Rule of 72

In one sentence: The rule of 72 says that dividing 72 by a yearly percentage rate gives a quick estimate of the number of years it takes money to double.

Read it out loud: “t is approximately equal to 72 divided by R. t is the number of years for money to double, and R is the yearly interest rate written as a plain percent, so 6 for 6 percent.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Percent

A percent is a number out of 100, and the sign % means divide by 100. In this rule the rate is written as a plain percent number, so 6% is written 6. The other pages in this group use decimals such as 0.06, so check which form a formula expects before you start.

Approximately equal (≈)

The symbol $\approx$ means approximately equal to. It says the two sides are close but not exactly the same. The rule gives a close estimate, so we write $t \approx 12$, not $t = 12$, because the exact answer for 6% is 11.9 years.

Natural logarithm (ln)

The natural logarithm, written ln, answers the question: what power must the number e be raised to, to get this number? So $\ln 2 \approx 0.693147$ means e to the power 0.693147 gives 2. A calculator has an ln key. You need it only for the exact doubling time, not for the rule itself.

Doubling

To double an amount is to make it twice as large. If 1000 dollars becomes 2000 dollars, the money has doubled. The doubling time is the number of years this takes at a steady yearly rate, and the rule of 72 estimates it.

What each symbol means

The formula piece by piece

$R$

The yearly rate in percent, written as a plain number. For 6% a year, R is 6. A bigger R means faster growth, so the money doubles sooner.

$72$

A fixed number chosen for the rule, close to 69.3 and easy to divide. It has many small divisors such as 2, 3, 4, 6, 8, 9 and 12, which makes the division quick to do in your head.

$\frac{72}{R}$

Divide 72 by the rate, which gives the doubling time in years. For R equal to 6 this gives 12, and for R equal to 12 it gives 6, so the faster rate doubles money in half the time.

$t \approx \frac{72}{R}$

The whole rule in one sentence: the years to double are approximately 72 divided by the yearly rate in percent.

What the formula is doing, and why

The rule of 72 is a quick mental shortcut for growth. Money that grows by a fixed percentage each year doubles after a certain number of years, and the rule gives that number by dividing 72 by the rate. You do not need a calculator, and the answer is close enough to plan with. The formula is $t \approx \frac{72}{R}$, where $R$ is the yearly rate in percent.

Why divide? The faster the rate, the sooner money doubles, so the doubling time must shrink as the rate grows. Dividing by $R$ does exactly this: a rate twice as large gives a time half as long. At 6% the rule says 12 years, and at 12% it says 6 years.

Where the 72 comes from. For a rate $r$ written as a decimal, the exact doubling time is $t = \frac{\ln 2}{\ln(1 + r)}$, where ln is the natural logarithm. For small rates, $\ln 2$ is about 0.693 and $\ln(1 + r)$ is about $r$, so $t$ is about 0.693 divided by $r$. Writing the rate in percent multiplies it by 100, so the number becomes 69.3 divided by $R$. The rule uses 72 instead, because 72 has many small divisors, such as 2, 3, 4, 6, 8, 9 and 12, and it fits yearly compounding at moderate rates a little better.

How good is it? The rule is most useful for moderate rates, and it becomes less accurate for very high or very low rates. The example compares it with the exact answer, so you can see the size of the error for yourself.

Working backwards. The same rule can estimate the rate you need. If you want money to double in 10 years, you need a rate of about 72 divided by 10, which is 7.2% per year.

An everyday picture

Think of a quick estimate at the checkout, such as rounding a price to the nearest 10 dollars to see whether you can afford something. You do one easy step in your head, and the answer is close enough to make a decision, even though a careful calculation would give a slightly different number. The rule of 72 gives a rough, fast answer for doubling time in the same way.

Worked example: How long does 1000 dollars take to reach 2000 dollars at 6%?

A savings account pays 6% per year, with interest added each year. We want to know how many years it takes for 1000 dollars to grow into 2000 dollars, which is doubling. In the rule, the rate is $R = 6$, and we want the doubling time $t$ in years.

1. Write the rule and put in the rate. The rule says the years to double equal 72 divided by the yearly rate in percent. The rate is 6 percent, so we write $t \approx \frac{72}{6}$. We use 6 here, not 0.06, because the rule works with the plain percent number.
2. Divide. Dividing 72 by 6 gives $t \approx 12$ years. The rule therefore says that 1000 dollars at 6% a year should reach about 2000 dollars after 12 years.
3. Check the answer by growing the money. Each year the balance is multiplied by 1.06, so after 12 years the balance is $1000 \times 1.06^{12} \approx 2012.20$ dollars. That is more than 2000 dollars, so the money has doubled a little before the 12th year is over, which agrees with the rule.
4. Find the exact doubling time. The exact answer uses logarithms: $t = \frac{\ln 2}{\ln 1.06}$, where $\ln 2 \approx 0.693147$ and $\ln 1.06 \approx 0.058269$. Dividing gives $t \approx 11.90$ years. The rule’s 12 years is about 0.10 years too many, which is roughly 1.2 months.
5. Try a higher rate. At 18% per year, the rule gives $\frac{72}{18} = 4$ years, but the exact answer is $\frac{\ln 2}{\ln 1.18} \approx 4.19$ years. The gap is now about 0.19 years. The rule is still close, but it is less close than at 6%, which is the pattern the explanation described.

Answer: At 6% a year, 1000 dollars takes about 12 years to double to 2000 dollars by the rule, and about 11.9 years exactly. The rule is a fast estimate, and it is close enough at moderate rates to be useful for planning.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

• Using 0.06 instead of 6. The rule takes the rate as a plain percent number, so 6% goes in as 6. With 0.06 the rule gives 72 divided by 0.06, which is 1200 years, far too long.
• Treating the estimate as the exact answer. In the example the rule is about 0.10 years off at 6%. When you need the precise time, use the logarithm formula instead.
• Using the rule for simple interest. Simple interest does not compound, so money doubles in 100 divided by the rate, in years. The 72 rule belongs to compound growth.
• Turning the rule around the wrong way. Years equal 72 divided by the rate, so a rate of 6 gives 12 years. Dividing the rate by 72 gives 0.083, which is not a number of years.

Where you will meet it

• Comparing savings accounts and investments at a glance, to see how quickly money could double at each quoted yearly rate.
• Estimating how long prices take to double when inflation runs at a steady yearly rate, a common quick check in economics classes.
• Seeing how fast a credit card debt grows when no payments are made, since unpaid interest is added to the balance.
• Judging population or business growth, where a steady yearly percentage increase doubles the size over time.

Questions people ask

Why 72 and not 70 or 69.3?

69.3 is the exact figure for continuous growth in percent terms. But 72 has more small divisors, so it is easier to divide in your head, and it is a closer match for yearly compounding at moderate rates.

Can I use the rule to find the rate?

Yes. Turn it around: the rate in percent is about 72 divided by the number of years. To double money in 9 years, you need about 72 divided by 9, which is 8% a year. The exact rate for a 9-year doubling is about 8.0% a year, so the estimate is very close.

Does the rule work for tripling or other goals?

The same idea works, but the number changes. For tripling, the exact time is $\frac{\ln 3}{\ln(1 + r)}$, and in percent form the constant is about 110, not 72. Use the exact logarithm formula for those cases.

Does it work when interest is added monthly?

It is an estimate built for yearly compounding. Monthly compounding makes money grow a little faster, so the true doubling time is a little shorter than the rule gives, and the rule’s answer runs slightly long.

Related formulas

• Compound Interest: Compound interest says a savings balance grows faster over time, because each round of interest is added to the balance and then earns interest of its own.
• Continuous Compounding: Continuous compounding says money grows as if interest were added every instant, and the number e, about 2.718, sets the size of that growth.
• Simple Interest: Simple interest says the interest is the starting amount times the yearly rate times the number of years, and interest never earns interest of its own.

Sources

1. Rule of 72, Wikipedia
2. Doubling time, Wikipedia