Doubling Time Calculator
Find out how long it takes an investment, population, or any growing quantity to double. Uses the Rule of 72 for quick estimates and the exact formula for precision.
How to Use
- Choose a mode: Use "From Growth Rate" to find how long doubling takes, or "From Doubling Time" to find the required rate.
- Enter the value: Type the growth rate percentage or the number of years.
- Compare results: See both the exact logarithmic answer and the Rule of 72 approximation side by side.
What You'll Discover
- Exact doubling time — in "From Growth Rate" mode, the precise number of years for a value to double at a constant annual rate, using ln(2) ÷ ln(1 + r).
- Rule of 72 estimate — the quick mental-math answer, 72 ÷ rate, shown next to the exact figure so you can see how good the shortcut is.
- Difference — the gap between the exact answer and the Rule of 72, in years. It is smallest near 8% and grows at very low or very high rates.
- Required growth rate — in "From Doubling Time" mode, the exact annual rate needed to double in a given number of years, (21/T − 1) × 100, plus the Rule of 72 estimate and the difference in percentage points.
The calculator assumes one compounding step per period and a steady rate. The rate must be above zero; a zero or negative rate never doubles, so the results stay blank.
Formula
Exact Doubling Time
The precise formula uses the natural logarithm:
T = ln(2) / ln(1 + r/100)
Where T is the doubling time in years and r is the annual growth rate as a percentage. For example, at 7%: T = ln(2) / ln(1.07) ≈ 0.6931 / 0.0677 ≈ 10.24 years. Any logarithm base works, as long as you use the same base on the top and bottom.
Rule of 72
The Rule of 72 is a handy mental shortcut — simply divide 72 by the annual growth rate:
T ≈ 72 / r
At 7%: T ≈ 72 / 7 ≈ 10.29 years, only 0.04 years more than the exact answer. Between 6% and 10% the rule stays within about 0.1 years of the exact value, and it is almost perfect near 8%.
Required Growth Rate from Doubling Time
r = (2^(1/T) − 1) × 100%
Rule of 72 approximation: r ≈ 72 / T. To double in 10 years you need 20.1 − 1 ≈ 7.18% a year; the rule says 7.2%.
Examples
Example 1 — Investment at 7% a year
An investment growing at a steady 7% a year doubles in ln(2) / ln(1.07) ≈ 10.24 years. Rule of 72: 72 / 7 ≈ 10.29 years. $10,000 becomes about $20,000 after 10.24 years and about $40,000 after twice that time.
Example 2 — High-growth investment
A venture fund targets 25% annual returns. Exact doubling time: ln(2) / ln(1.25) ≈ 3.11 years. Rule of 72: 72 / 25 = 2.88 years, about 0.23 years too short. The rule gets less accurate as rates rise.
Example 3 — Inflation context
At 3% annual inflation, prices double — and the purchasing power of money halves — in ln(2) / ln(1.03) ≈ 23.45 years. Rule of 72: 72 / 3 = 24 years.
Example 4 — Population growth
A population growing at 2% per year doubles in ln(2) / ln(1.02) ≈ 35.0 years. Rule of 72: 72 / 2 = 36 years.
How Accurate Is the Rule of 72? Exact vs Rule of 72 vs Rule of 70
The Rule of 72 is an estimate, and its error depends on the rate. The table below compares the exact doubling time with three shortcuts. All figures are in years and assume annual compounding, except the last column, which assumes continuous compounding.
| Annual rate | Exact (years) | Rule of 72 | Rule of 70 | Continuous, ln(2) ÷ r |
|---|---|---|---|---|
| 1% | 69.66 | 72.00 | 70.00 | 69.31 |
| 2% | 35.00 | 36.00 | 35.00 | 34.66 |
| 3% | 23.45 | 24.00 | 23.33 | 23.10 |
| 5% | 14.21 | 14.40 | 14.00 | 13.86 |
| 6% | 11.90 | 12.00 | 11.67 | 11.55 |
| 8% | 9.01 | 9.00 | 8.75 | 8.66 |
| 10% | 7.27 | 7.20 | 7.00 | 6.93 |
| 12% | 6.12 | 6.00 | 5.83 | 5.78 |
| 15% | 4.96 | 4.80 | 4.67 | 4.62 |
| 20% | 3.80 | 3.60 | 3.50 | 3.47 |
| 50% | 1.71 | 1.44 | 1.40 | 1.39 |
| 100% | 1.00 | 0.72 | 0.70 | 0.69 |
Three patterns stand out. Below about 7.85%, the Rule of 72 slightly overestimates the doubling time; above it, the rule underestimates. The Rule of 70 is closer for low rates such as 1–3%, which is why it is popular for population and economic growth. And the Rule of 69.3 (100 × ln 2 ≈ 69.3) is exact only for continuous compounding.
For quick planning, any of these is fine. For a contract, a loan or a report, use the exact figure the calculator shows.
Compounding Frequency, Periods and Halving Time
More frequent compounding shortens doubling time
The calculator treats the rate as compounding once per period. If interest compounds monthly, use the monthly rate instead. At a 6% nominal annual rate compounded monthly, the doubling time is ln(2) ÷ (12 × ln(1.005)) ≈ 11.58 years, versus 11.90 years with annual compounding. With continuous compounding it is ln(2) ÷ 0.06 ≈ 11.55 years.
Rates per month, per day or per hour
The formula does not care about the length of a period. If you enter a monthly growth rate, the answer is a number of months, even though the result label says years. The same applies to daily or hourly rates, such as the growth of a cell culture.
Tripling, quadrupling and tenfold growth
Replace the 2 in the formula with the multiple you want: T = ln(k) ÷ ln(1 + r). At 7% a year, a value triples in about 16.24 years, quadruples in about 20.49 years (exactly two doubling times) and grows tenfold in about 34.03 years. The mental shortcuts are the Rule of 114 for tripling and the Rule of 144 for quadrupling.
Halving time for shrinking values
A value that falls by a fixed percentage each period has a halving time, the counterpart of doubling time: T = ln(0.5) ÷ ln(1 − r). A value losing 3% a year halves in about 22.76 years; at −7% a year it halves in about 9.55 years. This calculator only accepts positive rates, so work halving time out by hand with this formula.
How Long to Double If...
How long does it take to double my money at 5% interest?
At 5% a year, compounded annually, money doubles in about 14.21 years. The exact formula gives ln(2) ÷ ln(1.05) ≈ 14.21. The Rule of 72 says 72 ÷ 5 = 14.4 years, about 0.19 years too long.
How long does it take to double my money at 8%?
At 8% a year, money doubles in about 9.01 years. The Rule of 72 gives exactly 9 years, which is almost spot on. 8% is close to the rate where the rule and the exact formula meet.
How long does it take to double at 10%?
At 10% a year, a value doubles in about 7.27 years. The calculation is ln(2) ÷ ln(1.10) ≈ 7.27. The Rule of 72 says 7.2 years, so it is about 0.07 years short.
How long does it take to double at 12%?
At 12% a year, doubling takes about 6.12 years. The Rule of 72 gives 6 years, about 0.12 years short. Above roughly 8%, the rule always comes in a little under the exact answer.
How long does it take to double at 1%?
At 1% a year, doubling takes about 69.66 years. The Rule of 72 says 72 years, more than two years too long. At low rates the Rule of 70 (70 years) or 69.3 is a much better estimate.
How long until prices double if inflation is 3%?
At 3% inflation, prices double in about 23.45 years. In that time, a dollar loses half of its purchasing power. The Rule of 72 gives 24 years.
How long does it take a population growing 1.5% a year to double?
A population growing 1.5% a year doubles in about 46.56 years. The calculation is ln(2) ÷ ln(1.015). Real populations rarely hold one growth rate for that long, so treat it as a projection, not a forecast.
How long does credit card debt at 20% APR take to double?
With no payments, a balance at 20% APR compounded monthly doubles in about 3.49 years. The monthly rate is 20% ÷ 12 ≈ 1.667%, and ln(2) ÷ (12 × ln(1.01667)) ≈ 3.49 years. With annual compounding it would take about 3.80 years.
What growth rate doubles a value in 10 years?
You need about 7.18% a year to double in 10 years. Switch to "From Doubling Time" and enter 10: (21/10 − 1) × 100 ≈ 7.18%. The Rule of 72 says 7.2%.
What growth rate doubles a value in 5 years?
Doubling in 5 years takes about 14.87% a year. The calculation is (21/5 − 1) × 100 ≈ 14.87%. The Rule of 72 gives 14.4%, about half a percentage point too low.
People Also Search For
How to calculate doubling time by hand
Divide ln(2), which is about 0.6931, by ln(1 + r), where r is the rate as a decimal. At 6%: ln(1.06) ≈ 0.05827, and 0.6931 ÷ 0.05827 ≈ 11.90 years. Any calculator with a log key works, as long as you use the same log on top and bottom.
Rule of 72 vs Rule of 70
Use 70 for low rates (about 1–4%) and 72 for typical investment rates (about 6–10%). At 2%, the Rule of 70 gives exactly the 35-year answer, while 72 gives 36. At 10%, the Rule of 72 gives 7.2 years against an exact 7.27, while 70 gives 7.0.
How to calculate cell doubling time
Use Td = t × ln(2) ÷ ln(N ÷ N0), where t is the elapsed time and N0 and N are the starting and ending counts. If a culture grows from 100,000 to 800,000 cells in 72 hours, that is three doublings, so the doubling time is 24 hours. This works for any exponential growth measured at two points.
Doubling time from a growth rate per month
Enter the monthly rate and read the result as months. A business growing 5% a month doubles in about 14.21 months. Divide by 12 to convert to years (about 1.18 years).
What is the doubling time formula for continuous growth?
For continuous growth, T = ln(2) ÷ k, where k is the continuous rate as a decimal. At k = 0.05, T ≈ 13.86. This is why 69.3 divided by the rate is exact for continuous compounding, and it is the form used in biology and physics.
Is doubling time the same as half-life?
They are mirror images: doubling time applies to growth, half-life to decay. Both come from the same exponential formula. Half-life uses ln(0.5) in place of ln(2) and a shrinking factor (1 − r) in place of (1 + r).
Frequently Asked Questions
What is the Rule of 72?
The Rule of 72 is a mental math shortcut to estimate how long it takes an investment to double. Divide 72 by the annual interest or growth rate. For example, at 6% per year: 72 / 6 = 12 years, against an exact 11.90 years. The rule is most accurate for rates between about 6% and 10% and becomes less precise at very high or very low rates.
Why 72 and not 70 or 69?
The exact constant for continuous compounding is 69.3 (100 × ln 2). With annual compounding the best constant rises with the rate, and 72 fits typical rates near 8% well. 72 is also divisible by many common rates (2, 3, 4, 6, 8, 9, 12, 18, 24, 36), which makes mental math easier. Below about 7.85% the Rule of 72 slightly overestimates doubling time, and above that it slightly underestimates it.
Does the Rule of 72 work for any growth scenario?
It works for any quantity growing at a constant compound rate, including investments, population, debt, prices (inflation), or data volumes. The rule applies whenever growth is exponential (each period's growth is based on the current total, not just the original). It does not apply to simple interest or to growth that adds a fixed amount each period.
How does compounding frequency affect doubling time?
This calculator assumes annual compounding. With more frequent compounding (monthly, daily), the effective annual rate is slightly higher, so the doubling time is slightly shorter. At 6%, doubling takes 11.90 years with annual compounding, 11.58 years with monthly compounding and 11.55 years with continuous compounding. For continuous compounding, the exact formula is T = ln(2) / r, where r is the nominal rate as a decimal.
Can I use this for debt growth?
Yes. If a debt grows at 20% a year, compounded annually, and you make no payments, it doubles in ln(2) / ln(1.20) ≈ 3.80 years. The Rule of 72 gives 72 / 20 = 3.6 years. Credit cards usually compound monthly, which shortens this to about 3.49 years. High-interest debt compounds against you just as effectively as investments compound for you.
Does the starting amount change the doubling time?
No. Doubling time depends only on the growth rate. $100 and $1,000,000 both take about 10.24 years to double at 7% a year. That is why the calculator asks only for the rate.
What happens if I enter a negative or zero growth rate?
A value that is not growing never doubles, so the calculator shows no result. For a shrinking value, calculate the halving time instead with T = ln(0.5) / ln(1 − r). At a 3% annual decline, that is about 22.76 years.
Can I use a monthly or daily growth rate?
Yes. Enter the rate per period and read the answer in the same period. A 2% monthly rate gives about 35.0, which means 35 months. The result label says years because the default use is annual rates.
How long does it take to triple or quadruple?
Quadrupling takes exactly two doubling times. Tripling takes ln(3) / ln(1 + r). At 7% a year, a value triples in about 16.24 years and quadruples in about 20.49 years. The Rule of 114 (tripling) and Rule of 144 (quadrupling) are the matching shortcuts.
Why is the Difference row larger at high rates?
The Rule of 72 is derived from an approximation that holds best for small rates. As the rate grows, ln(1 + r) moves further from r, so the rule drifts. At 50% the rule gives 1.44 years against an exact 1.71 years, a gap of about 0.27 years.
Is my data stored or sent anywhere?
No. All calculations run in your browser. Nothing you type is sent to a server or saved, and no account is needed.
Last reviewed: November 2026 — formulas and worked examples verified.