Rule of 72 Calculator
Rule of 72 Calculator
Find how long money takes to double, triple or grow to any multiple at an annual rate, or the rate you need, with the exact logarithm beside the Rule of 72, 70 and 69.3 shortcuts. See what inflation, fees and compounding do to the answer.
Balance over time
Exact years = ln(multiple) ÷ ln(1 + rate ÷ 100). The Rules of 72, 70 and 69.3 give 72, 70 or 69.3 ÷ rate, multiplied by log base 2 of the multiple when it is not 2. Rate needed = multiple to the power (1 ÷ years) − 1. Real rate = (1 + return) ÷ (1 + inflation) − 1; after a fee it is (1 + return) × (1 − fee) − 1. A constant rate is assumed and growth is never guaranteed. The CSV, image and printed report contain the values you entered, nothing preloaded.
How to use this tool
Choose what to work out. “How long money takes to grow” takes an annual rate and a goal: double, triple, quadruple, ten times, another multiple, or a starting and a target amount. “The annual rate needed” takes the years you have instead. “Doubling after inflation or fees” gives the real or net rate. “Doubling by compounding” compares annual, monthly, daily and continuous compounding, and “Rule of 72 error by rate” shows how far the shortcut drifts. Every result gives the exact years first, with the Rule of 72, 70 and 69.3 beside it.
The input people most often get wrong is the rate. Type 8 for 8%, not 0.08: the calculator reads 0.08 as eight hundredths of one percent and answers with a doubling time of about 867 years. The second is the kind of rate. If an account quotes a rate compounded monthly or daily, use the compounding mode, and in the inflation and fee mode enter the return before the deduction, not after it.
What the calculation is doing
Money growing at a yearly rate r is multiplied by 1 + r each year. It reaches M times its start after n years when (1 + r) to the power n equals M, and taking logarithms gives n = ln(M) ÷ ln(1 + r). For doubling at 8% that is 0.693147 ÷ 0.076961 = 9.006 years. The Rule of 72 replaces the division with 72 ÷ r, where r is the rate in percent, and gives 9.
The shortcut works because ln(1 + r) is close to r when r is small, so n is close to ln(2) ÷ r, and 100 × ln(2) is 69.3. For any other multiple, scale by the number of doublings, which is log base 2 of M: tripling uses 72 × 1.585, about 114, and quadrupling is two doublings, 144. Working backwards, the rate needed is M to the power (1 ÷ n) minus 1, and the Rule of 72 gives 72 ÷ n for a doubling.
Inflation and fees reuse the same maths on a net rate. After inflation f the real rate is (1 + r) ÷ (1 + f) − 1. After an annual fee it is (1 + r) × (1 − fee) − 1, the form that reproduces the SEC fee example quoted below. Compounding m times a year replaces the annual factor with (1 + r ÷ m) to the power m, and continuous compounding gives exactly ln(2) ÷ r.
Why 72 works, and where it stops working
The natural logarithm of 2 is 0.693147. If growth were continuous, doubling would take exactly 69.3 ÷ r years at any rate. Interest added once a year is slightly slower than continuous growth, so the constant the rule needs drifts upward with the rate. That constant, p × ln(2) ÷ ln(1 + p ÷ 100) for a rate of p percent, is 69.66 at 1%, 72.05 at 8% and 76.04 at 20%. The Rule of 72 is exact at 7.85% and stays within 1% of the truth from 5.74% to 10.01%, which fits the range of 6% to 10% given in Stanford’s course note on the rule.
Why 72 and not 71 or 73? Because 72 divides evenly by 1, 2, 3, 4, 6, 8, 9 and 12, which covers the rates people actually meet: 72 ÷ 6 = 12 and 72 ÷ 9 = 8 need no calculator. Stanford’s note and Wikipedia both give that reason: 72 is a convenient numerator. The number 70 divides by 2, 5, 7 and 10 and is closer to the truth at low rates. Of the three constants in the tool, 69.3 is nearest below 1%, 70 from 1% to 4.9%, and 72 above 4.9%.
Outside the middle range the errors grow quickly. At 1% the rule says 72 years against an exact 69.66, which is 3.4% too long. At 20% it says 3.6 years against 3.802, which is 5.3% too short, and at 50% it says 1.44 against 1.71. At 100% it gives 0.72 years when money doubles in exactly one, 28% short. Rutgers Cooperative Extension calls the rule accurate below about 20%, and the table below shows what that allows: an error near 5%.
There is a simple fix. Stanford’s note suggests adjusting 72 by 1 for every three points the rate is away from 8%, so 70 at 2%, 74 at 14% and 76 at 20%. Checked against the exact answer at every whole rate from 1% to 30%, that adjusted rule is never off by more than 0.1%. The tool lists it as “Adjusted rule”.
Compounding frequency moves the answer too. At 8% a year, doubling takes 9.006 years compounded annually, 8.693 monthly, 8.665 daily and 8.664 continuously. The Rule of 72 suits annual compounding. The Rule of 69.3 gives 8.6625, within 0.03% of the continuous figure, so it is the better shortcut for daily or continuous compounding.
What a point of return, inflation and fees are worth
One point of return sounds small and is not. Money doubles in 17.67 years at 4% and 14.21 years at 5%, which is 3.47 years sooner. From 7% to 8% the saving is 1.24 years, and from 9% to 10% it is 0.77. Over 30 years, $10,000 at a steady 6% becomes $57,435 and at 7% becomes $76,123, about a third more.
The same arithmetic runs in reverse for prices. At 3% inflation, prices double and cash loses half its buying power in 23.45 years; the rule gives 24, and Washington State’s Department of Retirement Systems quotes that figure for 3%. Even at the Bank of Canada’s 2% target, idle cash loses half its buying power in 35 years.
A return has to beat inflation to grow in real terms. A 7% return with 3% inflation is a real rate of 3.88%, not 4%, because 1.07 ÷ 1.03 − 1 = 0.03883. Buying power doubles in 18.19 years, against 10.24 years for the dollars. Subtracting first, 7 − 3 = 4 and 72 ÷ 4 = 18, is within 1.1% of that, which is good enough for a quick check.
Fees act like a lower return, and they compound against you. A 1% annual fee on an 8% return leaves 6.92% and stretches doubling from 9.006 to 10.36 years. The SEC’s Investor.gov page on fees shows the effect in dollars: $100,000 growing at 4% for 20 years reaches about $208,000 with a 0.25% ongoing fee, $198,000 with 0.50% and $179,000 with 1.00%. The tool’s fee mode reproduces those figures to the nearest thousand. This is not investment advice, and growth rates are never guaranteed.
Doubling time by rate, and how far the shortcuts are off
Exact years to double at each annual rate, with the Rule of 72 estimate and the error of each of the three rules. A positive error means the rule gives too many years and a negative one too few.
| Rate | Exact years | Rule of 72 | Error of 72 | Error of 70 | Error of 69.3 |
|---|---|---|---|---|---|
| 1% | 69.66 | 72.00 | +3.36% | +0.49% | −0.52% |
| 2% | 35.00 | 36.00 | +2.85% | −0.01% | −1.01% |
| 3% | 23.45 | 24.00 | +2.35% | −0.50% | −1.49% |
| 4% | 17.67 | 18.00 | +1.85% | −0.98% | −1.97% |
| 5% | 14.21 | 14.40 | +1.36% | −1.45% | −2.44% |
| 6% | 11.90 | 12.00 | +0.88% | −1.93% | −2.91% |
| 7% | 10.24 | 10.29 | +0.40% | −2.39% | −3.37% |
| 8% | 9.006 | 9.000 | −0.07% | −2.85% | −3.82% |
| 9% | 8.043 | 8.000 | −0.54% | −3.30% | −4.27% |
| 10% | 7.273 | 7.200 | −1.00% | −3.75% | −4.71% |
| 12% | 6.116 | 6.000 | −1.90% | −4.63% | −5.58% |
| 15% | 4.959 | 4.800 | −3.22% | −5.90% | −6.85% |
| 18% | 4.188 | 4.000 | −4.49% | −7.14% | −8.07% |
| 20% | 3.802 | 3.600 | −5.31% | −7.94% | −8.86% |
| 25% | 3.106 | 2.880 | −7.28% | −9.86% | −10.8% |
| 30% | 2.642 | 2.400 | −9.16% | −11.7% | −12.6% |
| 40% | 2.060 | 1.800 | −12.6% | −15.1% | −15.9% |
| 50% | 1.710 | 1.440 | −15.8% | −18.1% | −18.9% |
| 100% | 1.000 | 0.720 | −28.0% | −30.0% | −30.7% |
A worked example you can check on paper
Doubling at 8%, the tool’s starting values. ln(2) = 0.693147 and ln(1.08) = 0.076961, so n = 0.693147 ÷ 0.076961 = 9.006 years, which is 9 years and 0.08 of a month. The Rule of 72 gives 72 ÷ 8 = 9, which is 0.07% low. The Rule of 70 gives 8.75 (−2.85%) and the Rule of 69.3 gives 8.66 (−3.82%), so 72 is the closest of the three.
Check by compounding. 1.08 to the power 9 is 1.999, just short of 2, and 1.08 to the power 10 is 2.159, so the balance passes double during year 10. The extra 0.006 of a year, about two days, closes the gap.
Rate needed to double in 10 years: 2 to the power (1 ÷ 10) is 1.071773, so the rate is 7.177%. The Rule of 72 says 72 ÷ 10 = 7.2%, which is 0.32% high, and at 7.2% the balance reaches 2.0042 times its start after 10 years.
Halving at 3% inflation: (1 ÷ 1.03) to the power n = 0.5 gives n = ln(2) ÷ ln(1.03) = 0.693147 ÷ 0.029559 = 23.45 years. The rule says 72 ÷ 3 = 24, 2.35% high.
Mistakes that catch people out
Typing the rate as a decimal. Enter 8 for 8%. The box reads 0.08 as 0.08%, and the doubling time comes out near 867 years.
Trusting 72 at every rate. Between about 6% and 10% it is excellent. At 1% it runs 3.4% long and at 20% 5.3% short, so use 70 at low rates, the adjusted rule at high ones, or read the exact figure.
Ignoring inflation and fees. A 7% return is not 7% growth in buying power. After 3% inflation it is a real 3.88% and doubles in 18.19 years, not 10.24. A 1% fee adds 1.79 years to a 7% doubling.
Using 72 for tripling. The plain rule answers doubling only. Tripling at 6% takes 18.85 years, not 12, and the scaled rule 114 ÷ 6 gives 19.02. The tool scales the rule for any multiple you enter.
Reading the result as a promise. The calculator assumes one rate in every year. Real returns rise and fall, so the years show what the maths implies, not what will happen.
Popular calculations
Each opens the calculator with those figures filled in, and they can be changed once it is open. Rates are compounded once a year unless the compounding group says otherwise.
Doubling time by rate
Rate needed to double
Triple, ten times and target amounts
Inflation and fees
Compounding frequency
How accurate the Rule of 72 is
How this was checked
The closed form was compared with brute force. 1,440 pairs of rate and multiple, from 0.5% to 60% and from 1.5 times to 25 times, were compounded year by year and again month by month until the balance reached the target, and the year and month found always enclosed the formula’s answer. Another 252 combinations were run period by period for annual, semiannual, quarterly, monthly, weekly and daily compounding. The error of the Rule of 72 was checked at every whole rate from 1% to 100% against the hand derived expression 72 × ln(1 + p ÷ 100) ÷ (p × ln 2) − 1, and agreed to rounding error. The point where the error is zero, 7.85%, was confirmed by a separate bisection.
Published figures were reproduced. Stanford’s EE204 note gives $100 at 9% doubling in 8.0432 years. Wikipedia’s table gives 69.661, 9.006 and 8.043 years at 1%, 8% and 9%, with the Rule of 70 and 69.3 columns. The Federal Reserve Bank of St. Louis works $5,000 at 8% for 36 years to about $79,840 against the rule’s $80,000, and the calculation here gives $79,841. The SEC’s fee figures were matched above, and the 24 years quoted for 3% inflation is the rule’s answer, against an exact 23.45. Every address in the lists above was opened and checked for a distinct title, the right answer, a chart that renders, and clean exports.
Common questions
How does the Rule of 72 work?
Divide 72 by the annual rate in percent to estimate the years for money to double. At 8%, 72 ÷ 8 = 9 years, against an exact 9.006. Money that doubles in n years at rate r satisfies (1 + r) to the power n = 2, so n = ln(2) ÷ ln(1 + r), and ln(1 + r) is close to r when r is small.
Why is it 72 and not 70 or 69.3?
72 divides evenly by 1, 2, 3, 4, 6, 8, 9 and 12, so the sums are easy at common rates, and it sits close to the constant the exact answer needs between about 6% and 10%. The Rule of 70 is closer below about 5%, and 69.3 fits continuous compounding.
How accurate is the Rule of 72?
It is exact at 7.85%, within 1% of the true doubling time from 5.74% to 10.01% and within 2% from 3.70% to 12.22%. Below that range it runs long, by 3.36% at 1%. Above it, it runs short, by 5.31% at 20% and 28% at 100%.
How do I use the Rule of 72 for inflation?
Divide 72 by the inflation rate to find the years for prices to double, which is also the time for cash to lose half its buying power. At 3% the rule gives 24 years and the exact figure is 23.45; at 2% it gives 36 and 35.00; at 4% it gives 18 and 17.67. For an investment, take inflation off the return first, or use the doubling after inflation mode.
What is the Rule of 72 for tripling or ten times growth?
Multiply 72 by the number of doublings, which is log base 2 of the multiple. Tripling is 72 × 1.585, about 114, divided by the rate. Ten times is 72 × 3.32 = 239. At 7%, growth of ten times takes 34.03 years and the scaled rule says 34.17.
Does compounding frequency change the Rule of 72?
Slightly. At 8% a year, doubling takes 9.006 years compounded annually, 8.693 monthly, 8.665 daily and 8.664 continuously. The Rule of 72 suits annual compounding, and the Rule of 69.3 fits continuous compounding, giving 8.6625 years against an exact 8.664.
Related Calculators and Reference
- CAGR Calculator: the growth rate you actually earned, which is the rate this tool asks for.
- Inflation Calculator: what a past or future amount buys once prices have moved.
- Simple and Compound Interest Calculator with Contributions: add regular deposits to the doubling picture.
- APY Calculator: turn a quoted rate and compounding into the yield a year really earns.
- Present Value Calculator: what a future amount is worth today at a chosen rate.
- CD Calculator: see what a certificate of deposit earns over a fixed term.
- Federal Reserve Bank of St. Louis: How Does Compound Interest Work? The Rule of 72 explained with a $5,000 example.