The rule of 72: how many years until your money doubles
The rule of 72 estimates how many years money takes to double at a fixed yearly rate. Divide 72 by the rate in percent: at 6% a year, 72 ÷ 6 = 12 years. Across the usual range from 2% to 20% the answer is within about a year of the exact figure. The rule ignores fees, tax and inflation, so it describes how growth compounds, not what you will keep.
Compound interest calculatorWhat will my savings grow to?OpenWhere the number comes from
Doubling means the balance is multiplied by 2. If the yearly rate is r, after t years the multiplier is (1 + r) to the power t. Set that equal to 2 and take logarithms, and you get t = ln 2 ÷ ln(1 + r). The natural logarithm of 2 is about 0.693. For small rates, ln(1 + r) is very close to r, so t is roughly 0.693 ÷ r. Write the rate as a percentage instead, and that becomes about 69.3 ÷ the rate in percent.
So the figure that matches the mathematics most closely is 69.3, not 72. The number 69.3 is exact for continuous compounding, which means interest added every instant. Using 69 is a convenient near-match for that case. At a rate of 1%, the exact doubling time is 69.7 years. The rule gives 72, about 3% too long, while 69.3 is about half a percent short.
The doubling time does not depend on the amount. Starting with 1,000 or 1,000,000 at 6% takes the same 11.9 years to double with yearly compounding, because the multiplier is the same in both cases.
The rule uses 72 because it is easier to divide with. It has more factors than 69 or 70. It divides evenly by 2, 3, 4, 6, 8, 9 and 12, so the common rates of 3%, 4%, 6%, 8%, 9% and 12% give whole numbers of years. For yearly compounding near 8%, 72 is also a good fit: the exact figure at 8% is 9.0 years, and the rule gives 9.
How accurate it is
| Yearly rate | Exact years | Rule of 72 | Rule minus exact |
|---|---|---|---|
| 2% | 35.0 | 36.0 | +1.0 |
| 6% | 11.9 | 12.0 | +0.1 |
| 10% | 7.3 | 7.2 | −0.1 |
| 20% | 3.8 | 3.6 | −0.2 |
The exact figures use ln 2 ÷ ln(1 + rate), rounded to one decimal place. At 6% the rule is within a tenth of a year. At 2% it overstates by about a year, because 72 is larger than the 69.3 that fits small rates. At 20% it understates by about 5%. The approximation ln(1 + r) ≈ r gets worse as r grows: ln(1.2) is 0.182, not 0.2, so the true doubling time is longer than the rule suggests.
Checking it with the calculators
Open the compound interest calculator. Enter 10,000 as the starting amount, leave the monthly contribution at 0, enter 6% as the rate and set the years to 15. The year-by-year table shows when the balance first passes 20,000.
The calculator compounds monthly by default. Monthly compounding doubles 6% money a little sooner, at about 11.6 years instead of 11.9. Both settings first show a balance above 20,000 in the year-12 row of the table, but the monthly setting gets there a few months earlier.
To check one year by hand, use the percentage calculator in its add mode. Adding 6% to 10,000 gives 10,600. Adding 6% to that gives 11,236. Each year multiplies the balance by 1.06, and the step-by-step shows the same thing. If the rule and the exact table ever disagree, trust the table. The rule is for a quick estimate in your head. The starting amount does not change the timing. With 50,000 instead of 10,000, the balance still passes 100,000 in the year-12 row, because doubling is a multiplier, not a fixed sum.
Reversing the rule
The rule also works backwards. To double money in 10 years, divide 72 by 10 to get 7.2% a year. The exact rate is 2 to the power of 1/10, minus 1, which is 7.18%. The rule is a quick sense check: a return of 7.2% a year doubles money in about ten years, and a return of 3% takes roughly 23 to 24 years.
The same reasoning answers a goal stated as a number of years. To double money in six years, divide 72 by 6 to get 12% a year. The exact rate is 2 to the power of 1/6, minus 1, which is 12.25%. The rule lands within a quarter of a point, which is close enough for planning a savings goal, though not for setting a contract rate.
Edge cases
- Fees, tax and inflation. Fees lower the rate you really get. Tax lowers the return you keep. Inflation lowers what the money buys when it doubles. A 6% return with 2% inflation leaves growth of about 4% in real terms, and the rule then points to about 18 years instead of 12.
- Losses. The rule is written for growth. A fall of 10% followed by a rise of 10% does not return you to the start: 100 × 0.9 × 1.1 is 99. The percentage mistakes guide explains why each change applies to a different base.
- Very high rates. Above about 20% the rule drifts further from the exact figure. At 30% it gives 2.4 years, against an exact 2.6.
- Regular deposits. The rule assumes one lump sum and no further deposits. Regular contributions grow faster, and the monthly savings guide works through that case.
- Estimates, not guarantees. Savings rates and investment returns change. The rule uses the rate you put in, so treat the result as an estimate of the arithmetic, not a promise. Check whether a quoted rate is a yearly or a monthly figure, and how often it compounds, before you apply the rule.
