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Doubling Time Calculator
Calculate doubling time from a growth rate using the Rule of 72, exact formula, and continuous compounding. Also tripling time, comparison table across rates, and exponential growth curve visual.
Doubling Time Calculator
As a percentage (e.g. 6 for 6%)
Starting amount for growth curve
Exact Doubling Time⧉
11.90
ln(2)/ln(1 + 0.0600) periods
Rule of 72⧉
12.00
72 / 6.00 — error: 0.88%
Continuous Compounding⧉
11.55
ln(2) / 0.0600 — error: 2.89%
Rule of 69.3⧉
11.55
69.3 / 6.00 — most accurate for low rates
Tripling Time⧉
18.85
ln(3)/ln(1+r) — Rule of 114: 19.00
10× Time⧉
39.52
ln(10)/ln(1+r) — time to grow 10-fold
Exponential Growth Over 5 Doublings
Red dashed = doubling milestones at t = 11.9, 23.8, …
Time to Multiply
2× (double)11.90 periods
3× (triple)18.85 periods
4× (quadruple)23.79 periods
10× (ten-fold)39.52 periods
Doubling Time by Growth Rate
| Rate (%) | Exact | Rule of 72 | Error (%) | Tripling |
|---|---|---|---|---|
| 0.5% | 138.98 | 144.00 | 3.62% | 220.27 |
| 1% | 69.66 | 72.00 | 3.36% | 110.41 |
| 2% | 35.00 | 36.00 | 2.85% | 55.48 |
| 3% | 23.45 | 24.00 | 2.35% | 37.17 |
| 4% | 17.67 | 18.00 | 1.85% | 28.01 |
| 5% | 14.21 | 14.40 | 1.36% | 22.52 |
| 6% | 11.90 | 12.00 | 0.88% | 18.85 |
| 7% | 10.24 | 10.29 | 0.40% | 16.24 |
| 8% | 9.01 | 9.00 | 0.07% | 14.27 |
| 9% | 8.04 | 8.00 | 0.54% | 12.75 |
| 10% | 7.27 | 7.20 | 1.00% | 11.53 |
| 12% | 6.12 | 6.00 | 1.90% | 9.69 |
| 15% | 4.96 | 4.80 | 3.22% | 7.86 |
| 18% | 4.19 | 4.00 | 4.49% | 6.64 |
| 20% | 3.80 | 3.60 | 5.31% | 6.03 |
| 25% | 3.11 | 2.88 | 7.28% | 4.92 |
| 30% | 2.64 | 2.40 | 9.16% | 4.19 |
| 50% | 1.71 | 1.44 | 15.77% | 2.71 |
Planning notes, formulas, and examples
About the Doubling Time Calculator
Doubling time is the period required for a quantity to double at a constant growth rate. The famous Rule of 72 gives a quick estimate: doubling time ≈ 72 / (growth rate in percent). For example, an investment growing at 6% per year doubles in about 72/6 = 12 years. The exact formula is ln(2)/ln(1 + r), and for continuous compounding it simplifies to ln(2)/r. This concept applies far beyond finance — population growth, bacterial cultures, inflation, Moore's Law, and radioactive decay (half-life is the reverse). This calculator computes the doubling time using all three methods (Rule of 72, exact discrete, continuous), plus the tripling time (ln(3)/ln(1+r)), Rule of 69.3 and Rule of 70 variants, and a growth rate comparison table. An exponential curve visual shows how your quantity grows over multiple doubling periods. Presets cover common growth rates from savings accounts to hypergrowth startups.
When This Page Helps
The Rule of 72 is a great mental shortcut, but how accurate is it? This calculator compares the Rule of 72 with the exact formula, Rule of 70, Rule of 69.3, and continuous compounding — showing the percentage error for each approximation. It also extends to tripling time (Rule of 114), quadrupling, and 10× time, with an exponential growth curve visualizing your investment or population over multiple doubling periods.
How to Use the Inputs
- Enter the growth rate as a percentage (e.g., 6 for 6%) or switch to decimal mode.
- Click a preset like "6% (stocks)" or "3% (inflation)" to load a common rate.
- Review the exact doubling time, Rule of 72, Rule of 70, Rule of 69.3, and continuous results.
- See the tripling time, quadrupling time, and 10× time in the additional output cards.
- Enter an initial value and number of doublings to generate the exponential growth curve.
- Examine the rate comparison table showing doubling times across 18 different growth rates.
- Adjust Precision to control the number of decimal places displayed.
Formula used
Exact (discrete): t = ln(2) / ln(1 + r)
Rule of 72: t ≈ 72 / (r × 100)
Continuous: t = ln(2) / r
Tripling time: t = ln(3) / ln(1 + r)
Rule of 69.3: t ≈ 69.3 / (r × 100)Example Calculation
Result: Exact: 11.90 years, Rule of 72: 12 years
At 6% annual growth, the exact doubling time is ln(2)/ln(1.06) ≈ 11.90 years. The Rule of 72 gives 72/6 = 12 years.
Tips & Best Practices
- Check that all inputs use the same scale and assumptions before trusting the result.
- Compare the answer with the worked example or a rough estimate to catch entry mistakes.
Why the Rule of 72 Works
The exact doubling time is ln(2)/ln(1+r). For small r, ln(1+r) ≈ r, so doubling time ≈ ln(2)/r = 0.693/r. Multiplying by 100 gives 69.3/(r%). The number 72 is used instead of 69.3 because 72 has many divisors (1,2,3,4,6,8,9,12,18,24,36,72), making mental division easy. The slight overestimate from 69.3 to 72 actually compensates for the second-order term in the Taylor expansion of ln(1+r), making the Rule of 72 surprisingly accurate for rates between 2% and 20%.
Applications Beyond Finance
Doubling time applies to any exponential growth: bacterial colonies doubling every 20 minutes, world population doubling every ~50 years historically, computer transistor counts doubling every ~2 years (Moore's Law), and nuclear chain reactions. In medicine, tumor doubling time is a key metric for assessing cancer aggressiveness. In inflation analysis, the Rule of 72 tells you when prices will double: at 3% inflation, purchasing power halves in 24 years.
Tripling Time and Beyond
The tripling time formula replaces ln(2) with ln(3): t₃ = ln(3)/ln(1+r). The mental math shortcut is the Rule of 114: tripling time ≈ 114/(r%). For quadrupling, use ln(4) = 2×ln(2), giving exactly twice the doubling time. For 10× growth, use ln(10)/ln(1+r). These extended formulas are useful for long-term financial planning, epidemiological modeling, and any scenario where you need to project beyond a single doubling.
Sources & Methodology
Last updated:
Frequently Asked Questions
A mental math shortcut: divide 72 by the growth rate (as a percent) to estimate doubling time. It works best for rates between 2% and 20%.
72 is used because it has many divisors (1,2,3,4,6,8,9,12) making mental division easy. The mathematically precise number is 69.3 (= 100×ln2).
Doubling time = ln(2) / ln(1 + r), where r is the growth rate as a decimal. This assumes discrete compounding per period.
Very accurate for rates 2–20%. At 8%, it gives 9.0 years vs exact 9.01. At 1% it overestimates slightly; at 50% it underestimates.
Replace ln(2) with ln(3): tripling time = ln(3)/ln(1+r). The Rule of 114 is the tripling equivalent of the Rule of 72.
Yes — for a decay rate r, the half-life equals ln(2)/ln(1+r) using the same formula. Negative rates give a half-life instead of doubling time.
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