Doubling Time & Half-Life

Business Calculus · Axiom Academy

LESSON Doubling Time and Half-Life Key metrics for understanding how fast quantities grow or decay In business and finance, we often want to know: "How long until this investment doubles?" or "How long until this equipment loses half its value?" These questions are answered by two powerful concepts: Time required for a growing quantity to become twice its initial value Time required for a decaying quantity to become half its initial value Both concepts are closely related and use the same mathematical principles—they're just mirror images of each other! Let's derive the formula for doubling time using continuous growth: A(t) = A_0 e^ rt We want to find when A(t) = 2A_0 Doubling Time Formula (Continuous Growth) where r is the continuous growth rate as a decimal Notice that A_0 (the initial amount) doesn't appear in the formula! Doubling time depends only on the growth rate, not on how much you start with. The Rule of 72: A Quick Approximation Financial professionals use a handy shortcut called the Rule of 72 to estimate doubling time without a calculator. where r is expressed as a percentage (not a decimal) An investment earns 8% annual interest (compounded continuously). How long to double? Why does this work? Since (2) 0.693 and 0.693 100 69.3 , we'd get a "Rule of 69.3." But 72 is easier to divide (by 2, 3, 4, 6, 8, 9, 12...) and also accounts for the slight difference between continuous and annual compounding.

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