ABC Tools

The Rule of 72, Explained

After reading this you can estimate how long an investment takes to double in your head, know how far off that estimate is, and compute the exact answer when the shortcut is not good enough.

What the rule of 72 is

The rule of 72 is a mental shortcut for one question: how many years does it take for money growing at a fixed rate to double? You divide 72 by the annual rate written as a whole number. At 8% per year, 72 / 8 = 9 years. At 6%, 72 / 6 = 12 years. That is the whole method.

Here is the hook. A friend tells you a fund has returned 9% a year. Without a calculator, you know it doubles in about 72 / 9 = 8 years. Double again in another 8 years, quadruple in 16, eight times over in 24. So 30 years of 9% turns roughly into 10x, and you worked that out at the dinner table.

The catch is that the true doubling time comes from a logarithm, not a division. The rule of 72 is an approximation of that exact answer. It is close in the range most people care about, and this article shows exactly how close.

When to use it and when not to

Use the rule of 72 for quick sanity checks and rough planning. If you want to feel the shape of compound growth, or catch a claim that sounds wrong, the shortcut is enough. It also works in reverse: if you need money to double in 10 years, you need roughly 72 / 10 = 7.2% per year.

Do not use it for contracts, loan schedules, or any number someone will hold you to. It ignores fees, taxes, irregular contributions, and compounding that happens more often than yearly. For those, reach for an exact tool such as the Compound Interest Calculator or the Time Value of Money Solver.

The rule assumes a single fixed rate that never changes and no money added or removed. Real portfolios do none of these. Treat every rule-of-72 answer as an order-of-magnitude estimate, not a promise.

The formula and why 72

Start from exact compounding. If you invest an amount and it grows at rate r per period, after n periods you hold (1+r)^n times the start. Doubling means that factor equals 2:

(1+r)^n = 2

Take natural logs of both sides and solve for n. This is the exact doubling time.

n = \frac{\ln 2}{\ln(1+r)}

Here n is the number of years and r is the rate as a decimal, so 7% is 0.07. The value \ln 2 \approx 0.6931.

For small r, the denominator \ln(1+r) \approx r. So the exact answer is close to 0.6931 / r. Writing the rate as a percentage p where r = p/100, that becomes 69.31 / p. So the "true" constant near zero is about 69.3, not 72.

n \approx \frac{72}{p}

Why 72 instead of 69.3? Two reasons. First, 72 has many whole divisors (1, 2, 3, 4, 6, 8, 9, 12), which makes the division easy in your head. Second, the extra bit above 69.3 corrects for the fact that \ln(1+r) is smaller than r, which pushes the true answer up as the rate rises. The number 72 lands the approximation right in the middle of common rates. It is most accurate near 8%.

A worked example at 7%

Reproducing the demo (rate = 7%)

The tool's demo uses the default rate of 7%. Here is what it computes.

  1. Rule of 72 estimate: 72 / 7 = 10.29 years.
  2. Exact answer: convert to a decimal, r = 0.07.
  3. Compute the denominator: \ln(1.07) = 0.06766.
  4. Divide: 0.6931 / 0.06766 = 10.24 years.
  5. Compare: the estimate 10.29 is only 0.05 years high, about 18 days over a 10-year span, an error of roughly 0.5%.

Check the doubling by hand. At 7% for 10.24 years: 1.07^{10.24} = 2.00. At the rule's 10.29 years: 1.07^{10.29} = 2.003. Both land essentially on double, which is why the shortcut is safe here.

How accurate the estimate is across rates

The error is not constant. It is smallest near 8% and grows as you move away in either direction. The table shows the estimate, the exact value, and the gap.

Rule of 72 versus the exact doubling time
Rate72 / rateExact yearsError (years)Error (%)
2%36.0035.00+1.00+2.86
4%18.0017.67+0.33+1.87
6%12.0011.90+0.10+0.84
8%9.0009.006-0.006-0.07
12%6.0006.116-0.116-1.90
20%3.6003.802-0.202-5.31

Read the sign carefully. Below 8% the rule slightly overstates the doubling time (it says you wait longer than you really do). Above 8% it understates it. At 20% the shortcut is off by about a fifth of a year, so switch to the exact formula for high rates.

The error crosses zero near 8% (marked). It is positive for low rates and negative for high rates, and its size grows as you move away from 8%.

Without JavaScript, use the table above: at each rate you can read the rule-of-72 estimate, the exact doubling time, and the gap between them.

Interpreting and extending the result

Once you have a doubling time, chain it. Each doubling multiplies the balance by 2, so the multiple after k doublings is 2^k. At 8% (double every 9 years): 9 years is 2x, 18 years is 4x, 27 years is 8x, 36 years is 16x. Notice how the last 9 years add far more absolute value than the first, which is the point people mean when they say compounding rewards patience.

You can also flip the rule to find a required rate. If a goal must double in 6 years, you need about 72 / 6 = 12% per year. Whether 12% is realistic is a separate question the rule cannot answer.

For tripling instead of doubling, use 114 instead of 72 (since \ln 3 \approx 1.0986 and 1.0986 \times 104 \approx 114 after the same accuracy adjustment). For quadrupling, just double the doubling time.

Common mistakes

The errors below all come from misreading what the rule takes as input.

Using a decimal instead of a whole percent
The formula is 72 divided by the rate as a whole number. At 7% you divide by 7, not by 0.07. Dividing by 0.07 gives 1029 years, which should tell you something went wrong.
Applying it to a rate that is not annual
If your rate is monthly, the answer comes out in months. A 1% monthly rate doubles in about 72 months, not 72 years. Keep the time unit consistent with the rate's period.
Ignoring compounding frequency
The rule works on an effective periodic rate. A nominal 6% compounded monthly is not the same as 6% once a year. Convert first with the APY / Effective Rate Calculator, then apply 72 to the effective figure.
Trusting it at extreme rates
At 25% the rule is off by about 7%. Above 15% or so, use the exact formula.
Assuming a fixed rate exists
Stock returns vary year to year. A 7% average is not 7% every year, and the doubling time from a bumpy path can differ from the smooth-rate estimate.

Related tools

The rule of 72 answers one narrow question. These tools handle the rest:

Frequently asked questions

Why is it 72 and not 70?

The pure small-rate constant is about 69.3, so 70 is technically more accurate near 0%. But 72 divides cleanly by 2, 3, 4, 6, 8, 9, and 12, which makes mental math easy, and the slightly larger value corrects for the curve at typical rates. 72 is most accurate around 8%.

Does the rule work for continuous compounding?

For continuous growth the exact doubling time is \ln 2 / r = 0.6931 / r, so the right constant is 69.3, not 72. At 8% continuous, doubling takes 69.31 / 8 = 8.66 years.

Can I use it for inflation eating my money?

Yes, in reverse. At 3% inflation, prices double in about 72 / 3 = 24 years, meaning your cash halves in purchasing power over the same span. The exact answer is 23.4 years.

What rate should I plug in?

Use whatever rate you want to test. This guide does not recommend any figure. Long-run averages are uncertain and vary by asset, period, and fees, so treat any single number as an assumption you are exploring, not a forecast.

This article explains an approximation for education only. It is not financial advice. Before making money decisions, talk to a qualified adviser.

Try it yourself in the tool

Related tools

Compound Interest Calculator

Growth of a lump sum plus monthly contributions.

Savings Goal Calculator

Monthly saving needed to hit a target by a date.

APY / Effective Rate Calculator

Nominal ↔ effective annual rate at any compounding.

Deposit / CD Calculator

Maturity value of a fixed-term deposit.

Savings Withdrawal Calculator

How long a balance lasts under monthly withdrawals.

Simple Interest Calculator

Interest without compounding, vs the compound result.