The Rule of 72 estimates how many years money takes to double at a steady yearly interest rate: divide 72 by the rate.
Doubling calculator
What do you want to find?
Type a yearly rate. We show the Rule of 72 estimate beside the exact answer and tell you how far apart they are.
%
The interest or growth rate for one year, as a percent. Type 8 for 8%. Rates above zero and up to 100 work.
8%
1%30%
More options
Rule72 is the usual number. 70 and 69.3 fit lower rates better.
$
The one-time amount that starts growing. The doubling time does not depend on it, but the dollar results do.
The age when the money first goes in.
The age when the second person puts in the same amount. It has to come after the first starting age.
The age where both amounts are compared.
Your result
At 8% a year, money doubles in about
9 years
72 ÷ 8 = 9. The exact answer is 9.0 years, so the shortcut runs 0.1% short.
Doubling ladder
$1,000at 15
$2,000at 24
$4,000at 33
$8,000at 42
$16,000at 51
$32,000at 60
15203040506065
Age
Doubling ladder. At 8% a year, $1,000 from age 15 reaches $2,000 at age 24, $4,000 at age 33, $8,000 at age 42, $16,000 at age 51, $32,000 at age 60.
Doubling
Age
Amount
Start
15
$1,000
1
24
$2,000
2
33
$4,000
3
42
$8,000
4
51
$16,000
5
60
$32,000
Each step up is one doubling, placed at the age it happens.
Rule of 729 years
Exact9.0 years
Shortcut error-0.1%
How Does the Rule of 72 Work?
The Rule of 72 works by dividing 72 by the yearly rate, written as a whole number, and the answer is about how many years money takes to double. Here is the same thing in steps, then how far off it gets.
Write the rate as a whole number: 8 for 8%, not 0.08.
Divide 72 by it: 72 ÷ 8 = 9.
Read the answer as years: about 9 years until the money has doubled.
To check it, find the exact answer, ln 2 ÷ ln(1 + rate). At 8% that is 9.006 years.
The Rule of 72 estimate against the exact doubling time with yearly compounding. A plus sign means the shortcut says longer than the exact answer, and a minus sign means shorter. The rates are examples.
Yearly rate
Rule of 72 (years)
Exact (years)
Shortcut error
4%
18.0
17.7
+1.9%
6%
12.0
11.9
+0.9%
8%
9.0
9.0
-0.1%
10%
7.2
7.3
-1.0%
12%
6.0
6.1
-1.9%
The Math Behind the Rule of 72
Why does dividing by 72 work, and where does it slip?
Dividing by 72 works because 72 ÷ rate lands close to the exact doubling time, ln 2 ÷ ln(1 + rate), which is hard to do in your head, and it slips at very low and very high rates. Rapunzl built this calculator to show the shortcut beside the exact answer. The two stay within 1% of each other for yearly rates from about 5.7% to 10%, and they match best near 7.85%. Below that band the shortcut says doubling takes a little longer than it does, and above it a little less. At 20% a year it is 5.3% short, and it is off by more than 20% once the rate passes about 65%.
How much does starting at 15 instead of 25 change things?
Starting at 15 instead of 25 leaves about 2.2 times as much money at 65 in our example. Rapunzl's Saving Versus Investing module covers compounding growth, and the race on this page puts numbers on it. Two people each put in $1,000 at an assumed 8% a year and add nothing more. One starts at 15, the other at 25, and we check both at 65. The early starter ends with about $46,900 and the later one with about $21,700. The head start is 10 years, about 1.1 extra doublings, and money that doubles early gets doubled again, which is where the gap comes from. The 8% is a rate we picked because 72 divides by it evenly. It is not a forecast.
Does debt double the same way?
Yes, debt doubles the same way: interest on an unpaid balance compounds the way savings interest does, except the growing balance is a cost. Rapunzl's module The Power & Risks Of Credit covers APR and how cards charge interest. At an example APR of 24% with interest added monthly and nothing paid, a balance doubles in about 2.9 years, or 35 months. The shortcut says 3.0 and yearly compounding says 3.2. Rates that high are where the shortcut runs short, so the debt mode leads with the monthly answer. The 24% is an example, not a quote for any card.
For Teachers
Project the calculator on the board in a large-type view, then hand out a printed practice set with the answer key on page 2. The set code gives each class period its own problems. For a full lesson, pair the calculator with our Saving Versus Investing module. Our Rule of 72 worksheet post has a ready-made sheet with teacher notes, and the credit module covers the debt side.
Teacher tools
Two things for the front of the room: a big-screen view for the board and a practice set you can print.
Any word or number. The same code always prints the same practice set.
From 1 to 20. The default is 5.
Prints 5 problems, with the answer key on page 2. Enter the same set code later to print the same set again.
It depends on the yearly rate and not on the amount. Divide 72 by the rate for a quick estimate. At 4%, 6%, 8% and 12%, that is about 18, 12, 9 and 6 years. The exact answers are 17.7, 11.9, 9.0 and 6.1 years. The same time applies to $1,000 or $100,000. These rates are examples, not predictions.
Divide 72 by the yearly rate, written as a whole number. At 6% that is 72 ÷ 6 = 12 years, at 8% it is 9, and at 12% it is 6. To go the other way, divide 72 by the years: doubling in 9 years takes about 8% a year. In a spreadsheet with the rate in cell A1 as a decimal such as 0.08, the shortcut is =72/(100*A1) and the exact answer is =LN(2)/LN(1+A1).
Yes, within limits. It lands within 1% of the exact doubling time for yearly rates from about 5.7% to 10%, and it matches best near 7.85%. Outside that band it drifts. At 2% it runs about 2.8% long, at 20% about 5.3% short, and at 100% it says 0.72 years when the exact answer is 1 year. At a zero or negative rate money never doubles, so there is nothing to estimate.
At an assumed 12% a year, $100,000 doubles in about 6 years by the Rule of 72, or 6.1 years exactly. The starting amount does not change that. $50,000 left alone for 20 years at the same assumed rate grows to about $482,300, a little over 3 doublings. Both answers assume a steady rate. Real investments rise and fall, and 12% here is a number we typed in, not a forecast.
The Rule of 70 divides 70 by the rate instead of 72, and the Rule of 69.3 uses 69.3, the exact figure for continuous compounding. The Rule of 72 is popular because 72 divides evenly by 1, 2, 3, 4, 6, 8, 9 and 12, which makes mental math easy, and it fits best near 7.85%. The Rule of 70 fits lower rates better and is exact for yearly compounding near 2%. At 8% the three rules say 9, 8.8 and 8.7 years. Exact yearly compounding gives 9.0, and continuous compounding gives 8.7.
The earliest known written mention is in Luca Pacioli's Summa de arithmetica, printed in Venice in 1494, according to Wikipedia's Rule of 72 article. Pacioli stated the rule without explaining it, which suggests it is older. It works because money doubles when (1 + rate) raised to the number of years equals 2, so the years equal ln 2 ÷ ln(1 + rate). For small rates that is close to 69.3 ÷ rate, and nudging the number up to 72 fits rates near 8%, where the exact answer works out to 72.05 ÷ rate.
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