Why Does the Rule of 72 Work?

Want to know how long it takes to double your money at 8% interest? Divide 72 by 8. The answer is 9 years. The exact answer, worked out with logarithms, is 9.006 years.

That's remarkably accurate for mental math. The Rule of 72 has been used by merchants, bankers and investors for more than 500 years. But where does 72 come from? It isn't the "true" constant at all. That honor goes to 69.3. The reason people use 72 anyway is part calculus, part convenience.

72 Is Deliberately "Wrong," and That Makes It More Accurate

If you derive the doubling rule mathematically, the number you get is ln(2) × 100 ≈ 69.3. So why don't we use a "Rule of 69.3"?

Because for the interest rates people actually care about, roughly 4% to 12%, 69.3 gives answers that are consistently too small. Bumping the constant up to 72 cancels that error almost perfectly around 8%. And 72 is far easier to divide in your head.

The Exact Doubling Time

With annual compounding at rate r (as a decimal), money grows by a factor of (1 + r) each year. It doubles after t years when:

(1 + r)^t = 2

Take the natural log of both sides and solve:

t × ln(1 + r) = ln(2)
t = ln(2) / ln(1 + r)

That's the exact formula. At 8%: ln(2)/ln(1.08) = 0.6931/0.07696 ≈ 9.006 years. Compute logs with the logarithm calculator or look them up in the natural logarithm table.

The Approximation Step

For small r, the natural log has a simple approximation from its Taylor series:

ln(1 + r) = r − r²/2 + r³/3 − … ≈ r

Replace ln(1 + r) with r:

t ≈ ln(2) / r ≈ 0.693 / r

If R is the rate as a percentage (so r = R/100):

t ≈ 69.3 / R

That's the Rule of 69.3, and it's exact for continuous compounding, where money grows as e^(rt).

Why 72 Beats 69.3 for Annual Compounding

The approximation ln(1 + r) ≈ r slightly overstates ln(1 + r), because it ignores the −r²/2 term. That makes 69.3/R come out too low for annually compounded interest.

Including the next term gives a better estimate:

t ≈ 69.3/R + 0.35

Around R = 8%, that's 8.66 + 0.35 ≈ 9.01 years, almost exactly right. Since 0.35 × 8 ≈ 2.8, a constant of about 72 absorbs the correction. Here's how the rules compare:

The Rule of 72 is within about 0.1 years of the exact answer from 4% to 12%.

Why 72 Is So Convenient

The other reason for 72 is arithmetic. 72 has 12 divisors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72. That means the most common interest rates divide it evenly. See its properties on the page for 72.

By contrast, 69 has only four divisors (1, 3, 23, 69), and 70 has eight. The "Rule of 70" is sometimes used for population growth and inflation because it's slightly more accurate at low rates like 1–3%.

For higher rates, a common refinement is to add 1 to the constant for every 3 points above 8%: at 20%, use 72 + 4 = 76, giving 3.8 years (exact: 3.80).

An Insider Reference: Luca Pacioli, 1494

The earliest known written reference to the rule appears in Luca Pacioli's 1494 book Summa de arithmetica, geometria, proportioni et proportionalita, a landmark mathematics textbook that also contains the first published description of double-entry bookkeeping.

Pacioli presented the rule without any derivation, as a practical tip for merchants. Since logarithms weren't invented until John Napier's work in 1614, the rule was likely discovered empirically. Merchants noticed the pattern long before anyone could explain it.

Uses Beyond Investing

The rule works for anything that grows or shrinks at a steady percentage:

• Inflation: at 3% inflation, prices double in about 72/3 = 24 years
• Debt: a credit card balance at 24% doubles in about 3 years if unpaid
• Economic growth: an economy growing at 6% a year doubles in about 12 years
• Halving: something shrinking 9% a year halves in roughly 8 years

Explore the growth directly with the compound interest calculator.

Two Concepts Worth Knowing

Natural Logarithm

The natural logarithm ln(x) is the logarithm with base e ≈ 2.71828. It's the natural language of growth: ln(2) ≈ 0.693 is the "doubling constant" behind every doubling rule.

Taylor Series

A Taylor series approximates a function with a polynomial. ln(1 + r) ≈ r − r²/2 is the first two terms. Rules of thumb in finance and physics often come from keeping just one or two terms. See the calculus formulas.

Quick Answer: Why Does the Rule of 72 Work?

Money doubles when (1 + r)ᵗ = 2, so t = ln(2)/ln(1 + r). For small rates, ln(1 + r) ≈ r, giving t ≈ 69.3/R. Because annual compounding makes that slightly too low, and 72 has many divisors, 72 is used instead. It's very accurate for interest rates between about 4% and 12%.

Try Them Yourself

• Compound Interest Calculator: verify doubling times
• Logarithm Calculator: compute ln(2)/ln(1 + r) exactly
• Natural Logarithm Table: look up ln values
• Number 72: why its divisors make it ideal
• Financial Formulas: compound growth formulas
• The Mathematics of Compound Interest: why time matters so much

Take your savings rate, divide 72 by it, and ask how many doublings you'll get before retirement. That single number says more than most financial projections.