Rule of 72: formula, calculator and worked example

What the symbols mean

$t$: the approximate number of years to double (years)
$r$: the annual interest rate as a decimal (6% = 0.06), so 100r is the rate in percent

When to use it

Use it for quick estimates: at 6% money doubles in about 12 years, at 9% in about 8. It also works for anything growing at a steady percentage (prices with inflation, populations). It is most accurate for rates around 6% to 10%; the calculator shows the exact doubling time alongside.

Worked example

Roughly how long does money take to double at 6% a year?

Start from the formula.
$t = \frac{72}{100 r}$
Substitute $r$ = $0.06$ and work it out.
$t = \frac{72}{100 \cdot 0.06}$
$= \frac{72}{6}$
$= 12 years$
Compare the exact doubling time with yearly compounding, from (1 + r)ᵗ = 2:
$t_{exact} = \frac{\ln (2)}{\ln (1 + r)}$
$= \frac{\ln (2)}{\ln (1 + 0.06)}$
$= \frac{\ln (2)}{\ln (1.06)}$
$\approx \frac{0.693147}{0.0582689}$
$\approx 11.8957 years$
The rule of 72 is off by $0.104 years$ .

Answer About 12 years (exactly 11.9 years with yearly compounding)

Why it works

The exact doubling time is $ln 2/ln (1 + r)$ . For small r, $ln (1 + r) ≈ r$ , giving about $0.693/r$ , or 69.3 divided by the rate in percent. 72 is used instead because it has many divisors and corrects for the approximation at typical rates.

Common mistakes

Dividing 72 by 0.06 instead of by 6: use the rate in percent.
Trusting it at high rates: at 50% the rule says 1.44 years but the exact answer is 1.71.
Forgetting it is an approximation; for exact answers use logarithms.

Calculator

What the symbols mean

When to use it

Worked example

Why it works

Common mistakes