Guide · 5 min read
The Rule of 72: Where It Works and Where It Fails
The shortcut is a mental arithmetic trick that happens to be close to the truth across the rates people actually use. Knowing how close is what makes it useful rather than misleading.
At 7% the rule says 10.3 years and the exact answer is 10.2 — half a month apart. At 3% the gap opens to 6.6 months, and the shortcut starts overstating the time.
The rule says: divide 72 by the annual percentage return and you get the number of years it takes to double. At 7%, that is 72 ÷ 7 = 10.3 years.
The exact answer comes from the compound formula. Solve (1 + r)^t = 2 for t and you get t = ln(2) / ln(1 + r). At 7%, that is 10.24 years. The shortcut is off by about half a month over more than a decade.
Why 72 and not 69.3
The value that makes the rule exact for small rates is 100 × ln(2), which is about 69.3. Using 69.3 gives a better answer at 2% and a worse one at 10%.
Seventy-two wins for a different reason: it divides evenly by the rates people actually use. 72 ÷ 6 = 12, 72 ÷ 8 = 9, 72 ÷ 9 = 8. Mental arithmetic that comes out clean gets used; arithmetic that does not, does not.
Where the error grows
At 3%, the rule gives 24.0 years and the exact answer is 23.4 — a gap of 6.6 months, in the direction of overstating the wait. That matters for a savings account, where 3% is a realistic rate and the horizon is long.
At 10% the error reverses: the rule gives 7.2 years and the exact answer is 7.3, so it now understates by 0.9 months. The crossover happens somewhere around 8%.
Where it stops being appropriate
The rule is a sanity check, not a calculation. Its value is that it lets you judge an offer or a projection in your head, in the second before you decide whether it deserves a closer look.
- Variable returns. The rule assumes a constant rate; a portfolio that returns 15%, then −10%, then 8% does not behave like one returning 4.3% every year.
- Additions during the period. The rule describes a lump sum left alone; it says nothing about a balance that receives monthly deposits.
- Very high rates, where the linear approximation breaks down badly — at 50% the rule says 1.4 years and the exact answer is 1.7.
- Inflation-adjusted "real" doubling, which needs the real return, not the nominal one.
Frequently asked questions
Does the rule work for debt?
Yes, and it is often more useful there. A credit card at 24% doubles an unpaid balance in about three years. That is a more arresting fact than the annual rate.
Should I use 72 or 69.3?
Use 72 for mental arithmetic at ordinary rates. Use the exact formula whenever the number will inform a decision, because at that point the calculator is available and there is no reason to approximate.
What about tripling?
The same idea with a different numerator: roughly 114 divided by the rate. The approximation degrades faster here, so treat it as a rougher guide than the doubling rule.