The Rule of 72: how fast money doubles

How close the shortcut comes

The exact time for money to double, if it grows at a rate r each year and the growth is reinvested, is ln(2) / ln(1 + r), where ln is the natural logarithm. Few people can do that in their head, which is the point of the rule. Here is how the two compare:

Exact figures assume the rate compounds once a year, which is how an APY or an average annual return is quoted. A negative difference means the rule is too quick; a positive one means it is too slow. The last column is the number you would divide by to get the exact answer at that rate.

Below about 8% the rule says a little longer than the truth, and above it a little shorter. At 1% it is 2.3 years too slow over a 70-year wait, which is still only 3% off. At 24% it is 0.22 years too fast. For the rates most savers and investors see, the error is a matter of weeks or months.

Why 72, and when to use 69.3 or 70

The exact divisor for very small rates is 100 times ln(2), or about 69.3. As the rate rises, the right divisor creeps up, as the last column shows, and it passes 72 at about 7.85%. So 72 is a deliberate rounding up that makes the rule most accurate around 6% to 10%, where most long-run returns sit. It also helps that 72 divides evenly by 1, 2, 3, 4, 6, 8, 9 and 12, so the arithmetic is easy.

• Rule of 69.3 is exact when interest compounds continuously: at 8% that is 69.3 / 8 = 8.66 years. Daily compounding, which many banks and card issuers use, is almost the same: a 4% rate compounded daily (an APR, not an APY) doubles in 17.33 years, against 69.3 / 4 = 17.32.
• Rule of 70 is a rounder version of 69.3 that economists often use for low rates such as inflation or population growth. At 3.74% inflation it gives 18.7 years, closer than 72's 19.3 years to the exact 18.9 years.

If your rate is an APY (savings accounts and CDs quote one), compounding is already counted, so use 72. If it is an APR compounded daily, 69.3 or 70 is closer.

An old rule

The earliest known written version is in Luca Pacioli's Summa de arithmetica, a mathematics textbook printed in Venice in 1494. In a problem about how long capital takes to double, Pacioli divides 72 by an interest rate of 6% to get 12 years (the exact answer is 11.9). He gives the rule without explaining it, which suggests merchants were already using it; logarithms, the tool that explains it, were not published until Napier's work in 1614.

Doubling at today's rates

The same arithmetic applied to rates you can actually earn or owe now:

Deposit rates are FDIC national averages, weighted by deposits, as of Oct 19, 2026; online high-yield accounts often pay far more than the savings average, so check your own APY. The Treasury yield is the 10-year constant maturity rate (Oct 30, 2026), treated here as a yearly rate. The 7% and 9% returns are the assumptions our investment calculator uses for those mixes, not promises: stock returns vary widely from year to year. The card APR is the Federal Reserve's average for accounts assessed interest (third quarter of 2026), compounded daily.

At the national average savings rate of 0.38%, money takes about 183 years to double, longer than most working lives. At the 9% stock assumption it doubles in about 8 years, so $10,000 becomes roughly $20,000 after 8 years, $40,000 after 16 and $80,000 after 24. That is why time matters so much: each extra doubling period adds more than all the earlier ones combined. The I bonds vs savings vs CDs guide compares safe places for cash.

The same math on debt and inflation

Credit card balances

Card issuers charge a daily rate of the APR divided by 365 (or 360), and unpaid interest is added to the balance, so it compounds. At the 22.36% average APR that works out to 25.05% a year. The Rule of 72 says a balance left untouched would double in 3.2 years; the exact answer with daily compounding is 3.1 years, almost exactly what the Rule of 69.3 gives (3.1 years). A $5,000 balance would be about $15,288 after five years. In practice, missing payments also brings late fees and can trigger a higher penalty APR, so the real growth would be faster. The credit card interest guide and the credit card payoff calculator show how payments change the picture.

Inflation

Prices rising 3.74% a year (the latest 12-month change in the consumer price index, September 2026) double in about 18.9 years. Seen from the other side, cash under a mattress loses half its buying power over the same span. At that pace, $100 of today's prices costs about $144 in 10 years and $208 in 20, and $100 kept in cash buys what $69 and $48 buy today. Subtract inflation from your rate before applying the rule to see how fast your buying power, not just your balance, doubles. The inflation calculator shows what past dollars are worth now, and the raise and inflation guide applies it to pay.

Tripling and quadrupling: the Rules of 114 and 144

The same idea works for other multiples. Divide 114 by the rate to estimate the years to triple, and 144 for the years to quadruple (which is simply two doublings, 2 x 72):

Exact figures with yearly compounding: ln(3) / ln(1 + r) and ln(4) / ln(1 + r).

Like 72, both are tuned for rates near 8%: there, 114 gives 14.25 years to triple against an exact 14.27, and 144 gives 18.00 to quadruple against 18.01.

What the rule leaves out

• A steady rate. The rule assumes the same return every year. Savings rates change with the Federal Reserve, and stock returns swing widely, so a doubling time for stocks is an average, not a schedule.
• Reinvested growth. It assumes interest and dividends stay invested. Money you withdraw, or simple interest that is not added to the balance, grows more slowly. New deposits are separate; the compound interest calculator adds them.
• Fees. Use the return after costs. At 9%, the rule says $10,000 doubles three times in 24 years, to $80,000 (exactly, $79,111). A 1% yearly fee cuts the net return to about 7.91%, stretches each doubling to 9.1 years and leaves $62,225, about $16,886 less.
• Taxes and inflation. Interest taxed each year compounds at the after-tax rate, and inflation erodes what the doubled balance buys. The investment calculator includes fees, taxes and inflation together.