer for every example rate from 4% to 12%, and is most accurate near 8%. In years, the gap is biggest at low rates.
Rule of 72 vs 70 vs 69.3
Some people divide by 70 or 69.3 instead of 72. All three give an estimate.
- 72 is used because it divides evenly by many numbers: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24 and 36. That makes the math easy to do in your head.
- 70 is close to 72 and divides evenly by 2, 5, 7 and 10.
- 69.3 comes from ln 2, which is about 0.693. It matches continuous compounding, where interest is added all the time rather than once a year. So next to a yearly exact answer it looks less accurate than it is.
Example at 6%: rule of 72, 12.00 years; rule of 70, 11.67 years; rule of 69.3, 11.55 years. The exact answer with yearly compounding is 11.90 years. With continuous compounding, the exact answer is ln 2 ÷ 0.06 = 11.55 years, the same as the rule of 69.3. For education only, not financial advice.
How often interest is added changes the exact answer. To turn a rate that compounds monthly or daily into its yearly equivalent, try the APY calculator.
Using it backwards
The rule also works the other way. Divide 72 by a number of years to estimate the yearly rate that doubles an amount in that time. This is arithmetic, not a rate you can expect to get.
| Years | Rule of 72 | Exact, compounded yearly |
|---|---|---|
| 12 | 6.00% | 5.95% |
| 10 | 7.20% | 7.18% |
| 1 | 72.00% | 100.00% |
The exact rate is 21/years − 1. Over 1 year, an amount doubles only at 100% a year, so the rule's example rate of 72% is far off. The rule is meant for lower rates and longer times.
For education only, not financial advice.
How this calculator works
The two formulas.
- Years to double, exact: ln 2 ÷ ln(1 + r).
- Rate that doubles in a given number of years, exact: 21/years − 1.
- The rules divide 72, 70 or 69.3 by the rate, or 72 by the years.
Yearly compounding. "Exact" means a fixed rate added once a year. The rule of 72 is usually stated for a yearly rate. For a rate that compounds more often, the APY calculator gives the yearly equivalent to enter here.
Rounding. Years and rates are rounded to 2 decimal places, with halves rounded up.
"Off by" and "Rule minus exact". Both are the difference of the two figures as shown, so the numbers on screen always add up. When the two figures shown are equal, the calculator says they agree to 2 decimal places instead.
"% off". The gap as a share of the exact doubling time, worked out before rounding, then rounded to 1 decimal place.
At 0%. An amount never doubles, so the calculator says so instead of showing a number.
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For education only, not financial advice. The rule of 72 is an estimate. The 6% starting rate is an example, and both answers assume a fixed rate every year. Actual returns vary and are not guaranteed.