Rule of 72 Calculator

Ask anyone who works with money how long it takes savings to double at 9 percent, and you will often get an answer in a second: eight years. They divided 72 by 9. This habit, the Rule of 72, is a good example of a shortcut that is nearly right, easy to remember and useful in conversation. The calculator checks it against the exact answer and shows where it is reliable and where it drifts.

The rule and its origin

The exact time for money to double at a compound rate r is ln 2 ÷ ln(1 + r). For small rates ln(1 + r) is close to r, so the time is close to 0.693 ÷ r, or 69.3 divided by the rate in percent. Using 72 instead of 69.3 makes the estimate fit rates around 8 percent better, and 72 has the practical advantage of being divisible by many everyday rates.

A worked example

At 9% the exact doubling time is 8.04 years and the Rule of 72 says 8.00 years, off by 0.04 years or 0.5%. For tripling at 12%, the exact time is 9.69 years and the shortcut, using 114, gives 9.51, a difference of 0.18 years. Going the other way, to double money in 6 years the exact rate needed is 12.246% a year, against 12.000% from 72 divided by 6. The table lists the exact and shortcut times at common rates: at 2% it is 35.00 years against 36.00, and at 24% it is 3.22 against 3.00.

How accurate is it?

The error is smallest around 6 to 10 percent, where the rule is within about a tenth of a year of the exact time. At very low rates it overstates by one or two years because 72 is too large, and at very high rates it understates the time, because ln(1 + r) falls further below r. If you need an exact figure, the page gives it. If you are making a back-of-the-envelope estimate in a meeting, the shortcut is good enough for the rates most people deal with.

Other multiples

The constant for a growth factor m is 72 × log₂(m). For doubling that is 72, for tripling about 114.1, for quadrupling 144 and for ten times about 239. Enter the multiple in the box, 2 for double, 3 for triple, 10 for ten times, and the table headings change to match. These constants let you estimate how long money takes to triple at 8 percent (114 ÷ 8 is about 14 years) without a calculator.

Using it for rates and for decay

Reversing the rule gives the rate you need: to double in 8 years you need about 72 ÷ 8 = 9 percent. It also works for things that shrink. Inflation of 6 percent halves the purchasing power of money in about 12 years, and a fund charging 2 percent a year takes about 36 years to eat half of your money in fees alone, if nothing else changed.

Edge cases

The growth factor must be greater than 1. Rates are entered as percentages above zero; a negative rate has no doubling time. The page is accurate for any rate, even though the shortcut loses precision at extremes, and it shows the error so you know how far to trust the mental version.

Limits

All the formulas assume a steady rate compounded once a year. Real investments rise and fall, and fees and taxes change the rate you actually earn. The result describes arithmetic, not expected performance.

Related tools

• Compound Interest Calculator projects balances year by year for a chosen rate.
• CAGR Calculator finds the steady rate from two actual values.
• Inflation Calculator applies the same doubling logic to prices.
• Simple Interest Calculator contrasts the doubling time when interest does not compound.