The Rule of 72
A 400-year-old mental-math shortcut for estimating doubling time.
⚡ Quick Doubling-Time Calculator
The Rule of 72 says: divide 72 by your annual interest rate (as a whole number) to estimate how many years it takes for your money to double.
At 6% per year, money doubles in about 72 / 6 = 12 years. At 9%, about 8 years. At 12%, about 6 years. At 1%, about 72 years. No calculator needed.
Where does 72 come from?
The actual doubling time for compounding is ln(2) / ln(1 + r), where r is the annual rate. For small rates (say, below about 20%), the natural log expansion of ln(1 + r) is close to r − r²/2 + …. The Rule of 72 is the first-order approximation ln(2) / r ≈ 0.693 / r, multiplied by 100 to keep the math in percent. 0.693 × 100 = 69.3, which rounds to 72 for a reason that turns out to be convenience: 72 has many small integer factors (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36), so it divides cleanly by the most common interest rates.
When is it accurate?
The Rule of 72 is most accurate in the 6%–9% range, where the error is under 1% (and remains within ~2% across 5%–12%). Outside that range it drifts:
| Rate | Rule of 72 | Exact (ln 2 / ln(1+r)) | Error |
|---|---|---|---|
| 1% | 72.0 yr | 69.7 yr | +2.3 yr (+3.4%) |
| 3% | 24.0 yr | 23.4 yr | +0.6 yr (+2.5%) |
| 4% | 18.0 yr | 17.7 yr | +0.3 yr (+1.8%) |
| 6% | 12.0 yr | 11.9 yr | +0.1 yr (+0.9%) |
| 8% | 9.0 yr | 9.0 yr | -0.0 yr (-0.1%) |
| 9% | 8.0 yr | 8.0 yr | -0.0 yr (-0.5%) |
| 10% | 7.2 yr | 7.3 yr | -0.1 yr (-1.0%) |
| 12% | 6.0 yr | 6.1 yr | -0.1 yr (-1.9%) |
| 15% | 4.8 yr | 4.96 yr | -0.16 yr (-3.3%) |
| 20% | 3.6 yr | 3.80 yr | -0.20 yr (-5.3%) |
Key doubling milestones for common interest rates include: 72 / 3 = 24 years at 3%, 72 / 4 = 18 years at 4%, 72 / 6 = 12 years at 6%, 72 / 8 = 9 years at 8%, 72 / 9 = 8 years at 9%, and 72 / 12 = 6 years at 12%.
Below 5% the rule overestimates doubling time (money doubles faster than the rule predicts). Above 15% it underestimates (money doubles faster still — exponential growth accelerates). At very low rates, a variant — Rule of 69.3 — is mathematically tighter, but 72 is what you'll see quoted.
The 72 / 70 / 69.3 variants at 8%
Three closely related shortcuts coexist; at 8% each lands within 0.05 years of the exact logarithmic answer, and each has a different best use:
| Variant | Formula | Years at 8% | Best use |
|---|---|---|---|
| Rule of 72 | 72 / r | 9 yrs | Classic mental math at common 6–10% rates |
| Rule of 70 | 70 / r | 8.75 yrs | Rough long-horizon estimates (e.g., inflation doubling) |
| Rule of 69.3 | 69.3 / r + 0.35 | 9.0125 yrs | Annual compounding / actuarial-grade estimate |
| Exact doubling | ln 2 / ln(1 + r) | 9.0065 yrs | Logarithmic benchmark (no shortcut) |
For tripling money, none of the 72-family shortcuts applies — the corresponding constants are 114 (or 115): 114 / 8 = 14.25 years at 8%, versus the exact ln 3 / ln(1.08) = 14.2749 years.
When is it useful?
- Sanity-check marketing claims. "Earn 12% annually and double in 6 years" — is the doubling-time promise consistent with the rate? Rule of 72 lets you check in seconds.
- Estimate inflation impact. At 3% inflation, prices double in about 24 years. Useful for thinking about long-term planning horizons.
- Compare investments quickly. 8% doubles in 9 years; 10% doubles in 7 years. That's a 2-year head start — meaningful over a 30-year career.
- Debt. The same rule applies to debts that compound. At 20% APR (credit-card territory), balances double in under 4 years.
Calculating doubling time on TI-84 and in Excel
While the Rule of 72 is meant for mental arithmetic, you can verify doubling times directly on a TI-84 calculator or in Excel:
- TI-84 Calculator Steps:
- Home Screen: Type
72 / 8and pressENTERto get 9 years. - Exact Formula: Type
ln(2) / ln(1 + 0.08)and pressENTERto get 9.006 years. - TVM Solver: Press
APPS, choose1:Finance…, and select1:TVM Solver…. SetI% = 8,PV = -1,PMT = 0,FV = 2,P/Y = 1,C/Y = 1. Solve forNto find 9.01 years.
- Home Screen: Type
- Excel Formulas:
- Quick shortcut: Enter
=72/8or=72/A1to return 9 years. - Exact financial function: Enter
=NPER(8%, 0, -1, 2)to calculate the exact periods needed for $1 to grow to $2, returning 9.01 years.
- Quick shortcut: Enter
Try it yourself
Open the Compound Interest Calculator, type a starting principal of 1000, a growth rate of 8, and a number of periods equal to 72/8 = 9. The "Future Value" should land close to 2000. It won't be exact (8% compounding is closer to 1.999), but it will be close enough to plan with.
For an annuity (you add money every period), the math is different — there's no simple Rule-of-72 analog, and you should use the calculator instead of guessing.
Caveats
- The Rule of 72 assumes continuous compounding-like growth. Real-world nominal annual rates usually compound monthly or quarterly, which slightly shifts the answer.
- It assumes a constant rate. Variable-rate investments (variable annuities, some bonds) will deviate.
- It ignores fees and taxes. A "7% return" with a 1% annual fee is more like 6%, and Rule of 72 on 6% gives 12 years, not 10.3.
- It only handles the first doubling. To go from $1,000 to $4,000 you need two doublings — so roughly 2× the doubling time, not 4×.
Related tools
- Compound Interest Calculator — exact future value with optional periodic contributions.
- Present Value Calculator — discount a future amount to today.
- Loan Calculator — monthly payments and full amortization schedule.
- NPV / IRR Calculator — discount-cash-flow analysis for any series.
Frequently asked questions
What is the Rule of 72 and how does it work?
The Rule of 72 is a mental-math rule of thumb used to estimate how many years it will take for an investment to double in value at a fixed annual interest rate. You simply divide 72 by the annual percentage rate (e.g., 72 / 8 = 9 years).
Why use 72 instead of 69.3 or 70?
While 69.3 is mathematically closer for continuously compounded growth, 72 has many small integer divisors (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36), making it much easier to divide in your head across common interest rates.
How long does it take for money to double at 8%?
At an 8% annual return, money doubles in approximately 9 years (72 / 8 = 9). The exact logarithmic formula ln(2) / ln(1.08) yields 9.006 years, showing the Rule of 72 is exceptionally accurate at 8%.
Can the Rule of 72 be used for inflation and debt?
Yes. At 3% inflation, the purchasing power of money drops in half in about 24 years (72 / 3 = 24). Conversely, for high-interest debts like credit cards at 18% APR, the debt balance would double in about 4 years (72 / 18 = 4) if no payments were made.
Do the 72/70/69.3 variants work for tripling?
No — tripling needs its own constants. At 8%, money triples in about 114 / 8 = 14.25 years (115 / 8 = 14.375), while the exact logarithmic answer is ln(3) / ln(1.08) = 14.27 years. The 72, 70, and 69.3 shortcuts estimate doubling only.
What exactly does the Rule of 72 estimate — the rate or the time?
It estimates the time in years for the money to double at a fixed annual interest rate — for example, 72 / 8 = 9 years at 8%. A classic exam-style trap: the statement "the Rule of 72 estimates the time needed to double an interest rate" is false — the amount doubles, never the rate. In multiple-choice form, the rule is used to find the doubling time of an investment at a constant rate.
When should I use 69.3 instead of 72?
For continuously compounded growth the exact constant is ln(2) × 100 ≈ 69.3, so doubling time is 69.3 / r: at 8%, 69.3 / 8 = 8.66 years (exact: ln(2) / 0.08 = 8.6643 years). For annual compounding, add a small correction — 69.3 / 8 + 0.35 = 9.0125 years — which nearly matches the true 9.0065 years.