Rule of 72 / Doubling Time Calculator
How long does it take for something growing at a fixed annual rate to double? The exact doubling time is ln 2 ÷ ln(1 + r); the rule of 72 approximates it as 72 ÷ r (in percent) — a handy mental shortcut that is most accurate around 8% and good to within a few percent for typical rates. At 6% money doubles in ~11.9 years (rule says 12); at 10% in ~7.3 years (rule says 7.2). See the exact time, the rule-of-72 estimate and its error, the tripling time (rule of 114), a general "time to multiply by ×N" solver, and a comparison table across rates. Everything runs locally in your browser.
Doubling & tripling
Time to multiply by ×N
Rule of 72 vs exact, across rates
The math. For growth at rate r per period (as a decimal), the value after t periods is (1+r)t; setting that equal to a target factor F gives the exact time t = ln F ÷ ln(1 + r). For doubling, F = 2 so t = ln 2 ÷ ln(1 + r) ≈ 0.693 ÷ ln(1 + r). The rule of 72 replaces ln(1+r) with its first-order approximation r (in percent) and rounds the constant 100·ln 2 ≈ 69.3 up to 72 — chosen because it divides evenly by 1, 2, 3, 4, 6, 8, 9, 12 and lands closest to exact for common rates near 8%. At very low rates the rule slightly overestimates (since 72 > 69.3); at high rates it underestimates (the ln(1+r) ≈ r approximation breaks down); the error is zero near 7.85%. For continuous compounding the exact doubling time is ln 2 ÷ r = the "rule of 69.3". Tripling uses ln 3 and the rule of 114 (100·ln 3 ≈ 109.9, rounded to 114). Pairs with the Compound Interest, Inflation, Loan, and Margin / Markup calculators. Everything runs locally — nothing leaves your browser.