Exponential Decay

Exponential decay is what happens when a quantity loses a fixed percentage, not a fixed amount, each period. Losing 20% means keeping 80%, so each step multiplies by $1 - r = 0.8$ and $A = A_0(1-r)^t$. The consequence that separates this from linear loss is that the amount lost shrinks along with the amount remaining: the first year of that 20% takes $6,000 off a $30,000 car, the fifth year takes only $2,458. The curve steepens nowhere and flattens forever, approaching zero without ever arriving.

A worked instance: that car, at 20% a year, is worth $30000 \times 0.8^5 = \$9{,}830$ after five years. The same shape governs coffee cooling toward room temperature, a drug clearing between doses, and chlorine residual fading in a treated tank: in each case something is leaving at a rate proportional to how much is still there, which is the physical condition that produces this curve.

The natural way to describe such a process is by its half-life, the time to fall to 50%. Setting $A/A_0 = 0.5$ and solving gives $t_{1/2} = \ln(0.5)/\ln(1-r)$, so 20% a year corresponds to a half-life of 3.1 years. For small rates there is a serviceable shortcut mirroring the Rule of 72: divide 70 by the percentage rate. The solver's rearrangements do the general version of this: solving for $t$ answers "how long until it reaches this level?", and solving for $r$ extracts the per-period rate from two measurements, which is how you find the decay constant of a real system rather than assuming one.

Three errors, and the third is the one that matters. First, $r$ is a decimal: 20% is 0.20, and entering 20 makes $1 - r = -19$, producing an answer that flips sign every period and is obviously wrong if you look at it. Second, the rate and the time must share a clock: 20% per year with $t$ in months is off by a factor no amount of care elsewhere will recover. Third, and worth stating flatly: percentages do not add. Losing 20% five times does not lose 100%; it leaves 32.8%. And a 20% drop followed by a 20% rise does not return you to where you started: it leaves 96%, because the rise is calculated on the smaller base. That asymmetry is why an investment that falls 50% needs a 100% gain to recover, and it catches people in every field where percentages get quoted casually.