{
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  "question": "Using the Rule of 72 and the exact logarithmic formula, how many years does it take to double an investment growing at 8% per year?",
  "answer": "Approximately 9.00 years (Rule of 72); exactly 9.01 years",
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    {
      "label": "Principles of Corporate Finance",
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    }
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    "value": "Brealey, Myers, Allen, Principles of Corporate Finance, 13th ed., Chapter 3, box on the Rule of 72",
    "retrieved_utc": "2026-09-30T19:31:29Z"
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  "method_deck": {
    "inputs": [
      {
        "name": "annual_rate_percent",
        "type": "number",
        "unit": "%",
        "required": true,
        "description": "Annual compound growth rate",
        "min": 0.0001
      }
    ],
    "outputs": [
      {
        "name": "approx_years",
        "unit": "years",
        "description": "Rule of 72 approximation: 72 / rate"
      },
      {
        "name": "exact_years",
        "unit": "years",
        "description": "Exact doubling time using logarithms"
      }
    ],
    "steps": [
      "Rule of 72 approximation: approx_years = 72 / annual_rate_percent.",
      "Convert the rate to a decimal: r = annual_rate_percent / 100.",
      "Exact doubling time: exact_years = ln(2) / ln(1 + r).",
      "Round approx_years and exact_years to 2 decimals."
    ],
    "constants": [
      {
        "name": "rule_of_72_numerator",
        "value": 72,
        "unit": "dimensionless",
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      }
    ],
    "examples": [
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        "inputs": {
          "annual_rate_percent": 8
        },
        "expected": {
          "answer": "Approximately 9.00 years (Rule of 72); exactly 9.01 years"
        },
        "check": "hand computation: approx = 72/8 = 9.00; exact = ln(2)/ln(1.08) = 0.693147.../0.076961... = 9.0065..., rounds to 9.01"
      },
      {
        "name": "test vector: low rate",
        "inputs": {
          "annual_rate_percent": 3
        },
        "expected": {
          "answer": "Approximately 24.00 years (Rule of 72); exactly 23.45 years"
        },
        "check": "hand computation: approx = 72/3 = 24.00; exact = ln(2)/ln(1.03) = 0.693147.../0.029559... = 23.4498..., rounds to 23.45"
      }
    ],
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          "value": "Brealey, Myers, Allen, Principles of Corporate Finance, 13th ed., Chapter 3, box on the Rule of 72",
          "retrieved_utc": "2026-09-30T19:31:29Z"
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