Rule of 72 calculator

What the rule of 72 is

The rule of 72 is a mental shortcut for compound growth. Divide 72 by the annual percentage rate and the answer is, approximately, the number of years it takes something growing at that rate to double. It has been used since at least the fifteenth century (Luca Pacioli described it in 1494) because it lets you reason about compounding without a calculator.

It applies to anything that grows by a steady percentage: an investment, a savings balance, a debt at a fixed interest rate, a population, prices under inflation. At 6% a year, 72 ÷ 6 = 12 years to double. At 9%, 8 years. At 12%, 6 years. And because doubling twice is quadrupling, the same number tells you that at 6% money roughly quadruples in 24 years and grows eightfold in 36.

Run backwards, it answers the opposite question. If you want something to double in eight years, you need about 72 ÷ 8 = 9% a year. That is the second mode in the calculator above.

The exact formula, and why 72 works

Something growing at rate r a year doubles when (1 + r)t = 2. Solving for t:

Exact years to double = ln(2) ÷ ln(1 + r)

For small rates, ln(1 + r) is close to r, and ln(2) is 0.693. So the doubling time is close to 0.693 ÷ r, or 69.3 ÷ the rate in percent. That is the rule of 69.3, and it is exact for continuous compounding.

For yearly compounding at everyday rates, ln(1 + r) is a little smaller than r, which pushes the true answer up. Replacing 69.3 with 72 corrects for that around 8%, and 72 has the practical advantage of dividing cleanly by 1, 2, 3, 4, 6, 8, 9 and 12. The rule of 70 sits in between and is popular in economics for low growth rates, such as GDP growth or inflation of 1% to 3%.

The table under the calculator shows the error at each rate from 1% to 20%. The rule overstates the doubling time at low rates (+2.34 years at 1%, +0.55 at 3%), is almost spot on at 8% (−0.01), and understates it at high rates (−0.20 years at 20%). The rate-needed mode uses the exact form 21/t − 1.

Worked examples (hypothetical)

Doubling time at 8%. A hypothetical $10,000 growing at a constant 8% a year: 72 ÷ 8 = 9.00 years. The exact figure is ln(2) ÷ ln(1.08) = 9.01 years, an error of −0.01 years (−0.1%). The rule of 70 says 8.75 years and the rule of 69.3 says 8.66, both further out at this rate. Tripling (to $30,000) takes 14.27 years.

Rate needed to double in 10 years. 72 ÷ 10 = 7.20% a year. The exact rate is 2^1/10 − 1 = 7.18%, a difference of 0.02 percentage points.

At 3%. 72 ÷ 3 = 24.00 years, against an exact 23.45 years. The shortcut is 0.55 years long here, which is why economists dealing in low rates often prefer 70 (23.33 years).

These figures assume the rate never changes. No investment produces a constant return, and a rate that held in the past does not predict the future. The useful lesson is the shape of the curve, not a date on which your money will have doubled.

Using the rule of 72 to check claims

The rule's real value is speed. It lets you test a number in your head before you believe it, and three quick uses cover most situations.

Comparing rates. The difference between 4% and 8% sounds like four points. In doubling time it is the difference between 18 years and 9, so over 36 years one sum doubles twice (four times the money) and the other doubles four times (sixteen times). Small differences in rate become large differences in outcome over long horizons, in both directions.

Spotting impossible promises. If someone advertises 6% a month, 72 ÷ 6 says the money doubles every 12 months on the monthly rate alone, and compounding that for a year is just over doubling (1.06 to the power of 12 is 2.01). Five years of that would turn $1,000 into about $33,000. Claims that imply money doubling every few months for years on end describe arithmetic, not anything a real market has delivered. The compounding calculator lays the same sums out period by period.

Working backwards from a goal. If you hope to double a sum in six years, 72 ÷ 6 says you need about 12% a year, every year. Comparing that with long-run market returns of the order of 6% to 10% a year in nominal terms, with large swings between years, tells you how much risk such a goal implies. Past returns do not predict future ones, so treat any such comparison as a reality check, not a plan.

Inflation: the rule of 72 in reverse

Inflation compounds against the purchasing power of money in exactly the same way growth compounds for it. At a steady 3% inflation, prices double in about 23.4 years, which means a fixed sum buys half as much. At 6% that happens in 11.9 years; at 10%, in 7.3 years. Enter an inflation rate in the calculator and the last line shows the halving time.

This is why cash held for decades loses value even when its nominal amount never falls, and why a return should be judged after inflation. A 5% return during 3% inflation is roughly 1.9% of real growth (1.05 ÷ 1.03 − 1), and the real doubling time is closer to 36 years than 14.

The same logic applies to costs that compound against you. Fees of 1% a year look small, but over 72 years they would take half of everything; over a 30-year horizon they take roughly a quarter of the ending value. For full growth schedules, see the compounding calculator and the CAGR calculator.

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