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How the rule of 72 works

The rule of 72 estimates how long money takes to double: divide 72 by the annual growth rate in percentage points. At 7 percent it gives 10.2857 years against an exact 10.2448, about 15 days long. It is sharpest near 8 percent, where 72/8 is 9 against an exact 9.0065.

Rule of 72 doubling time

10.29 years

At 7.00% a year, compounding needs 10.24 years, so the shortcut runs 15 days long.

Exact doubling time
10.24 years
The shortcut overstates by
15 days
Error against the exact time
0.40%
%

The rate the balance actually grows by over a full year. A nominal rate that compounds monthly has to be converted first.

On this page

In short

  • The shortcut is t≈72/rt \approx 72 / r. At 7 percent that is 10.2857 years. The exact wait is ln⁡2/ln⁡(1.07)=10.2448\ln 2 / \ln(1.07) = 10.2448 years, so the rule runs 0.0409 of a year long, about 15 days.
  • At 8 percent, 72 divides evenly: 9 years against an exact 9.0065. The gap is 0.0065 of a year, a little over two days. This is where the rule is at its best.
  • At 20 percent the shortcut says 3.6 years. The exact wait is 3.8018. The rule is 0.2018 of a year short, which is the drift that shows up at high rates.
  • The 72 is a convenience that sits near 100×ln⁡2≈69.3100 \times \ln 2 \approx 69.3, nudged up so that it divides cleanly at the rates people actually quote.
  • It is a doubling-time estimate under compound interest, not a return and not a promise.

A shortcut for a logarithm

Money doubles when (1+r/100)t=2(1 + r/100)^t = 2. Solve for tt and you have a logarithm:

t=ln⁡2ln⁡(1+r/100)t = \frac{\ln 2}{\ln(1 + r/100)}

The rule of 72 replaces that with a division:

t≈72rt \approx \frac{72}{r}

At 7 percent the shortcut is 72/7=10.285772 / 7 = 10.2857 years. The exact wait is 0.6931472/0.0676586=10.24480.6931472 / 0.0676586 = 10.2448 years. The gap is 0.0409 of a year, about 15 days on a wait of more than a decade.

Nobody knows a future growth rate to four decimals, so at 7 percent the approximation costs nothing you could act on. The rule of 72 calculator on this page prints both numbers so you can see the gap rather than trust the rhyme.

This is compound interest asked as a wait. How compound interest works is the balance over time. This page is the time to double.

The shortcut answers a wait. The compound interest explorer answers the balance at that wait: drag the years toward the doubling time and read what the curve has done.

Why 72, and why it is sharpest near 8 percent

100×ln⁡2100 \times \ln 2 is about 69.3, which is the constant that would make the shortcut exact for a tiny rate. 72 is that number nudged up so that it divides cleanly by 6, 8, 9 and 12, the rates a class actually quotes.

At 8 percent, 72/8=972 / 8 = 9 years exactly as a division. The logarithm is 9.0065 years. The gap is 0.0065 of a year, a little over two days. That is the textbook case, and it is why 72 beat 69 in the mnemonic: the error is smallest around the rates the mnemonic is used for.

Where the estimate drifts

At 20 percent the shortcut says 72/20=3.672 / 20 = 3.6 years. The exact wait is 3.8018 years. The rule is 0.2018 of a year short, about ten weeks on a wait of under four years.

The rule is a linear stand-in for a curve. It runs a little long at moderate rates and a little short at high ones. Use it to size a wait in your head. Use the logarithm when the wait is the answer you will publish.

It also assumes a constant compound rate. A return that jumps around compounds at less than its average, which is a different page: volatility drag.

What this page is not doing

It is not a return, not a savings-goal payment, and not a claim that money will double. The three sheets are 7 percent (10.2857 against 10.2448), 8 percent (9 against 9.0065), and 20 percent (3.6 against 3.8018). This is educational material, not financial advice.

Worked examples

Doubling at 7 percent a year

An investment is expected to compound at 7 percent a year. What does the rule of 72 say, and how close is it to the exact answer?

  1. Divide 72 by the rate in percentage points: 72/7=10.285772 / 7 = 10.2857 years.
  2. Now the exact version. Doubling means (1.07)t=2(1.07)^t = 2, so t=ln⁡2/ln⁡1.07t = \ln 2 / \ln 1.07.
  3. ln⁡2=0.6931472\ln 2 = 0.6931472 and ln⁡1.07=0.0676586\ln 1.07 = 0.0676586, so t=0.6931472/0.0676586=10.2448t = 0.6931472 / 0.0676586 = 10.2448 years.
  4. Subtract to size the error: 10.2857−10.2448=0.040910.2857 - 10.2448 = 0.0409 of a year, which is about 15 days.

The rule of 72 says 10.2857 years and the exact answer is 10.2448 years. The shortcut is 0.0409 of a year long, about 15 days on a wait of more than a decade, and 0.4 percent of the answer. Nobody knows a future growth rate that precisely, so at this rate the approximation costs nothing.

8 percent, where the rule is at its best

8 percent is the textbook case, because 72 divides by 8 exactly. How good is the estimate right there?

  1. The shortcut: 72/8=972 / 8 = 9 years, no remainder.
  2. The exact answer: t=ln⁡2/ln⁡1.08=0.6931472/0.0769610=9.0065t = \ln 2 / \ln 1.08 = 0.6931472 / 0.0769610 = 9.0065 years.
  3. The gap is 9−9.0065=−0.00659 - 9.0065 = -0.0065 of a year, a little over 2 days.
  4. The sign has flipped. At 7 percent the shortcut was long; at 8 percent it is short.

The rule says 9 years and compounding takes 9.0065 years, so the estimate is 0.0065 of a year short, a bit over 2 days. The two methods cross just under 7.85 percent, which is why anything from 7 to 9 percent lands within about two weeks of the truth.

20 percent, where the gap shows

A balance compounds at 20 percent a year, the kind of rate that turns up on card debt rather than on a savings account. What does each method say now?

  1. The shortcut: 72/20=3.672 / 20 = 3.6 years.
  2. The exact answer: t=ln⁡2/ln⁡1.2=0.6931472/0.1823216=3.8018t = \ln 2 / \ln 1.2 = 0.6931472 / 0.1823216 = 3.8018 years.
  3. The gap is 3.6−3.8018=−0.20183.6 - 3.8018 = -0.2018 of a year, about 74 days.
  4. As a share of the wait, 0.2018/3.80180.2018 / 3.8018 is 5.3 percent, more than ten times the 0.4 percent error at 7 percent.

The rule says 3.6 years and compounding needs 3.8018 years, so the shortcut is 0.2018 of a year fast, about 74 days. At high rates it always promises doubling sooner than it arrives, and the gap grows with the rate. This is the end of the range where you should stop dividing and take the logs.

Common questions

Why 72 and not 69 or 70?

Because 72 divides cleanly by the rates people quote, and because the error is smallest near 8 percent, where 72/8 is 9 years against an exact 9.0065. The constant that would match a tiny rate is 100 ln 2, about 69.3. The mnemonic chose the nearby integer that is easy to divide.

Can I use it for inflation?

As a head-arithmetic estimate of how long a price level takes to double, yes: 72 divided by the inflation rate in percentage points. It is the same logarithm. It is not a forecast of the inflation rate.

Does the rule work for simple interest?

No. Simple interest doubles when rt = 1, so t = 1/r. At 7 percent that is about 14.3 years, not 10.3. The rule of 72 is a compound-interest shortcut.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.