Definition: Exponential decay occurs when a quantity is multiplied by the same factor between 0 and 1 in each equal time period, so it drops quickly at first and then slowly.
How does decay differ from growth?
The multiplier is smaller than 1. Each period removes the same percentage of what is left, not the same amount. The graph falls steeply, then flattens toward the x-axis without touching it.
What is the model?
y = a(1 – r)t, where a is the starting amount, r is the decay rate, and t counts periods. The decay factor is 1 – r.
Example. A medicine level of 80 mg falls by 15% per hour. After 5 hours: 80(0.85)5. Since 0.855 is about 0.4437, the level is about 35.5 mg. Half-life problems are decay with factor 0.5 per half-life.
What mistakes are frequent?
Using 1 + r instead of 1 – r flips decay into growth. Another is subtracting 15% five times from the original, which treats the loss as constant: 80 – 5(12) = 20, a wrong answer. Remember that the base must stay between 0 and 1, and that the quantity never reaches exactly zero in this model.