Rule Of 72 Calculator

What this rule of 72 calculator does

Divide 72 by your interest rate and you get, roughly, how many years it takes your money to double. That is the whole rule. It is the most useful piece of mental arithmetic in personal finance and you can do it in a queue.

This calculator gives you two numbers rather than one. It shows you what the shortcut says, and it shows you the exact answer worked out properly with logarithms, side by side.

Most rule of 72 calculators only do the first bit, which is odd when you think about it, because they are performing a division you could have done in your head and hiding the real answer they were perfectly capable of computing. Seeing both is how you learn when to trust the shortcut and when to stop.

Runs entirely in your browser. Nothing stored, nothing sent.

How to use it

One field. Type your annual rate as a percentage, so 8 rather than 0.08, and press Calculate.

You get back:

• Exact answer. Worked out as ln(2) ÷ ln(1 + r), which is the real doubling time under annual compounding.
• Rule of 72 estimate. Simply 72 ÷ r.

Press Reset to clear it. Whatever rate you use, it needs to be a compound rate, because the whole rule assumes your returns are being reinvested. On simple interest money doubles when t = 1 ÷ r instead, which at 5 percent is a flat 20 years rather than 14.

Where the number 72 actually comes from

Fair question, and the honest answer is more interesting than most explanations let on. Because mathematically, 72 is the wrong number.

Start from the real thing. Your money doubles when (1 + r) raised to the power of t equals 2. Take logs of both sides and you get:

t = ln(2) ÷ ln(1 + r)

Now, when r is small, ln(1 + r) is very close to r itself. So the whole thing collapses to t = ln(2) ÷ r. And ln(2) is 0.6931. Multiply through by 100 to work in percentages and the natural constant sitting at the heart of this is:

69.3147

Not 72. So why does everybody say 72?

Two reasons, and they are both good ones. The first is that 69.3 is the constant for continuous compounding. Most money compounds annually or monthly, not continuously, and that pushes the right divisor up a bit. Around the rates people actually care about, the ideal divisor drifts from about 69.7 at 1 percent to 72 at 8 percent to 73.4 at 12 percent. So 72 sits comfortably in the middle of the useful range.

The second reason is the one I like. Look at what divides cleanly into 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36. Eleven whole number divisors before you even reach 72 itself. Now look at 69: just 1, 3 and 23.

The entire point of a rule of thumb is that you can do it standing up without a calculator. A number that divides neatly by 2, 3, 4, 6, 8, 9 and 12 is worth far more in practice than a number that is 4 percent more accurate and impossible to divide. Precision lost, usability won. Whoever settled on 72 understood exactly what the tool was for.

How close the shortcut gets

Here is the estimate against the truth, across the range of rates you might plausibly meet:

Read the last column and the pattern is clear. Below about 8 percent the shortcut is generous and tells you doubling takes longer than it does. Above 8 percent it is optimistic and tells you doubling happens sooner. The crossover, where 72 divided by the rate is almost exactly right, sits at about 7.85 percent.

Between roughly 4 and 15 percent the error stays under two months, which is nothing. Below 2 percent it starts to matter, and above 25 percent you should stop using it entirely, which is fine because nobody needs a shortcut to know that money doubling every two and a half years is remarkable.

So the shortcut is at its best precisely where ordinary investing lives. That is not luck, it is why the number was chosen.

A worked example: 9 percent

Say a fund has averaged 9 percent a year and you want to know how long your money takes to double.

The shortcut: 72 ÷ 9 = 8 years. Which is a nice clean division, and part of why 9 percent is such a satisfying number to work with.

The exact answer: ln(2) ÷ ln(1.09) = 0.6931 ÷ 0.0862 = 8.04 years.

Two weeks apart, on an eight year forecast. You would not adjust a single decision because of it.

The useful part is what happens when you keep going. If it doubles every eight years, then across a forty year working life it doubles five times. Two, four, eight, sixteen, thirty two. So money you put away at twenty five turns into thirty two times as much by sixty five, before you have added a single further deposit.

That is the real point of the rule. Not the precision, but the fact that it turns compounding into something you can count on your fingers.

Running it the other way round

The rule works just as well in reverse, and this is the version people forget.

If you know how long you have, divide 72 by the years instead and you get the rate you would need:

Rate needed = 72 ÷ years to double

Want to double in 6 years? You need about 12 percent a year. In 10 years? About 7.2 percent. In 20 years? About 3.6 percent, which most decent savings accounts have managed at some point.

This is the version worth using when somebody is selling you something. If an offer promises to double your money in three years, that is 72 ÷ 3 = 24 percent a year, every year, compounding. Now you have a number you can be sceptical about, rather than a promise you have no handle on.

If you want the rate to hit a target that is not exactly double, our savings interest rate calculator solves for it properly.

Using it on things that are not investments

Anything growing at a steady percentage doubles on the same schedule, which makes this rule far more portable than its finance reputation suggests.

Inflation. At 6 percent, prices double in about 12 years. Run that against a pension or a salary that is not rising and you can see the problem immediately. This is probably the most sobering use of the rule.

Debt you are not paying down. A credit card at 24 percent doubles what you owe in roughly 3 years if you leave it. The rule is unpleasantly good at making that vivid.

Populations, users, subscribers. Anything compounding at a steady rate. A city growing 3 percent a year doubles in about 24 years, which is roughly one generation, and that is how long the infrastructure has to last.

For continuously compounding processes, some people switch to 69 or 70 rather than 72, since those are closer to the true constant of 69.31. In practice the difference is small enough that using 72 everywhere and remembering it runs a touch generous at low rates is perfectly good.

Five hundred years of this

The earliest known appearance of the rule is in Summa de arithmetica, geometria, proportioni et proportionalita, published in Venice in 1494 by Luca Pacioli, a Franciscan friar who taught mathematics, collaborated with Leonardo da Vinci, and is generally credited with the first printed description of double entry bookkeeping.

The interesting detail is that Pacioli states the rule and does not derive it. He simply uses it, in the manner of something everybody already knew. Which suggests merchants had been dividing things into 72 for a good while before anybody bothered to write it down.

Worth adding, since the internet insists otherwise, that Albert Einstein had nothing to do with it. The rule predates him by four centuries. The quote about compound interest being the most powerful force in the universe has never been traced to him either.

It has survived logarithm tables, slide rules, pocket calculators and spreadsheets, and people still use it. Not bad for a division.

Questions people ask

How long does it take to double money at 7 percent?

About 10.3 years. The shortcut says 72 ÷ 7 = 10.29, and the exact answer is 10.24, so the rule is doing very well at that rate.

Is the rule of 72 accurate?

Between roughly 4 and 15 percent it is within about two months of the truth. Outside that range it drifts, generously at low rates and optimistically at high ones. The calculator above shows you both numbers so you never have to guess.

Why do some people say rule of 69 or rule of 70?

Because 69.31 is the mathematically correct constant for continuous compounding. 70 is a rounded version popular in economics for growth rates. 72 wins in everyday use because it divides by so many numbers.

Does it work with simple interest?

No. Simple interest doubles when t = 1 ÷ r, so 5 percent takes 20 years rather than the 14.4 the rule would suggest. The rule assumes returns are reinvested.

Is there a rule for tripling?

Yes, the rule of 114, on exactly the same logic since 100 × ln(3) is about 109.9 and 114 divides more conveniently. Quadrupling is the rule of 144, which is just doubling twice.

What if my return varies year to year?

Use your average annual compound return rather than the arithmetic average of the yearly figures. Those two are not the same number, and the compound one is always the lower.

References

A note on sourcing. The exact doubling formula and the logarithmic approximation behind the rule are standard results in the time value of money literature. The historical attribution rests on the first known printed appearance of the rule in Pacioli's 1494 arithmetic.

1. Pacioli, L., Summa de arithmetica, geometria, proportioni et proportionalita, Venice: Paganino de Paganini, 1494.
2. OpenStax, Principles of Finance, Section 7.2, Time Value of Money Basics. https://openstax.org/books/principles-finance/pages/7-2-time-value-of-money-tvm-basics
3. Kellison, S. G., The Theory of Interest, 3rd edition, McGraw-Hill, 2008.
4. U.S. Securities and Exchange Commission, Office of Investor Education and Advocacy, Compound Interest Calculator. https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator