Rule of 72 calculator and doubling time

Published August 16, 2026

The rule of 72 estimates how long money takes to double: divide 72 by the annual growth rate in percentage points. At 7 percent a year it gives 10.2857 years against an exact 10.2448, so the shortcut runs about 15 days long. It is sharpest near 8 percent and drifts at very high and very low rates.

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%

Annual growth rate

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

The formula

t \approx \frac{72}{r} \qquad t = \frac{\ln 2}{\ln(1 + r/100)}

$r$ is the annual growth rate in percentage points, so 7 rather than 0.07, and $t$ is the number of years the money takes to double. The expression on the left is the shortcut. The one on the right is the exact answer it is standing in for.

What this calculator works out

Enter one annual growth rate. It returns the rule of 72 estimate, the exact doubling time worked out from compounding, and the gap between the two, both as a length of time and as a share of the exact answer.

No starting amount is needed, and that is the useful part of the idea. Doubling depends only on the rate, so one answer covers a savings balance, a pot nobody is topping up, a debt nobody is paying down, and the price level under steady inflation. Whatever the rate is applied to, it doubles at the same moment.

The condition hiding inside that is that the sum is left alone. Deposits arriving along the way pull the doubling forward and withdrawals push it back, and how far depends on the amounts, so a balance you are still paying into doubles sooner than anything on this page says.

Where the 72 comes from

Doubling is a compound interest question. A balance growing at a rate $r$ , written as a decimal, is multiplied by $1 + r$ every year, so it has doubled at the moment $(1 + r)^t = 2$ . Take logs of both sides and solve for time:

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

That is exact, and it is not a sum anyone does at a bus stop. The shortcut drops out of it in two steps. First, $\ln(1 + r)$ is close to $r$ itself when $r$ is small, which collapses the formula to $0.6931$ divided by the rate, or 69.31 divided by the rate in percentage points.

Second, 69.31 is the figure for growth ticking over continuously. Interest credited once a year lands a little behind that, so the numerator that fits annual compounding is slightly larger, and it climbs with the rate: 70.0 at 2 percent, 72.05 at 8 percent, 76.0 at 20 percent.

72 sits in the middle of the range people actually use, and it divides cleanly by 2, 3, 4, 6, 8, 9 and 12. That is the whole reason it won. 69.31 divides cleanly by nothing, and a shortcut you cannot do in your head is not a shortcut.

Where the estimate drifts

The two methods cross just under 7.85 percent. That is the single rate where the rule of 72 is exactly right. Everywhere else it is an approximation, and the error widens in both directions:

Annual rate	Rule of 72	Exact	The rule is
2 percent	36.00 years	35.00 years	1.00 year long
5 percent	14.40 years	14.21 years	0.19 years long
8 percent	9.00 years	9.01 years	0.01 years short
12 percent	6.00 years	6.12 years	0.12 years short
20 percent	3.60 years	3.80 years	0.20 years short

Between roughly 5 and 12 percent the error is weeks rather than years: about ten weeks at the 5 percent end, six weeks at the 12 percent end, and about two weeks either side of 8 percent. That is far smaller than the error in your guess at the rate. Outside that band it is worth doing properly. At 2 percent the shortcut adds a full year to a 35 year wait. At 20 percent it takes 74 days off a wait of under 4 years.

The direction is worth holding on to. Below the crossover the rule runs long and promises a wait that compounding beats. Above it the rule runs short, so the doubling lands later than the division said. What that costs depends on what is doubling: on a savings balance a short answer flatters the plan, while on a debt it overstates how fast the balance runs away, which is the harmless direction to be wrong in. The gap widens either side of the crossover, and it is worst measured in years at the bottom of the range and worst as a share of the answer at the top.

Using it for more than one doubling

A doubling is a unit you can stack. Two doublings is four times the money and three is eight times, so a rate that doubles a balance in about 9 years turns it into 4 times in about 18 years and 8 times in about 27. That is usually a faster mental route than any formula, and stacking the doublings shows where the growth sits: the third one adds 4 times the original stake by itself, more than the 3 times the first two added between them.

The same trick has cousins. Divide 114 by the rate for the time to triple, and 144 for four times, which is just two doublings back to back. At 8 percent, 114 divided by 8 is 14.25 years against an exact 14.27.

When the decision turns on the actual dates or the actual balance rather than on a rough sense of the wait, run the real schedule. The compound interest calculator gives the balance year by year, and how compound interest works covers why the curve steepens the way it does.

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?

Divide 72 by the rate in percentage points: $72 / 7 = 10.2857$ years.
Now the exact version. Doubling means $(1.07)^t = 2$ , so $t = \ln 2 / \ln 1.07$ .
$\ln 2 = 0.6931472$ and $\ln 1.07 = 0.0676586$ , so $t = 0.6931472 / 0.0676586 = 10.2448$ years.
Subtract to size the error: $10.2857 - 10.2448 = 0.0409$ 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?

The shortcut: $72 / 8 = 9$ years, no remainder.
The exact answer: $t = \ln 2 / \ln 1.08 = 0.6931472 / 0.0769610 = 9.0065$ years.
The gap is $9 - 9.0065 = -0.0065$ of a year, a little over 2 days.
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?

The shortcut: $72 / 20 = 3.6$ years.
The exact answer: $t = \ln 2 / \ln 1.2 = 0.6931472 / 0.1823216 = 3.8018$ years.
The gap is $3.6 - 3.8018 = -0.2018$ of a year, about 74 days.
As a share of the wait, $0.2018 / 3.8018$ 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.

Dividing 72 by the wrong rate

The arithmetic is one division, so almost every bad answer comes from the number going into it.

The first trap is compounding frequency. The rule wants the rate the balance actually grows by over a full year. A nominal rate quoted with monthly compounding grows the balance by more than the quoted figure, so the money doubles sooner than 72 divided by that quoted number. 12 percent quoted monthly is 12.68 percent a year in effect: the division says 6.00 years and the balance is really there in 5.81. Convert to an effective annual rate first, then divide. In the United States, deposit accounts are advertised with an APY, which already carries the within-year compounding, while borrowing is quoted as an APR, which does not. Other countries label the two differently, so the question to put to any quoted rate is whether the compounding is already inside it.

The second trap is inflation. A pot growing at 7 percent a year doubles in about 10 years measured in dollars, which is what this page answers. What it will buy is a different question, and the rate that answers it is the growth rate after inflation. Divide 72 by a nominal rate and you get the doubling of a number rather than the doubling of what the number is worth. The real return calculator does that adjustment properly, and it is not simple subtraction.

Used on the right rate the shortcut is a sanity check you can run in a conversation. Used on a nominal monthly rate it lands roughly a third of a year late at any rate you like, on top of whatever the shortcut was already out by.

Common questions

Why 72 and not 69.3?

69.31 is the mathematically clean number, since it comes straight from the natural log of 2, but it is the figure for growth compounding continuously and it divides evenly by nothing. Annual compounding needs a slightly larger numerator, rising from about 70.0 at 2 percent to about 72.05 at 8 percent. 72 is both close to right across the usual range and divisible by 2, 3, 4, 6, 8, 9 and 12. Some people use 70 for low rates, and it is a better fit down there.

Does it work for debt and for inflation as well?

Yes. Anything compounding at a steady rate obeys the same arithmetic, and none of it depends on the amount. A balance on a card nobody is paying down doubles on the same schedule a savings pot does at the same rate. Prices under steady inflation work the same way, and prices doubling is the same event as the purchasing power of a fixed sum halving.

Can I run it backwards to find the rate?

Yes, and it is just as quick. Divide 72 by the number of years instead. To double in 12 years you need about 72 divided by 12, which is 6 percent a year, against an exact requirement of 5.95 percent. The same accuracy pattern applies: the answer is closest for waits of roughly 6 to 14 years and drifts outside that.

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.