Rule of 72 vs exact doubling time
The rule of 72 estimates doubling time by dividing 72 by the annual rate in percentage points. The exact wait is ln 2 over ln(1+r). At 7 percent the shortcut is 10.2857 years and the exact wait is 10.2448 years, about 15 days apart.
| Rule of 72 | Exact doubling time | |
|---|---|---|
| Formula | 72 / r, with r in percentage points. | ln 2 / ln(1 + r/100). |
| At 7 percent | 10.2857 years. | 10.2448 years. The shortcut is 0.0409 of a year long, about 15 days. |
| At 8 percent | 9 years exactly. | 9.0065 years. The shortcut is 0.0065 of a year short, a bit over 2 days. |
| At 20 percent | 3.6 years. | 3.8018 years. The shortcut is 0.2018 of a year short, about 74 days. |
| Where they cross | Just under 7.85 percent, the rate where 72 is exactly right. | The same crossover. Below it the rule runs long. Above it the rule runs short. |
| What it is for | A wait you can do at a bus stop. | The date the balance actually doubles, if the rate holds and nothing is added or taken out. |
On this page
72 is a fitted numerator, not a law
Doubling means . The exact wait is a logarithm. The shortcut replaces that logarithm with 72 divided by the rate in percentage points, because 72 sits near the annual-compounding numerator and divides cleanly by 2, 3, 4, 6, 8, 9 and 12.
Between roughly 5 and 12 percent the error is weeks rather than years. At 7 percent it is about 15 days long. At 8 percent it is a bit over 2 days short. At 20 percent, a card-debt rate rather than a savings rate, it is about 74 days short.
How the rule of 72 works is the long form. The rule of 72 explorer is the two waits as marks on a line you can drag.
The sum has to be left alone
Deposits pull doubling forward and withdrawals push it back. The shortcut and the logarithm both assume a rate applied to a sum that is not being topped up or drawn down. How compound interest works is the curve when money is also being paid in. 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?
- Divide 72 by the rate in percentage points: years.
- Now the exact version. Doubling means , so .
- and , so years.
- Subtract to size the error: 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: years, no remainder.
- The exact answer: years.
- The gap is 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: years.
- The exact answer: years.
- The gap is of a year, about 74 days.
- As a share of the wait, 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
When is the rule of 72 good enough?
Near 8 percent, and more broadly between about 5 and 12 percent, where the error is weeks. Outside that band, especially at 2 percent or 20 percent, use the logarithm.
Does the rule work for inflation?
Yes, on the same terms: a steady rate applied to a sum left alone. At 3 percent the shortcut is 24 years. The exact wait is a little shorter. How inflation factors work is the factor itself.
What about tripling?
Divide 114 by the rate for a rough triple, 144 for four times. Those are cousins of 72, not this table. The exact wait is ln 3 / ln(1+r) and ln 4 / ln(1+r).
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.