What is the Rule of 72?

The Rule of 72 is a quick way to estimate how long it takes for an investment to double given a fixed annual rate of return (or interest rate).

📝 Formula:

72 / Annual Rate of Return (%) ≈ Years to Double

📊 Examples:

• At 6% return → 72 ÷ 6 = 12 years to double.
• At 8% return → 72 ÷ 8 = 9 years.
• At 12% return → 72 ÷ 12 = 6 years.

🔄 Reverse Use:

You can also use it to estimate the rate of return needed to double your money in a certain number of years: 72 / Years to Double ≈ Rate (%)

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✅ It’s just a rule of thumb — not exact, but surprisingly accurate for interest rates between about 5% and 12%.

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Early Origins

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The Rule of 72 doesn’t have a single inventor — it’s a mathematical shortcut that has been around for centuries. Here’s the background:

• 📜 Early Origins: The idea comes from natural logarithms and compound interest math, which date back to the work of Jacob Bernoulli in the late 1600s. He studied continuous compounding and discovered the constant e.
• 🧮 Why 72? The exact doubling time is based on: t = ln⁡(2) / ln⁡(1+r)t
• For small interest rates, this simplifies to ~ 72 ÷ r.
  The number 72 is used (instead of 70 or 69.3) because it divides neatly by many integers (2, 3, 4, 6, 8, 9, 12), making it practical for mental math.
• 📚 Popularization: The rule became widely known in the 20th century through personal finance educators, banks, and investment guides, not from one specific economist.

Bottom line. It’s rooted in Bernoulli’s work on compound interest, refined over time for convenience, then popularized as an easy rule of thumb.

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1️⃣ The Exact Math

The real formula for doubling time is: t = ln⁡(2) / ln⁡(1+r)

For small rates of return (rr), ln⁡(1+r)≈r

so: t ≈ 0.693 / r

👉 That’s why 69.3 is the mathematically precise constant (since ln(2) ≈ 0.693).

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2️⃣ Why 72?

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• Divisibility: 72 has many divisors (2, 3, 4, 6, 8, 9, 12). That makes mental math easier:
  • 72 ÷ 6 = 12 years
  • 72 ÷ 8 = 9 years
  • 72 ÷ 12 = 6 years
• Practical Accuracy: For interest rates between 6%–10% (the range most people care about), using 72 instead of 69.3 actually gives results closer to reality.

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3️⃣ Example Comparison

At 8% interest:

• Exact formula: t=ln⁡(2)÷ln⁡(1.08)≈9.006t = \ln(2) ÷ \ln(1.08) ≈ 9.006 years
• Rule of 69.3: 69.3÷8=8.6669.3 ÷ 8 = 8.66 years
• Rule of 72: 72÷8=9.0072 ÷ 8 = 9.00 years ✅ (closer!)

At 12% interest:

• Exact: 6.12 years
• Rule of 69.3: 5.78 years
• Rule of 72: 6.00 years ✅ (closer again)

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🔑 Takeaway

• 69.3 = mathematically exact.
• 72 = more accurate in common interest ranges + easier mental math.
  That’s why 72 “stuck” in finance education.

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