By: Joshua Diaz.
Finish that coffee in your hand and five hours later roughly half of its caffeine is gone from your bloodstream. Buy a delivery van and a year later roughly half of its book value has disappeared from the balance sheet. Neither one loses a fixed amount over time. Instead both lose a fixed percentage of whatever’s left. That’s the key: the rate of loss depends on what’s there, not on where you started. One equation can describe both even though one deals in milligrams of caffeine and the other in dollars of assets.
Any quantity that drains at a rate proportional to its own size has to obey the same first order differential equation: where Q is the amount remaining, t is time, and k is a constant describing how quickly the system loses. Solved, this becomes the exponential curve Q(t) = Q₀e^(−kt). What Q represents doesn’t matter to the equation; all it requires is that loss compounds on whatever currently exists.
Most drugs, caffeine among them, leave the body through first order elimination, where the amount cleared at any moment depends on how much is currently circulating rather than on some fixed daily quota (“Elimination Half-Life of Drugs,” 2025). Pharmacologists track this using the elimination half life: the time for the concentration to drop by 50%, with that same 50% drop repeating every half life thereafter, regardless of the dose. In most healthy adults, caffeine’s half life runs about five hours, though pregnancy, smoking, and genetics can shift it substantially (Sleep Foundation, 2025). Drink a 180 mg cup at 7 a.m., and roughly 90 mg is still circulating by noon; by 5 p.m., about 45 mg remains. The curve never reaches zero.
Accountants model a different kind of vanishing through the declining balance method of depreciation, where an asset loses a fixed percentage of its remaining book value each year rather than a fixed dollar amount. One formula treats each year’s value as the previous year’s value multiplied by a constant retention factor q, so that after n years, the remaining value is R(n) = K₀qⁿ, a sequence, the discrete exponential decay (An Introduction to Business Mathematics, 2015). Put a $30,000 delivery van on a 20% declining balance schedule and q = 0.80; solving qⁿ = 0.5 gives n ≈ 3.1, the van’s own half life, denominated in years instead of hours, dollars instead of milligrams.
That overlap is easy to miss, since each field teaches its version separately. Biology introduces half-life as something you need to know about drugs. Accounting introduces declining balance as a way to depreciate assets. Underneath the different terminology, a pharmacist and an accountant will be working with the same mathematical idea.
Neither model works perfectly in every situation, and the exceptions matter. Ethanol, for example, is generally modeled with zero order kinetics, meaning it’s eliminated at a constant rate rather than a rate proportional to the amount remaining. Depreciation schedules can also have limits or switch methods once an asset’s value gets small enough. Outside those cases, the basic. Proportional loss is loss, whether you’re measuring milligrams or dollars. Different fields, different units, the same equation, each time shrinking what’s left by a fixed fraction, without reaching zero.
References
“Eliminate half-life of drugs.” (2025). In StatPearls. StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK554498/
An introduction to business mathematics. (2015). arXiv. https://arxiv.org/abs/1509.04333
Sleep Foundation. (2025, July 16). How long does it take for caffeine to wear off? https://www.sleepfoundation.org/nutrition/how-long-does-it-take-caffeine-to-wear-of.
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