Choosing the right shortcut for estimating how long money takes to double depends on the compounding assumption. The Rule of 70, Rule of 72, and Rule of 69.3 all convert an assumed annual percentage rate into an approximate timeline, but they are not interchangeable exact formulas.
Quick Answer:
The Rule of 72 is usually the most convenient mental-math shortcut for moderate assumed rates, and it is especially close to the exact annual-compounding result around roughly 6% to 10%. The Rule of 70 is often closer at low annual growth rates. The Rule of 69.3 comes from continuous-compounding mathematics. For an exact result, choose the compounding assumption first and compare it with the Rule of 72 Calculator.
How we approached this analysis
This comparison treats the input as a nominal annual rate, not an APY or effective annual return. The main accuracy table compares the shortcuts with the exact annual-compounding formula. Continuous compounding is discussed separately because it uses a different exact formula.
TL;DR
- The Rule of 72 is easiest to use mentally and is close for moderate annual-compounding examples.
- The Rule of 70 is very close at low annual growth rates such as 2% to 4%.
- The Rule of 69.3 is tied to continuous compounding, not ordinary annual-compounding benchmarks.
- Exact calculations are precise only for the stated rate, timing, and compounding assumptions.
- None of these shortcuts predicts market returns or guarantees a future balance.
Why Do Three Different Rules Exist?
All three rules estimate:
How long will it take for money or prices to double?
The difference comes from the exact formula used for the compounding assumption.
For a nominal annual rate r written as a decimal:
- Annual compounding:
t = ln(2) ÷ ln(1 + r) - Periodic compounding:
t = ln(2) ÷ [n × ln(1 + r ÷ n)] - Continuous compounding:
t = ln(2) ÷ r
In these formulas, t is the doubling time in years and n is the number of compounding periods per year. If a rate is already an APY or effective annual return, it should not be divided across compounding periods again.
The shortcuts exist because most people do not calculate logarithms mentally. Instead, they divide a convenient number by the annual percentage rate:
- 70
- 72
- 69.3
Each number favors a different balance between convenience and fit.
Understanding the Mathematics Behind Each Rule
The Rule of 69.3 lines up with the continuous-compounding equation because:
100 × ln(2) ≈ 69.3147
When the rate is entered as a percentage, 69.3 ÷ rate is essentially the continuous-compounding result.
The Rule of 70 rounds that constant to a simpler number and often fits low annual growth rates. The Rule of 72 moves farther from the continuous constant, but gains easier divisibility and a better fit for many moderate annual-compounding examples.
That is why no rule is correct for every assumption. The right comparison depends on the compounding method.
Rule of 70
The Rule of 70 estimates doubling time as:
Doubling Time ≈ 70 ÷ Annual Growth Rate (%)
This shortcut is commonly used in economic illustrations because many variables grow at low annual rates.
Examples include inflation, GDP, population, and productivity.
Suppose inflation is assumed to average 2%.
The Rule of 70 estimates:
70 ÷ 2 = 35 years
With annual compounding at 2%, the exact result is also about 35.00 years, so the shortcut is very close in that scenario.
Rule of 72
The Rule of 72 uses:
Doubling Time ≈ 72 ÷ Annual Interest Rate (%)
The number 72 is useful because it divides cleanly by many common rates, including 2, 3, 4, 6, 8, 9, and 12.
| Annual Return | Rule of 72 Estimate |
|---|---|
| 4% | 18 years |
| 6% | 12 years |
| 8% | 9 years |
| 9% | 8 years |
| 12% | 6 years |
Illustrative - actual results vary.
For a broader rate-by-rate view, see How Long Does Money Take to Double at Different Interest Rates?.
Rule of 69.3
The Rule of 69.3 estimates doubling time as:
Doubling Time ≈ 69.3 ÷ Annual Growth Rate (%)
This rule comes directly from 100 × ln(2), so it closely matches the exact formula for continuous compounding.
Continuous compounding is useful in finance theory, but it is not the typical mechanism for ordinary consumer investment products.
Savings accounts and CDs may credit interest on a defined schedule, such as daily or monthly. Market investments, such as mutual funds and ETFs, usually do not have a guaranteed constant rate or fixed compounding frequency; their compounding effect depends on actual returns and reinvested distributions.
Because of that, the Rule of 69.3 is best understood as a continuous-compounding shortcut, not as a general investing rule.
Rule of 70 vs. Rule of 72 vs. Rule of 69.3
Before comparing numerical accuracy, it helps to separate the intended use of each shortcut.
| Rule | Common Use | Ease of Mental Math | Best Fit |
|---|---|---|---|
| Rule of 70 | Low-rate economic illustrations | Easy | Lower annual growth rates |
| Rule of 72 | Quick mental estimates | Excellent | Moderate annual-compounding examples |
| Rule of 69.3 | Continuous-compounding math | Moderate | Continuous compounding |
Illustrative - actual results vary.
These are approximations, not replacements for the relevant exact formula.
Why Is the Rule of 72 So Easy to Use?
If the Rule of 69.3 is mathematically tied to continuous compounding, why do many people remember the Rule of 72 instead? Usability is the main reason.
Imagine an assumed annual rate of 8%:
- Rule of 70 -> 8.75 years
- Rule of 72 -> 9 years
- Rule of 69.3 -> 8.66 years
Most people can instantly calculate 72 ÷ 8. That convenience makes the Rule of 72 a practical educational shortcut when the goal is a rough timeline rather than a complete projection.
Which Rule Is Most Accurate?
The answer depends on what you are measuring. The table below compares the three shortcuts with the exact result for annual compounding.
| Annual rate | Exact annual | Rule of 70 | Rule of 72 | Rule of 69.3 | Closest shortcut |
|---|---|---|---|---|---|
| 2% | 35.00 years | 35.00 | 36.00 | 34.65 | Rule of 70 |
| 4% | 17.67 years | 17.50 | 18.00 | 17.33 | Rule of 70 |
| 6% | 11.90 years | 11.67 | 12.00 | 11.55 | Rule of 72 |
| 8% | 9.01 years | 8.75 | 9.00 | 8.66 | Rule of 72 |
| 10% | 7.27 years | 7.00 | 7.20 | 6.93 | Rule of 72 |
| 12% | 6.12 years | 5.83 | 6.00 | 5.78 | Rule of 72 |
Illustrative - actual results vary.
The pattern is clear:
- The Rule of 70 is very close at low annual rates.
- The Rule of 72 becomes closer in the moderate annual-compounding range.
- The Rule of 69.3 should not be judged as a failed annual-compounding shortcut; it is designed around continuous compounding.
For compounding-frequency comparisons, see When Does the Rule of 72 Stop Being Accurate?.
Tradeoff Analysis: Simplicity vs. Precision
Choosing a shortcut is about matching the method to the assumption.
| If your priority is... | Better Choice | Why |
|---|---|---|
| Mental calculations | Rule of 72 | Divides evenly by many common rates |
| Low-rate economic illustration | Rule of 70 | Very close at lower annual rates |
| Continuous compounding | Rule of 69.3 | Matches the continuous-compounding constant |
| Detailed projection | Exact calculation | Models the stated rate, timing, and compounding assumptions more precisely |
Illustrative - actual results vary.
Even an exact calculation is only exact for the inputs provided. It does not account for volatility, changing rates, taxes, fees, or cash flows unless those assumptions are included.
When Should You Skip the Shortcut?
Use an exact compound-interest calculation when you need to model:
- recurring contributions,
- withdrawals,
- a specific time horizon,
- taxes or fees,
- different compounding frequencies,
- a stated nominal annual rate versus an APY or effective annual return.
For market investments, exact math can make the scenario internally consistent, but it cannot guarantee the assumed return.
Run Your Own Scenario
Instead of asking which shortcut is always "best," compare results under your chosen assumptions.
The Rule of 72 Calculator lets you:
- estimate doubling time using the Rule of 72,
- compare it with exact annual, periodic, or continuous compounding,
- test different assumed nominal annual rates,
- see how the difference changes in months.
You may also find these calculators helpful:
For the broader compounding cluster, visit the Compound Interest and Growth topic page.
Key Takeaways
- The Rule of 70 is very close at low annual growth rates.
- The Rule of 72 is the most convenient mental shortcut for many moderate annual-compounding examples.
- The Rule of 69.3 is tied to continuous compounding.
- Exact formulas are more precise for the stated assumptions, but they do not make uncertain returns reliable.
- Use a calculator when compounding frequency, cash flows, or planning decisions matter.
Frequently Asked Questions
Is the Rule of 72 more accurate than the Rule of 70?
For annual compounding, Rule of 70 is closer at 2% or 4%, while Rule of 72 is closer around 6% to 12%.
Why is the Rule of 70 used for inflation examples?
Inflation and other economic indicators are often illustrated with low annual growth rates, where Rule of 70 can be very close to the annual-compounding result.
Is the Rule of 69.3 always the most accurate?
No. It closely matches continuous compounding because it comes from 100 × ln(2). Against an annual-compounding benchmark, it can be less close.
Do mutual funds and ETFs have a fixed compounding frequency?
Usually no. Their results depend on market returns, price changes, fees, and reinvested distributions, not a guaranteed constant rate or fixed schedule.
Which rule should I memorize?
For moderate assumed annual rates, Rule of 72 is usually easiest. For low-rate economic illustrations, Rule of 70 may fit better. For continuous compounding, use Rule of 69.3 or the exact formula.
Can any of these rules predict investment returns?
No. They estimate doubling time under a constant assumed rate. They do not account for market volatility, taxes, fees, inflation, or changing returns.
Should I use a calculator instead of these shortcuts?
Use shortcuts for quick estimates. Use the relevant exact formula or a calculator when comparing scenarios, modeling cash flows, or making financial decisions.
This article is for informational purposes only and does not constitute financial advice. Please consult a qualified financial advisor before making financial decisions.