Rule of 72 Explained: Where the 72 Comes From

0:00The Rule of 72 — and why 72 is really 69.3 ▶

The Rule of 72: divide 72 by your interest rate, and you get the years until your money doubles. At 8%, 72 ÷ 8 = 9 years. But 72 is not a magic number — it is really 69.3, nudged up on purpose. So why 72, and not 70?

0:27The one idea you need ▶

All you need is one idea: at 8%, the whole account is multiplied by 1.08 every year, interest included. First, let's watch the rule at work.

0:36The rule at work: the staircase at 8% ▶

At 6% it would say twelve years; at 8%, nine. Say you put in $10,000 and it grows 8% a year. Guess where it lands after eighteen years. After nine years it is about $20,000, and after eighteen years about $40,000. Every step of that staircase takes the same nine years — and every step is twice as tall as the one before. So where does that 72 come from?

1:12Where it comes from: n = ln 2 / ln(1 + r) ▶

Here is where it comes from. Write P for the money you put in, say $10,000, and r for the rate, 0.08. After n years, P has become P × (1 + r)ⁿ. Doubling means that is twice P, so the growth factor itself has to reach two: (1 + r)ⁿ = 2. That n is stuck up in the exponent, and pulling it down is exactly what a logarithm does. Take the natural log of both sides, and n × ln(1 + r) = ln 2. So n = ln 2 ÷ ln(1 + r). The log of 2 is 0.693. For 8%, the log of 1.08 is 0.077. Divide, and n = 9.0 — the number the rule guessed for free. Now the trick: for small rates, the log of 1 + r is almost r — 0.077 against 0.08.

2:18Why 72 and not 69.3 — the number that fits ▶

Swap them, and n is roughly 0.693 ÷ r, or 69.3 ÷ the rate in percent. So the honest version is the rule of 69.3. The log of 1 + r is a touch below r, so the real constant sits above 69.3, and climbs with the rate. Multiply the exact years by the rate and you get that constant: 70 at 2%, 71 at 5%, 72 at 8%, 73 at 12%. The further from 8%, the further that number moves from 72: 69.7 at 1%, 77 at 24%. And 72 divides by 2, 3, 4, 6, 8, 9 and 12: you can do it in your head. Now how close is it in practice, with the exact answer next to the rule?

3:16Rule vs exact at 5%, 6%, 12% ▶

At 5%, the rule says 14.4 years; the exact answer is 14.2. At 6%, twelve years against 11.9; at 12%, six against 6.1. From about 4 to 15 percent, the rule is never more than four months off; at 2%, it is a full year. But the rule will multiply $10,000 by sixteen — in your head. The question is how far off it will be.

3:52Four doublings: $10,000 → $160,000 ▶

Four steps of nine years is 36 years, and four doublings is 2 × 2 × 2 × 2 — sixteen times. So the rule says your $10,000 becomes $160,000. The exact formula says 1.08 to the power 36 is 15.97. That is $159,682. So a number you can get in your head lands within a fifth of a percent of the truth. So, here is the whole story: divide 72 by the rate,

4:25The whole story ▶

and you have the years until your money doubles. The 72 is really the log of two, nudged up from 69.3. It even runs backwards: money that doubled in twelve years grew at 72 ÷ 12, six percent a year. And that's the math behind the money.

4:51What's next: APR vs APY ▶

Next time: APR vs APY, where the sign says 5%. But you get 5.12 — or 5.13 — and one word decides which. And for you: what rate would double your money in ten years? I'm the MoneyMathician — subscribe, and the next one will find you. Until then: do the math!