ALGEBRAPercentagesMathematics Calculator
📈

Doubling Time (Rule of 72)

Doubling time is how long it takes for a value to double at a given growth rate. The Rule of 72: years ≈ 72 / rate. At 7% annual, money doubles in ~10 years. Exact formula: ln(2) / ln(1 + r/n)^n for n compounding periods.

Concept Fundamentals
72 / rate
Rule of 72
~10 years
7% annual
~7.2 years
10% annual
ln(2)/ln(1+r)
Exact

Did our AI summary help? Let us know.

At 7% annual return, $1 becomes $2 in ~10 years, $4 in ~20 years. Rule of 72 works best for rates between 4% and 15%; outside that, use exact formula. Compounding frequency matters: monthly compounds faster than annual at same rate.

Key quantities
72 / rate
Rule of 72
Key relation
~10 years
7% annual
Key relation
~7.2 years
10% annual
Key relation
ln(2)/ln(1+r)
Exact
Key relation

Ready to run the numbers?

Why: Investors want to know when their money doubles. Population biologists model growth. The Rule of 72 is a mental shortcut; exact formulas account for compounding frequency (annual, monthly, continuous).

How: Rule of 72: doubling time ≈ 72 / annual rate (as %). Exact: t = ln(2) / (n × ln(1 + r/n)) for n compounding periods per year. Continuous: t = ln(2) / r.

At 7% annual return, $1 becomes $2 in ~10 years, $4 in ~20 years.Rule of 72 works best for rates between 4% and 15%; outside that, use exact formula.
Sources:Rule of 72Exponential Growth

Run the calculator when you are ready.

Exponential GrowthConstant percentage growth leads to doubling in a predictable time.

Quick Examples — Click to Load

Input Values

%
doubling_time
CALCULATED
$ calc --rate=7% --freq=annual
Doubling Time
10.24 years
Doubling Time (days)
3742 days
Rule of 72
10.29 years
Rule of 72 Error
+0.4%
Tripling Time
16.24 years
2× Time
10.24 years
Share:

Growth Curve (Initial to Doubled)

Rule of 72 vs Exact (Annual)

Step-by-Step Breakdown

INPUT
Growth rate
7%
ext{Annual} ext{rate}
Compounding
annual
(1+r/n)^(nt)
CALCULATION
Exact formula
ln(2) / ln(1 + r/100)
RESULT
Exact doubling time
10.24 years
Doubling time (days)
3742 days
Rule of 72
10.29 years
72 div r (approximation)
Rule of 72 accuracy
+0.4%
ext{Error} ext{vs} ext{exact}
ADVANCED
Tripling time
16.24 years
\text{ln}(3)/\text{ln}(1+r)
2× time
10.24 years
ln(2)/ln(1+r)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

— History

— Extended rules

Key Takeaways

  • • Rule of 72 is a quick mental shortcut: divide 72 by the growth rate to estimate doubling time in years
  • • Rule of 72 works best for rates between 5% and 10% — outside that range, use the exact formula
  • • Continuous compounding grows faster than discrete (annual/monthly) — doubling time is shorter
  • • For tripling, use Rule of 114 (114 ÷ r); for 10× use Rule of 230

Did You Know?

📐The Rule of 72 comes from ln(2) ≈ 0.693; 69.3/r is more accurate for continuous compounding, but 72 is easier to divideSource: Mathematics
📈At 7% annual return, your money doubles in ~10 years. At 10%, it doubles in ~7 years — small rate changes have huge long-term effectsSource: Investing
🌍World population grew ~1.1% annually in the 20th century — a doubling time of ~63 years. It actually doubled from 3.5B to 7B in ~40 yearsSource: Demographics
🦠Bacteria can double every 20 minutes under ideal conditions — that's a 50%+ growth rate per hour, explaining exponential outbreaksSource: Biology
💡Rule of 69.3 is exact for continuous compounding. The Rule of 72 uses 72 because it has many divisors (2,3,4,6,8,9,12) for mental mathSource: Finance
📉Negative growth (decay) also has a "half-life" — the time to halve. Formula: t = ln(0.5)/ln(1-r/100)Source: Physics

How Doubling Time Works

Doubling time is the period required for a quantity to double in size or value under exponential growth. It applies to investments, populations, bacteria, and any process that grows at a constant percentage rate.

Exact Formula (Annual Compounding)

t = ln(2) / ln(1 + r/100)

Where r is the annual growth rate in percent. For continuous compounding: t = ln(2) / (r/100) = 69.31/r years.

Rule of 72

Divide 72 by the growth rate to get approximate doubling time. At 8%, 72÷8 = 9 years. The exact answer is 9.01 years — remarkably close for such a simple rule.

Compounding Frequency

More frequent compounding (monthly, daily, continuous) means faster growth and shorter doubling time. Savings accounts compound daily; stocks assume annual compounding for simplicity.

Expert Tips

Rule of 69.3

For continuous compounding, use 69.3/r instead of 72/r. More accurate when growth is truly continuous (e.g., bacteria, radioactive decay).

Rule of 114 for Tripling

To estimate tripling time: 114 ÷ r. For 10× growth: 230 ÷ r. Same logic as Rule of 72, using ln(3) and ln(10).

When Rule of 72 Fails

At very low rates (<2%) or very high rates (>15%), the Rule of 72 error grows. Use the exact formula for precision.

Quick Mental Check

6% → 12 years, 8% → 9 years, 10% → 7.2 years. Memorize these for quick investment projections.

Frequently Asked Questions

What is the Rule of 72?

A simple approximation: divide 72 by your growth rate (in percent) to get the doubling time in years. At 8%, 72÷8 = 9 years. It works well for rates between 5% and 10%.

Why does compounding frequency matter?

More frequent compounding (monthly, daily, continuous) means interest earns interest more often, so you reach 2× faster. Continuous compounding is the theoretical limit.

How do I calculate tripling time?

Use t = ln(3)/ln(1+r/100) for annual compounding, or the Rule of 114: 114 ÷ r. For continuous: t = ln(3)/(r/100) = 109.9/r years.

When is the Rule of 72 most accurate?

Around 8% — the rule was chosen because 72 has many divisors and gives good results for typical investment rates (6–10%).

What about negative growth (decay)?

Use the same formula with a negative rate, or use half-life: t = ln(0.5)/ln(1-r/100). For small decay rates, Rule of 70 (70/|r|) approximates half-life.

Can I use this for population growth?

Yes. Population often grows at 1–3% annually. At 2%, doubling time is ~35 years. At 1.1%, it's ~63 years.

Quick Reference

72 ÷ r
Rule of 72 (approx)
69.3 ÷ r
Continuous (exact)
114 ÷ r
Tripling (approx)
ln(2)/ln(1+r)
Exact (annual)

Disclaimer: This calculator provides mathematical results for educational purposes. For investment decisions, tax planning, or financial advice, consult a qualified professional. Past growth rates do not guarantee future results.

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

Average Percentage Calculator

Compute average of percentages using simple mean, weighted average (by sample size or importance), geometric mean for multiplicative data, and harmonic mean...

Mathematics

Decimal to Percent Converter

Instant decimal-to-percent conversion with fraction equivalent display. Enter any decimal (e.g., 0.75) and get the percentage (75%) plus the simplified...

Mathematics

Fraction to Percent Calculator

Convert fractions to percentages with GCD simplification, mixed number support, and repeating decimal handling. Enter proper, improper, or mixed fractions...

Mathematics

Percentage Calculator

All-in-one percentage calculator with five modes: percent of a number, what percent is X of Y, percentage change, reverse percentage (find original from...

Mathematics

Percentage Change Calculator

Calculate percentage change between two values with multiple methods: standard relative change, CAGR for multi-year growth, log change for multiplicative...

Mathematics

Percentage Decrease Calculator

Calculate percentage decrease with support for successive decreases, half-life decay, and depreciation. Handles discounts, value decline, radioactive decay...

Mathematics