Rule of 72 Explained (2026): Formula, Math Proof, Full Rate Table
Years to double = 72 ÷ rate. 6%→12 yrs, 8%→9, 10%→7.2. Full math derivation from ln(2), accuracy band (where 72 drifts), 1%–25% lookup table, plus free calculator. (2026)
A mental math shortcut that estimates how many years it takes for an investment to double, by dividing 72 by the annual percentage return.
Example: At 8% annual return, money doubles every 9 years (72 ÷ 8 = 9).
A more accurate variant of the Rule of 72 for low rates (under 5%), using 70 as the numerator instead of 72. Closer to the true ln(2) × 100 ≈ 69.31.
Example: At 3% inflation, prices double every 70 ÷ 3 ≈ 23.3 years (Rule of 72 says 24; exact is 23.45).
A higher-accuracy variant for rates above ~15%, using 76 as the numerator. Compensates for the curvature of the exact ln(2)/ln(1+r) function at high rates.
Example: At 20% return, money doubles in 76 ÷ 20 = 3.8 years (Rule of 72 says 3.6; exact is 3.80).
Extension that estimates tripling time. Divide 114 by the rate to find years for money to grow 3×.
Example: At 8%, money triples in 114 ÷ 8 ≈ 14.25 years.
Extension that estimates quadrupling time. Divide 144 by the rate to find years for money to grow 4×.
Example: At 8%, money quadruples in 144 ÷ 8 = 18 years (two doublings).
The mathematically true doubling time at a continuous rate r: t = ln(2) / ln(1 + r). The Rule of 72 is an approximation of this formula.
Example: At 8% (r = 0.08): t = ln(2) / ln(1.08) ≈ 0.6931 / 0.0770 ≈ 9.006 years.
The Rule of 72 is the most useful mental math shortcut in personal finance. Divide 72 by your annual return rate, and you get the number of years it takes your money to double. At 8% return, money doubles every 9 years. At 12%, every 6 years. It works for any compounding investment — no spreadsheet required.
Quick reference: Rule of 72 by rate
Rule of 72 vs Rule of 70 vs Rule of 76 vs exact ln(2)/ln(1+r)
Drag the rate slider. See which mental-math rule is most accurate at that rate, and how long it really takes to double, triple, and 8× your money.
| Method | Years to double | Error vs exact | Best for |
|---|---|---|---|
| Rule of 70 70 ÷ 8.0 | 8.75 | -2.8% | Low rates (1–5%) |
| Rule of 72 72 ÷ 8.0 | 9.00 | -0.1% | Typical rates (6–10%)← winner here |
| Rule of 76 76 ÷ 8.0 | 9.50 | +5.5% | High rates (15%+) |
| Exact: ln(2) / ln(1+r) The mathematically true answer | 9.01 | 0.0% | Requires a calculator |
See accuracy band across all rates (1% – 25%)
| Rate | Rule 70 err | Rule 72 err | Rule 76 err |
|---|---|---|---|
| 1% | 0.5% | 3.4% | 9.1% |
| 2% | 0.0% | 2.8% | 8.6% |
| 3% | 0.5% | 2.3% | 8.0% |
| 5% | 1.5% | 1.4% | 7.0% |
| 7% | 2.4% | 0.4% | 6.0% |
| 8% | 2.8% | 0.1% | 5.5% |
| 10% | 3.7% | 1.0% | 4.5% |
| 12% | 4.6% | 1.9% | 3.5% |
| 15% | 5.9% | 3.2% | 2.2% |
| 20% | 7.9% | 5.3% | 0.0% |
| 25% | 9.9% | 7.3% | 2.1% |
Green = lowest error at that rate. Rule of 70 wins below 4%, Rule of 72 wins 5–12%, Rule of 76 wins above 18%.
- 2% → doubles in 36 years
- 4% → doubles in 18 years
- 6% → doubles in 12 years
- 7% → doubles in 10.3 years
- 8% → doubles in 9 years
- 10% → doubles in 7.2 years
- 12% → doubles in 6 years
- 15% → doubles in 4.8 years
- 20% → doubles in 3.6 years
- 24% (credit card APR) → doubles in 3 years
Key takeaways
- The Rule of 72 formula: years to double = 72 ÷ annual rate (entered as a whole-number percent, not a decimal).
- It is an approximation of the exact doubling formula ln(2) / ln(1 + r), which gives 69.31% as the "true" numerator for low rates.
- Accuracy: within 1% for rates between 4% and 12%, within 2% for 2–15%. Less accurate at extremes (1% or 20%+).
- Rule of 70 is more accurate for low rates (under 5%); Rule of 76 is more accurate at high rates (above 15%). Rule of 72 is the popular default because 72 has many integer divisors.
- Reverse application: 72 ÷ (years to double) gives the implied annual rate. If your portfolio doubled in 9 years, the implied return is 72 ÷ 9 = 8%.
- Variants: Rule of 114 for tripling time, Rule of 144 for quadrupling time.
Rule of 72 Formula & How to Calculate It
The formula is: Years to double = 72 ÷ annual rate (in percent). So 72 ÷ 6 = 12 years, 72 ÷ 9 = 8 years, 72 ÷ 4 = 18 years. Always use the whole-number percent (6, not 0.06). The approximation stays within 1–2% of the true compound doubling time for any rate between 4% and 15% — accurate enough for any real-world plan.
The math behind 72: derivation in three lines
The exact doubling time at compounding rate r is the solution to (1 + r)^t = 2. Taking the natural log of both sides: t × ln(1 + r) = ln(2), so t = ln(2) / ln(1 + r). For small r, ln(1 + r) ≈ r, so t ≈ ln(2) / r ≈ 0.6931 / r. Expressed as a percent (multiply numerator and denominator by 100): t ≈ 69.31 / R, where R is the rate in percent. The number 72 is used instead of 69.31 because it has many clean divisors (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72), making mental math far easier — at the cost of about 1% accuracy at typical investment rates.
Worked math: comparing the Rule of 72 to the exact formula
- 2% rate: Rule of 72 says 36 years; exact ln(2)/ln(1.02) = 35.00 years. Off by ~3%.
- 5% rate: Rule of 72 says 14.4 years; exact = 14.21 years. Off by ~1.3%.
- 8% rate: Rule of 72 says 9 years; exact = 9.01 years. Essentially exact.
- 10% rate: Rule of 72 says 7.2 years; exact = 7.27 years. Off by ~1%.
- 15% rate: Rule of 72 says 4.8 years; exact = 4.96 years. Off by ~3%.
- 24% rate (credit card APR): Rule of 72 says 3 years; exact = 3.22 years. Off by ~7%.
For every rate you will plausibly encounter in personal finance — 4% bonds, 5% HYSA, 7% real stock return, 10% nominal stock return — the Rule of 72 is accurate enough that no calculator is needed.
Step-by-step: how to use the Rule of 72
- Step 1 — Identify your annual return rate. For an index fund, that might be 8%. For a HYSA, 5%. For credit-card debt, 24%.
- Step 2 — Divide 72 by that rate. Use the whole number, not the decimal: 72 ÷ 8, not 72 ÷ 0.08.
- Step 3 — The result is the number of years it takes your money (or debt) to double.
- Step 4 — Double the answer to find when it quadruples, or triple it to find when it grows 8×.
Why 72?
The exact doubling time at rate r is ln(2) / ln(1+r), but that requires a calculator. 72 is close to 100 × ln(2) ≈ 69.3, and 72 has nicer divisors (2, 3, 4, 6, 8, 9, 12). It's chosen for mental math convenience, not mathematical purity.
Real-world examples
- S&P 500 historical 10% → doubles every 7.2 years. A $10K investment becomes $40K in ~14 years, $160K in ~29 years.
- High-yield savings at 5% → doubles every 14.4 years. Slow but beats 0% checking.
- Credit card debt at 24% → doubles every 3 years. This is why debt compounds into disaster.
- Inflation at 3% → halves purchasing power every 24 years. Same math, opposite direction.
Reversed: the Rule of 72 for inflation
The same formula tells you how fast money loses value. At 3% inflation, purchasing power halves every 24 years. At 7% inflation (like 2022), every 10 years. This is why holding cash over decades is mathematically dangerous.
The Rule of 114 (tripling) and 144 (quadrupling)
Extensions: divide 114 by rate for tripling time, 144 for quadrupling. At 8%, money triples in 14.25 years and quadruples in 18 years. Handy for quick wealth-trajectory estimates.
For a full lookup at every rate from 1% to 25%, see our Rule of 72 table with both the approximation and the exact ln(2)/ln(1+r) value side-by-side. Or plug any scenario into our compound interest calculator or monthly compound interest calculator to verify — the Rule of 72 will be within 1-2% of the true answer for almost any rate you care about. For daily-compounded HYSA modelling, use our daily compound interest calculator.
Frequently asked questions
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Sources & further reading
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