Rule of 72 Calculator (Doubling Time)
The Rule of 72 is a mental-math shortcut: under compound growth, the years to double your money ≈ 72 ÷ annual rate (percent). Enter a rate or a target period to compare the Rule-of-72 estimate against the exact logarithmic answer, with a compound-growth table alongside.
Enter a positive number (rate 0–100%, years 0–200).
Rule of 72 — years to double
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Exact answer (logarithmic)
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Compound growth over time
| Years | Multiple | Doublings |
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Assumes annual compounding, no withdrawals, and no inflation — for ballpark comparison only. Real-world returns vary and incur fees and taxes.
Formula
years ≈ 72 ÷ r(%) | exact: n = ln 2 ÷ ln(1 + r/100) | rate ≈ 72 ÷ n | exact: r = (2^(1/n) − 1) × 100
- · 72 is the popular constant. Strictly, ln 2 × 100 ≈ 69.3 is the precise constant for continuous compounding — 72 is preferred because it has many integer factors (1, 2, 3, 4, 6, 8, 9, 12), making it easy to divide in your head.
- · Small error: within roughly 4–10%, the Rule of 72 stays within 1% of the exact logarithmic answer. Error grows the further you stray from ~8%; above ~20% it noticeably over-estimates and you should use the exact formula.
- · Applies only to positive returns with annual compounding. Use the "Rule of 69" for continuous compounding and the "Rule of 70" for very small percentages such as inflation or population growth.
- · The rule only finds doubling time. For 3× or 10× use (ln k) ÷ ln(1 + r/100) — for example, "Rule of 114" approximates tripling time.
- · Useful for: retirement asset doubling, comparing fixed-deposit rates, gauging how fast inflation halves purchasing power, and back-of-envelope portfolio sizing.
- · Source: arithmetic approximation popularised in Burton G. Malkiel's "A Random Walk Down Wall Street" and standard introductory finance texts.
Frequently asked
Why 72 instead of 69 or 70?
Because 72 divides cleanly: 1, 2, 3, 4, 6, 8, 9 and 12 are all factors. Mathematically the "true" constant for continuous compounding is ln 2 × 100 ≈ 69.3, but we usually deal with annual compounding, and 72 is unbeatable for mental arithmetic — and still typically within 1% of the exact answer.
How accurate is the Rule of 72?
Highly accurate between 4–10% — typically within 1% of exact. At 8%: Rule says 9 years, exact ≈ 9.0065. At 6%: 12 years vs 11.896 years. Above 15–20% the rule drifts: at 25%, Rule of 72 says 2.88 years while exact is ≈ 3.106 years (i.e. it under-estimates the time). Use the exact formula above for very high rates.
Can I use it to estimate when inflation will double prices?
Yes, though for low rates (typical 2–4%) the "Rule of 70" is slightly more accurate. At 3% inflation: Rule of 72 = 24 years, Rule of 70 = 23.3 years, exact = 23.45 years. The same number tells you when purchasing power halves — because doubled prices means money buys half as much.
Can it work backwards — knowing the doubling time, find the required rate?
Yes — that is exactly the second mode of this calculator. The formula flips to r ≈ 72 ÷ n. To double in 10 years you need 72 ÷ 10 = 7.2% (exact ≈ 7.177%). To double in 20 years only 3.6% is required (exact ≈ 3.526%). In practice, remember to net out fees, taxes and inflation when planning.
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