Rule of 72 calculator

Divide 72 by a yearly growth rate and you get roughly how many years it takes to double. Here is how rough, next to the exact answer and the rules of 70 and 69.3.

At 6.00% a year, compounded yearly, money doubles in

11.90 years

exactly: ln 2 ÷ ln(1 + 0.0600).

Rule of 72: 12.00 +0.9%
Rule of 70: 11.67 −1.9%
Rule of 69.3: 11.55 −2.9%
Continuous compounding: 11.55

Closest at this rate: the rule of 72. To double in 10 years takes 7.177% a year; the rule of 72 says 7.200%.

rule of 72rule of 70rule of 69.3error vs exact

Years to double: exact and by each rule (closest in green)
Rate	Exact	72 ÷ rate	70 ÷ rate	69.3 ÷ rate
1%	69.66	72.00	70.00	69.30
2%	35.00	36.00	35.00	34.65
3%	23.45	24.00	23.33	23.10
4%	17.67	18.00	17.50	17.33
5%	14.21	14.40	14.00	13.86
6%	11.90	12.00	11.67	11.55
7%	10.24	10.29	10.00	9.90
8%	9.01	9.00	8.75	8.66
9%	8.04	8.00	7.78	7.70
10%	7.27	7.20	7.00	6.93
12%	6.12	6.00	5.83	5.78
15%	4.96	4.80	4.67	4.62
20%	3.80	3.60	3.50	3.47

Where 72 comes from

Money growing at a yearly rate r doubles when (1 + r)^n = 2, so the exact time is n = ln 2 ÷ ln(1 + r). For small rates ln(1 + r) is close to r, which gives 0.693 ÷ r: the rule of 69.3, exact for continuous compounding. With yearly compounding the true time runs a little longer, so of the three rules 69.3 is closest only below about 1%, 70 from about 1% to 5%, and 72 from about 5% up. It is exact near 7.85%, and 72 divides neatly by 2, 3, 4, 6, 8, 9 and 12.

Exact years to double = ln 2 ÷ ln(1 + r)Rate needed to double in n years = 2^(1 ÷ n) − 1

The rule is old: it appears in Luca Pacioli's Summa de arithmetica, printed in Venice in 1494.

Example: at 6% the exact time is ln 2 ÷ ln 1.06 = 11.90 years. 72 ÷ 6 = 12 is 0.9% too long; 70 ÷ 6 = 11.67 is 1.9% too short.

Sources

Investor.gov (U.S. Securities and Exchange Commission), glossary: compound interest, the growth the rule describes.
ICAEW Library, Summa de arithmetica (Luca Pacioli, Venice, 1494).

Questions

Does the rule work for inflation too?

Yes. At 3% inflation prices double in about 72 ÷ 3 = 24 years (exactly 23.4), after which money buys half as much.

What about monthly compounding?

At the same stated rate, monthly compounding grows a little faster, so doubling comes slightly sooner, closer to the rule of 69.3. For an exact answer enter the effective yearly rate.

Can I find the rate needed to double in a set time?

Yes, with the second field. The exact rate is 2^(1 ÷ years) − 1; the shortcut is 72 ÷ years.