What is the Rule of 72?

The Rule of 72 is a quick way to estimate doubling time: divide 72 by the growth rate percentage. For example, at 8% growth, it takes about 72/8 = 9 periods to double. Easy mental math; accurate within ~5% for typical rates 2-12%.

How accurate is the Rule of 72?

Most accurate for rates between 2% and 10%. At higher rates, becomes less precise. Exact formula t = ln(2) / ln(1 + r) always gives correct answer. For example, at 5%: Rule of 72 says 14.4 years, exact says 14.21 years (close). At 50%: Rule of 72 says 1.44 years, exact says 1.71 years (more off).

How long until my investment doubles?

Depends on growth rate. At 5%: ~14 years (Rule of 72). At 7%: ~10 years. At 10%: ~7 years. At 12%: ~6 years. Historical stock market (S&P 500 long-term ~10%): doubles every 7-10 years. Inflation-adjusted (real growth ~7%): doubles every ~10 years.

What's the doubling time for inflation?

2% inflation: ~36 years. 3% inflation: ~24 years. 5% inflation: ~14 years. 10% inflation: ~7 years (rare in developed countries, common in some emerging markets). US historical inflation ~2-3%, doubling every 24-36 years. Erosion of purchasing power compounds.

What's the formula for tripling time?

Replace ln(2) with ln(3) ≈ 1.0986 in exact formula: t_triple = ln(3) / ln(1 + r). Rule of 110 (approximation): t_triple ≈ 110 / rate%. For 7% growth: tripling time ≈ 110/7 ≈ 15.7 years (vs exact 16.2 years).

Why is the number 72?

Has many divisors (1,2,3,4,6,8,9,12,18,24,36,72) — easy mental division. Mathematically, exact value is ln(2)/r ≈ 0.693/r, expressed as 69.3/r%. 72 is close enough and easier to work with. Also, slightly overestimates doubling time at higher rates, giving conservative estimates.

How is doubling time used in epidemiology?

Tracking epidemic growth. Early COVID-19 doubling times of 3-5 days (15-25% daily growth) indicated rapid spread. Once interventions worked, doubling time stretched to weeks. Public health officials monitor doubling time to assess intervention effectiveness and forecast capacity needs.

Does inflation affect investment doubling?

Yes. Real (inflation-adjusted) growth = Nominal - Inflation (approximately). If investment grows 7% but inflation is 3%, real growth is 4%. Nominal doubling: ~10 years. Real doubling (purchasing power): ~17.7 years. Always check whether returns are nominal or real for accurate long-term planning.

Doubling Time Calculator

Enter a growth rate to find how long it takes for a value to double. Shows both the Rule of 72 approximation (72 / rate) and the exact doubling time using ln(2) / ln(1 + r).

Doubling time is the time it takes for a value growing at a constant rate to double. It's a powerful way to grasp exponential growth: instead of thinking about percentages, think about how often the value doubles. A 10% growth rate doubles every ~7 years. A 7% rate every ~10 years. A 3% rate every ~24 years. Small rate differences compound to large time differences.

The **Rule of 72** is a famous mental math shortcut: divide 72 by the growth rate percentage to estimate doubling time. For 6% growth: 72/6 = 12 years to double. Accurate within ~5% for typical rates (2-12%). Quick and useful for back-of-envelope investment, demographic, or economic estimates.

The **exact formula** is t = ln(2) / ln(1 + r), where r is the rate as a decimal. ln(2) ≈ 0.6931, so t ≈ 0.693 / r for small r. Compare to Rule of 72: 72 / (100 × r) = 0.72/r. Close to the exact value, but slightly larger — exact gives smaller doubling time for high rates.

Doubling time is essential for understanding exponential phenomena: - **Investments**: how long until your portfolio doubles? - **Population**: when will the city's population double? - **Inflation**: how long until prices double? - **Bacteria**: how often do they double in count? - **Technology**: how fast does processing power double? (Moore's Law: ~2 years)

Common applications: financial planning (retirement, savings), demographic projections, epidemiology (disease spread), business growth, Moore's Law analysis, biological population growth, and any context involving exponential change.

Inputs

Growth Rate (%)

Results

Rule of 72 (Approx)

10.29 periods

Exact Doubling Time

10.2448 periods

Approximation Error

0.0409 periods

Tripling Time

16.2376 periods

Quadrupling Time

20.4895 periods

10x Time

34.0324 periods

Last updated: May 29, 2026

Formula

**Rule of 72 (approximation):** Doubling time ≈ 72 / r (where r is rate as percentage) For r = 8%: doubling time ≈ 72/8 = 9 years. **Exact formula:** t = ln(2) / ln(1 + r) Where r is rate as decimal (e.g., 0.08 for 8%). t = 0.6931 / ln(1.08) = 0.6931 / 0.0770 ≈ 9.0 years. **For continuous compounding:** t = ln(2) / r ≈ 0.6931 / r Where r is annual rate as decimal. For r = 0.08 continuous: t = 0.6931/0.08 ≈ 8.66 years. **Worked example: 5% growth** Rule of 72: 72/5 = 14.4 years. Exact: t = ln(2) / ln(1.05) = 0.693/0.04879 ≈ 14.21 years. Close — Rule of 72 is slight overestimate. **Doubling time table:** | Rate | Rule of 72 | Exact (discrete) | Continuous | |---|---|---|---| | 0.5% | 144 | 139 | 139 | | 1% | 72 | 69.7 | 69.3 | | 2% | 36 | 35.0 | 34.7 | | 3% | 24 | 23.4 | 23.1 | | 5% | 14.4 | 14.2 | 13.9 | | 7% | 10.29 | 10.2 | 9.9 | | 10% | 7.2 | 7.3 | 6.9 | | 15% | 4.8 | 4.96 | 4.6 | | 20% | 3.6 | 3.80 | 3.5 | | 50% | 1.44 | 1.71 | 1.39 | Rule of 72 underestimates at high rates. **Common doubling times:** | Phenomenon | Doubling time | |---|---| | US economy GDP | ~25 years (3% growth) | | World population | ~60 years (1.2%) | | Moore's Law | ~2 years (~41% growth) | | Bacterial cell | 20 min - hours | | S&P 500 (long-term) | ~7 years (10%) | | US debt growth | varies, ~10-15 years recently | | Renewable energy | ~5-10 years | | Smartphone subscribers | ~3-7 years (early 2010s) | | Bitcoin price (volatile) | varies wildly | | Computer storage | ~1-2 years (current) | **Half-life (decay equivalent):** Same formula for decay: Half-life = ln(2) / (-ln(1 - decay_rate)) For 5% annual decline: Half-life = ln(2) / ln(1/0.95) = 0.693/0.0513 ≈ 13.51 years. **Tripling, quadrupling times:** | Target | Rule estimate | Formula | |---|---|---| | Double (× 2) | 72/r | ln(2)/ln(1+r) | | Triple (× 3) | 110/r | ln(3)/ln(1+r) | | Quadruple (× 4) | 144/r | ln(4)/ln(1+r) | | 10× (decuple) | 230/r | ln(10)/ln(1+r) | For 7% growth: - Double: ~10 years - Triple: ~16 years - Quadruple: ~21 years - 10×: ~34 years **Why Rule of 72?** 72 has many divisors (1,2,3,4,6,8,9,12,18,24,36,72), making mental division easy. Also accidentally close to exact formula: t ≈ ln(2) / r ≈ 0.6931 / r In percentage form: t × r ≈ 69.31, but 72 gives slight overestimate that's easier to compute and remember. For lower rates, "Rule of 69.3" is more accurate; for higher rates, "Rule of 70" is sometimes used. **Continuous vs discrete compounding:** Discrete (annual): t = ln(2) / ln(1 + r). Continuous: t = ln(2) / r. For small r: very similar. For large r: differ noticeably. Real-world investments (interest, growth) are usually discrete (compounded monthly, quarterly, or annually). **Worked example: investment doubling** $10,000 invested at 7% compound annual: Years to double: ln(2) / ln(1.07) = 0.693/0.0677 ≈ 10.24 years. After 10 years: 10,000 × 1.07^10 = $19,672 (almost doubled). After 11 years: $21,049 (just over double). Verifies ~10.24 year doubling. **Exponential growth context:** A value growing exponentially: V(t) = V₀ × (1 + r)^t After doubling time t_d: V = 2 × V₀. So: 2 × V₀ = V₀ × (1 + r)^(t_d). Solving: t_d = ln(2) / ln(1 + r). **Common applications:** - **Investment**: retirement planning timelines. - **Inflation**: how long for prices to double. - **Population**: demographic projections. - **Disease spread (early epidemic)**: doubling time of cases. - **Technology**: Moore's Law, internet growth. - **Wealth**: compounding savings. - **Renewable energy**: capacity growth. - **AI capability**: arguments about doubling rate. **COVID-19 example:** Early in pandemic, doubling time of cases was ~3-5 days in many countries (corresponding to ~14-25% daily growth). After interventions, doubling time stretched to weeks. Tracking doubling time was key public health metric. **Tax-deferred vs taxable:** Tax-deferred investments compound at full rate; taxable accounts compound at after-tax rate. For 7% growth taxed at 30%: after-tax rate ≈ 5%. Doubling: 7% → 10 years; 5% → 14 years. 4-year difference adds up over decades. **Inflation impact:** Real return = Nominal return - Inflation. If investment grows 7% but inflation is 3%, real doubling time uses 4%: Years to double real value: ln(2) / ln(1.04) ≈ 17.7 years (vs 10 years nominal). **Software:** - **Excel**: =LN(2)/LN(1+rate) - **Calculator**: Rule of 72 in head, or full formula. - **Spreadsheets**: easy formulas for projections. - **Financial software**: built-in doubling time analysis.

How to use this calculator

Enter growth rate as a percentage (e.g., 7 for 7%).
Calculator returns Rule of 72 estimate and exact formula result.
Rule of 72 quick: 72 / rate% = years to double.
Exact: t = ln(2) / ln(1 + rate/100).
For decay/half-life: similar formula with -ln(1 - rate).
For tripling: use ln(3) instead of ln(2).

Worked examples

Investment doubling

**Scenario:** $10,000 invested in S&P 500 (historical 10% nominal return). Years to double? **Calculation:** Rule of 72: 72/10 = 7.2 years. Exact: ln(2)/ln(1.10) ≈ 7.27 years. **Result:** ~7.3 years to double — very close to Rule of 72. After 30 years (4+ doublings): ~$174,500. Compound growth makes a $10K investment worth nearly $175K after a working career.

Inflation impact

**Scenario:** Annual inflation 3.5%. Time for prices to double? **Calculation:** Rule of 72: 72/3.5 ≈ 20.6 years. Exact: ln(2)/ln(1.035) ≈ 20.15 years. **Result:** ~20 years to double prices at 3.5% inflation. $100 today buys what $200 will buy in 20 years. Salary keeping up with inflation must double in same time. Why long-term financial planning must consider inflation.

Bacterial doubling

**Scenario:** E. coli bacteria double every 20 minutes under ideal conditions. **Calculation:** Per-period growth: 100% (doubles every 20 min). Per minute: (2)^(1/20) - 1 ≈ 0.0353 = 3.53%. Daily doublings: 24×60/20 = 72. **Result:** After 24 hours: starting from 1 bacterium, you'd have 2^72 ≈ 4.7 × 10²¹ bacteria — about 4.7 sextillion. Of course, real conditions limit growth (nutrients, space) so this doesn't happen indefinitely. Demonstrates the staggering power of exponential growth.

When to use this calculator

**Use doubling time for:**

- **Investment planning**: when does money double? - **Demographic projections**: population doubling estimates. - **Inflation analysis**: how fast prices erode value. - **Disease tracking**: epidemic doubling time as a key metric. - **Technology**: Moore's Law, internet, AI capability. - **Energy**: renewable capacity growth rates. - **Business**: revenue, user, customer growth. - **Education**: building intuition for exponential change.

**Rule of 72 advantages:**

- **Easy mental math**: 72/8 = 9, much faster than logarithms. - **Reasonable accuracy**: within 5% for typical rates (2-12%). - **Memorable**: simple to remember and explain. - **Round numbers**: works well with common rates.

**When exact formula matters:**

- **High precision** financial calculations. - **Very high or low rates** where Rule of 72 errs. - **Scientific work**: bacterial cultures, radioactivity. - **Multi-decade projections**: small errors compound.

**Common applications:**

- **Stock market**: ~7-10% historical = 7-10 year doubling. - **Real estate**: ~3-4% appreciation = 18-24 year doubling. - **Inflation**: 2-3% = 24-36 year doubling. - **GDP**: ~2-3% = 24-36 year doubling. - **Population (developing nations)**: 2-3% = 24-36 year doubling. - **Compound interest savings**: depends on rate.

**Doubling examples in nature:**

- **Bacterial cells**: 20 min - hours. - **Yeast**: 90 min. - **Cancer cell lines**: 24-48 hours. - **Insects** (in season): days to weeks. - **Most mammal populations**: years to decades.

**Comparing investments:**

10% CAGR doubles every 7 years. Over 35 years (5 doublings): 32× initial. 7% CAGR doubles every 10 years. 35 years (3.5 doublings): ~10× initial.

The 3% rate difference compounds to 3× difference over a career.

**Software:**

- **Excel/Sheets**: =LN(2)/LN(1+rate%/100) - **Python**: import math; math.log(2)/math.log(1+r) - **Calculator**: scientific calculator with ln function. - **Mental math**: Rule of 72 is the go-to.

**Pitfalls:**

- **Constant growth assumption**: real growth varies year to year. - **Inflation ignored**: nominal vs real doubling. - **Taxes ignored**: pre-tax vs after-tax. - **Rule of 72 limits**: less accurate at high rates. - **Compounding period**: annual vs monthly vs continuous differ. - **Negative rates**: gives "halving time" for decay. - **Rate vs cumulative**: doubling time refers to constant rate.

**Demographic transitions:**

Most countries show declining growth rates over time: - **Developing world**: 2-3% (24-36 year doubling). - **Developed world**: 0.5-1% (72-144 year doubling). - **Aging societies (Japan, parts of Europe)**: negative (declining population).

Doubling time helps project future populations under various assumptions.

**Exponential vs linear:**

People underestimate exponential growth. Doubling time helps: - $1 doubling daily for 30 days: $1 billion! ($2^30). - $1 doubling weekly for a year: $4 × 10^15 — absurd.

Exponential change always surprises if you only think in linear terms.

**Rule of 72 origin:**

Records suggest mathematicians have used variations of this rule for 500+ years. Luca Pacioli (1494) used it. Mathematically, exact value depends on rate but converges to ln(2)/r ≈ 0.693/r. The number 72 works well for typical rates (5-10%) and has many divisors.

**Pitfalls:**

- **Mistaking simple growth for compound**: simple is linear, not doubling. - **Forgetting to convert percentage**: 7% rate = 0.07 in formula. - **Treating future as guaranteed**: past CAGR doesn't predict future. - **Ignoring withdrawals**: doubling time assumes no money removed. - **Halving time confusion**: for decay, use different formula. - **Single-period vs compound**: critical for accurate projection.

Common mistakes to avoid

Confusing simple growth (linear) with compound (exponential).
Forgetting to convert percentage to decimal in exact formula.
Treating Rule of 72 as exact when high precision needed.
Ignoring inflation when projecting purchasing power.
Using doubling time without accounting for taxes.
Mixing continuous vs discrete compounding (small but real difference).
Assuming current growth rate continues indefinitely.
For decay/halving: forgetting to negate the rate in formula.

Frequently Asked Questions

Sources & further reading

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