How the Rule of 72 is calculated
The approximation divides a multiple-specific constant (72 for doubling, 114 for tripling, 144 for quadrupling) by the annual rate. The exact figure solves the compound-interest formula FV = PV(1+r)^n for n using logarithms.
- r — annual rate of return (%)
- N — approximation constant for the target multiple — 72 (2×), 114 (3×), 144 (4×)
- multiple — target multiple of the principal (2, 3, or 4)
More detail
Where the Rule of 72 is most accurate
The shortcut works because N ≈ 100 × ln(multiple), rounded to a number with many small divisors for easy mental math. It tracks the exact logarithmic answer closest at rates around 6–10% a year; outside that band — very low or very high rates — the gap between the approximate and exact outputs above widens.
Why 114 and 144, not just 72
72 only approximates doubling. Since 100 × ln(2) ≈ 69.3, 100 × ln(3) ≈ 109.9, and 100 × ln(4) ≈ 138.6, each target multiple needs its own constant — rounded up slightly to correct for real compounding — which is why tripling uses 114 and quadrupling uses 144 instead of scaling 72 directly.
Practical tip. The approximate figure is for quick mental math; when the stakes are real, read the exact (logarithmic) output above instead — the two only match closely in the 6–10% range.
Frequently asked questions
At 8% annual return, how long until my money doubles?
The Rule of 72 approximation gives 72 ÷ 8 = 9 years; the exact logarithmic formula gives about 9.01 years. Enter 8 for annual rate and 2× for target multiple above to see both figures.
Why does the Rule of 72 differ from the exact answer?
The Rule of 72 linearizes a logarithmic relationship, so it drifts further from the exact figure at very low (under ~5%) or very high (over ~20%) annual rates. The "Time to reach target (exact)" output above always computes ln(multiple) ÷ ln(1 + r/100) with no approximation error.
Can I use this for tripling or quadrupling, not just doubling?
Yes — set the target multiple to 3× to use the Rule of 114 (114 ÷ r) or 4× for the Rule of 144 (144 ÷ r); the calculator switches the constant automatically and shows which rule applied.
Where does the number 72 come from?
100 × ln(2) ≈ 69.3, but adjusting for real compound growth at typical 6–10% rates brings the best-fit constant closer to 72 — and 72 has many small divisors (1,2,3,4,6,8,9,12...), which makes it easy to divide in your head.
What happens if I enter a 0% or negative rate?
A 0% or negative annual rate can never double a principal, so the time to reach the target is undefined — both outputs show "—". Enter a rate greater than 0 to get a result.